Multivariate Fidelities
Theshani Nuradha Hemant K. Mishra Felix Leditzky Mark M. Wilde









Multivariate Fidelities
    Theshani NuradhaSchool of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14850, USA
  (pt388@cornell.edu, hemant.mishra@cornell.edu, wilde@cornell.edu).
Hemant K. Mishra[1]
Felix LeditzkyDepartment of Mathematics &
Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA
  (leditzky@illinois.edu)  
Mark M. Wilde[1]

     
=================================================================================================================================================================================================================================================================================================================================================================================================================================




The bivariate classical fidelity is a widely used measure of the similarity of two probability distributions. 
There exist a few extensions of the notion of the bivariate classical fidelity to more than two probability distributions; herein we call these extensions multivariate classical fidelities, with some examples being the Matusita multivariate fidelity and the average pairwise fidelity. Hitherto, quantum generalizations of multivariate classical fidelities have not been systematically explored, even though there are several well known generalizations of the bivariate classical fidelity to quantum states, such as the Uhlmann and Holevo fidelities. 
The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose three variants that reduce to the average pairwise fidelity for  commuting states: average pairwise z-fidelities, the multivariate semi-definite programming (SDP) fidelity, and  a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All three of these variants satisfy the following properties: 
(i) reduction to multivariate classical fidelities for commuting states, (ii)  the data-processing inequality, (iii) invariance under  permutations of the  states, (iv) its values are in the interval [0,1]; they are faithful, that is, their values are equal to one if and only if all the  states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the  states are mutually orthogonal to each other, (v)  direct-sum property, (vi)  joint concavity, and (vii) uniform continuity bounds under certain conditions. 
Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. Lastly, we introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis.










§ INTRODUCTION


Distinguishability and similarity are essential concepts that hold significance across all scientific disciplines. The fundamental toolkit for understanding these concepts revolves around measures of distinguishability and  similarity. 
This has led to several measures of distinguishability between probability distributions in classical information theory, such as the Kullback–Leibler divergence <cit.> and Rényi relative entropy <cit.>, and similarity measures such as the Bhattacharyya overlap <cit.>.
In quantum information theory, we have distinguishability measures between quantum states such as quantum relative entropy <cit.>, Petz–Rényi relative entropy <cit.>, and sandwiched Rényi relative entropy <cit.>, and information quantities derived from these measures, such as quantum mutual information <cit.>.  Moreover, the Uhlmann <cit.> and Holevo <cit.> fidelities  are similarity measures between two quantum states. 
All of the aforementioned distinguishability and similarity measures take into account two probability distributions in the classical case and two quantum states in the quantum case.

There exist a few extensions of the notion of the Bhattacharyya overlap or  bivariate classical fidelity to more than two probability distributions. Examples include the Matusita affinity <cit.> and the average pairwise classical fidelity; herein we call all such extensions  multivariate classical fidelities.
Hitherto, quantum generalizations of multivariate classical fidelities have not been systematically explored. 
Existing multivariate measures that do not reduce to classical multivariate fidelities are as follows: Holevo information of a tuple of states 
with uniform prior characterizes the correlations between the classical system and the quantum system (see (<ref>) for a precise definition).
To this end, Holevo information being approximately zero implies that the states in this tuple are similar. Multivariate Chernoff divergence is defined in <cit.> for commuting states while discussing generalizations to non-commuting states. 
Moreover, multivariate generalizations for Rényi divergences have been explored in <cit.> for general states and in <cit.> for commuting states. 





The main contribution of our paper is to introduce quantum generalizations of the Matusita affinity and average pairwise classical fidelity; here we call them multivariate quantum fidelities. We show that these multivariate fidelities satisfy several desirable properties that are natural generalizations of those of bivariate fidelities. Consequently, the multivariate quantum fidelities are bona fide measures of similarity between multiple states. See <Ref> for a more detailed discussion of our contributions.



 §.§ Classical and quantum bivariate fidelities




We first provide a brief overview of classical and quantum fidelities.

The bivariate classical fidelity quantifies the similarity of two discrete probability distributions p and q on a finite set ; it is defined as 

    F(p,q) ∑_x∈√(p(x) q(x)).

It satisfies 0 ≤ F(p,q) ≤ 1 for all distributions p and q,  it is equal to one if and only if p=q, and it is equal to zero if and only if the supports of p and q are disjoint. The above formulation can be naturally extended to commuting states as follows: for commuting states ρ and σ, define

    F(ρ,σ) ∑_x √(ρ(x) σ(x)),

where (ρ(x))_x and (σ(x))_x are the tuples of eigenvalues of ρ and σ, respectively, in their common eigenbasis.

In the same spirit of distinguishability measures, there is an infinite number  of similarity measures that generalize the bivariate classical fidelity to quantum states. Here we call them z-fidelities and  define them for z ≥ 1/2 and quantum states ρ and σ as

    F_z(ρ,σ) [  ( σ^1/4zρ^1/2zσ^1/4z)^z ] = ρ^1/4zσ^1/4z_2z^2z.

For every positive integer z, cyclicity of trace leads to the following expression:

    F_z(ρ, σ) = Tr[(ρ^1/2zσ^1/2z)^z].

The z-fidelities are obtained by fixing α=1/2 in the α-z Rényi relative entropies <cit.>. They reduce to the classical fidelity for commuting states, and each of them satisfies the data-processing inequality <cit.>; i.e., for all z≥ 1/2 and for every quantum channel  and pair of states ρ and σ, the following inequality holds:

    F_z(ρ,σ) ≤ F_z( (ρ), (σ) ).

The z-fidelities are monotonically decreasing and continuous in z <cit.>.
Notably, for z=1/2, the z-fidelity reduces to the Uhlmann fidelity <cit.>: 

    F_1/2(ρ,σ)=F(ρ,σ) √(ρ)√(σ)_1,

which was expounded upon in <cit.>. 
For z=1, the z-fidelity reduces to the Holevo fidelity <cit.>: 

    F_1(ρ,σ) = F_H(ρ, σ) [√(ρ)√(σ)].

The log-Euclidean fidelity is defined as the z→∞ limit of (<ref>), for which it is possible to obtain a closed-form expression by an application of the Lie–Trotter product formula. Indeed, for ε>0, define ρ(ε)ρ+ε I and σ(ε)σ+ε I. Then

    F_♭(ρ,σ)     lim_z→∞ F_z(ρ,σ) 
        = inf_ε >0Tr[exp(1/2 (lnρ(ε) + lnσ(ε))] 
        = lim_ε→  0^+Tr[exp(1/2 (lnρ(ε) + lnσ(ε))].

See <ref> for a short proof of (<ref>), and see <cit.> and <cit.> for a quantity that generalizes the log-Euclidean fidelity.

For two pure states |ψ⟩⟨ψ| and |ϕ⟩⟨ϕ|, the z-fidelity reduces to

    F(|ψ⟩⟨ψ|, |ϕ⟩⟨ϕ|) = | ⟨ψ | ϕ⟩ |^2z.

Operationally, this is the zth power of the probability for the state |ψ⟩⟨ψ| to pass a test for being the state |ϕ⟩⟨ϕ|.


The bivariate setting focuses on the similarity of two states. Going forward, the main goal of our paper is to identify measures for quantifying the similarity of a given tuple of states.
We focus on two main approaches to arrive at multivariate quantum fidelities, as shown in <ref>. The first approach is to generalize multivariate classical fidelities, which reduce to bivariate classical fidelity for two commuting states, to a tuple of non-commuting states. The second is to generalize bivariate quantum fidelities to multivariate quantum fidelities that reduce to the bivariate setting when two states are considered. We also note that some generalizations can be obtained by following either of the aforementioned approaches.








 §.§ Contributions




In this paper, we comprehensively study multivariate generalizations of bivariate fidelity, while establishing connections between different formulations. 

First, we recall some multivariate classical fidelities for commuting states, namely, the Matusita multivariate fidelity and average pairwise fidelity, and then we establish an inequality between them (<ref>). Thereafter we generalize these quantities to average k-wise fidelities (<ref>) and prove that they are ordered (<ref>). 

Next, we introduce quantum generalizations,  mainly focusing on three variants that reduce to average pairwise fidelity for commuting states: average pairwise fidelity using z-fidelities for z ≥  1/2  (<ref>); multivariate semi-definite programming (SDP) fidelity (<ref>); and secrecy-based multivariate fidelity (<ref>). The multivariate SDP fidelity is obtained by generalizing the SDP of Uhlmann fidelity from <cit.> to multiple states. The secrecy-based multivariate fidelity is inspired by an existing secrecy measure from <cit.>—the average fidelity between each state of the tuple and a fixed state, where it is maximized over all such fixed states (c.f., (<ref>) for the definition). 
With these definitions, we show that both SDP fidelity  and secrecy-based multivariate fidelity are sandwiched between the average pairwise Holevo fidelity (z=1) and the average pairwise Uhlmann fidelity (z=1/2), implying that all of these variants converge to average pairwise fidelities in the commuting case (see <ref>).


<ref>, <ref>,  and <ref> assert that all three of the aforementioned quantum variants satisfy a number of properties desired for a multivariate fidelity: (i) reduction to multivariate classical fidelities for commuting states; (ii) data processing; (iii) permutation invariance; (iv) faithfulness (i.e., it is equal to one if and only if the states are the same); (v) orthogonality (i.e., it is equal to zero if and only if the tuple of states forms an  orthogonal set); (vi) direct-sum property; and (vii) joint concavity. In addition, the average pairwise fidelity satisfies super-multiplicativity, whereas the SDP fidelity and secrecy-based multivariate fidelity do not satisfy this property in general. <ref> establishes that the multivariate SDP fidelity satisfies a uniform continuity bound, which follows by means of an alternative formulation for the SDP fidelity in <ref>. <ref> and <ref> state that the average pairwise Uhlmann and Holevo fidelities and secrecy-based multivariate fidelity also satisfy uniform continuity bounds, respectively. In addition, we define maximal and minimal extensions of multivariate classical fidelities (<ref> and <ref>) and analyse their properties. 

Lastly, we explore a quantum generalization of the Matusita multivariate fidelity, and we show that it is a special case of a multivariate log-Euclidean divergence. We call this variant the multivariate log-Euclidean fidelity.  We show that this satisfies most of the desirable properties of a multivariate fidelity in <ref>.
We give an operational interpretation of all multivariate log-Euclidean divergences in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis (<ref>), by making use of the main result of <cit.>. In addition, we define the oveloh information in (<ref>) and show that it is related to the multivariate log-Euclidean fidelity in (<ref>). In passing, we also derive a connection between the Holevo information and oveloh information in <ref>. We also define average k-wise log-Euclidean fidelities (<ref>) and show that they are ordered (<ref>), as a generalization of the aforementioned average k-wise classical fidelities.




 §.§ Paper organization


The rest of our paper is organized as follows.  In Section <ref>, we introduce notation and background needed to understand the rest of the paper. Section <ref> reviews and presents different formulations of the bivariate Uhlmann fidelity. Multivariate classical fidelities are defined and studied in Section <ref>. In Section <ref>, we introduce proposals for multivariate quantum fidelity,  focusing on four main variants,
their properties, relationships between different formulations, and operational interpretations of some of them.  
In the appendices, we provide mathematical proofs of  results presented in the paper.
Finally, Section <ref> provides concluding remarks and future directions.





§ NOTATIONS AND BACKGROUND
 


We begin by reviewing basic concepts from quantum information theory and refer the reader to <cit.> for more details. A quantum system R is identified with a finite-dimensional complex Hilbert space ℋ_R with inner product denoted in bra-ket notation as ⟨ψ|ϕ⟩ for |ψ⟩,|ϕ⟩∈_R. We denote the set of linear operators acting on ℋ_R by ℒ_R. The support of a linear operator X ∈ℒ_R is defined to be the orthogonal complement of its kernel, and we denote it by supp(X). We denote the Hermitian conjugate or adjoint of X by X^†, which is the unique linear operator acting on ℋ_R that satisfies 
⟨ψ| (X^† |ϕ⟩) = (X |ψ⟩)^† |ϕ⟩
for all | ψ⟩, | ϕ⟩∈ℋ_R.
Here, |ψ⟩^†≡⟨ψ| is the linear functional ℋ_R→ℂ given by |ϕ⟩↦⟨ψ|ϕ⟩.
The set ℒ_R is a linear space and has a Hilbert-space structure given by the Hilbert–Schmidt inner product, defined as ⟨ X, Y ⟩[X^† Y] for all X, Y ∈ℒ_R.
Given two quantum systems A and B, with respective Hilbert spaces ℋ_A and ℋ_B, the Hilbert space of the composite system AB is given by ℋ_ABℋ_A ⊗ℋ_B. For K_AB∈ℒ_AB, let [K_AB] denote the trace of K_AB, and let _A [K_AB] denote the partial trace of K over the system A.
We use the standard notation K_A ≡_B[K_AB] and K_B ≡_A[K_AB] to denote the marginals of K_AB. The trace norm  of an operator K is defined as K_1 [√(K^† K) ]. For Hermitian operators 
K and L, the notation K ≥ L indicates that K-L is a positive semi-definite (PSD) operator, while K > L indicates that K-L is a positive definite operator. Given a complex number z, we denote the real part of z by ℜ[z].


A quantum state of a system R is identified with a density operator ρ_R ∈ℒ_R, which is a PSD operator of unit trace. 
We denote the set of all density operators acting on ℋ_R by 𝒟_R. We shall also use the notations 𝒟 and ℒ to denote the sets of density operators and linear operators, respectively, when there is no ambiguity regarding the underlying quantum system. A quantum state ρ_R  is called a pure state if its rank is equal to one, and in this case, there exists a state vector | ψ⟩_R ∈ℋ_R such that ρ_R= | ψ⟩⟨ψ |_R. Otherwise, ρ_R is called a mixed state. By the spectral decomposition theorem, every state can be written as a convex combination of pure and mutually orthogonal states. 
A quantum channel from system A to system B is a linear, 
completely positive and trace-preserving (CPTP) map from ℒ_A to ℒ_B. 
A measurement of a quantum system R is described by a
positive operator-valued measure (POVM), which is defined as a tuple of PSD operators (M_y)_y ∈𝒴 satisfying ∑_y ∈𝒴 M_y= I_R, where I_R is the identity operator acting on ℋ_R and 𝒴 is a finite alphabet. The Born rule asserts that, when applying the above POVM to a state ρ, the probability of observing the outcome y is given by [M_y ρ] <cit.>.  Associated with any POVM (M_y)_y ∈𝒴 is a measurement or quantum-to-classical channel ℳ described as follows. Let 𝒦 be a complex Hilbert space of dimension |𝒴|, and let {|y ⟩: y ∈𝒴} be a fixed orthonormal basis of 𝒦. Given  an  input state ω, the output of the measurement channel is given by

    ℳ(ω) ∑_y ∈𝒴[ M_y ω]   |y⟩⟨ y|.


Finally, throughout our paper, we adopt the shorthand [r] {1, …, r}, and we let S_r denote the permutation group on [r].





§ MANY FORMS OF UHLMANN FIDELITY AND ITS PROPERTIES
 


In this section, we discuss several equivalent formulations for the Uhlmann fidelity, along with its various properties. This section also serves as a foundation for subsequent sections, in which we extend the Uhlmann fidelity to multivariate fidelities.


We adopt the following notations here: 𝒰_R denotes the set of unitary operators acting on ℋ_R;
POVM_A denotes the set of POVMs acting on system A; and ℒ_A^+⊂ℒ_A denotes the set of positive semi-definite operators.

Let us begin by recalling that the Uhlmann fidelity has several equivalent formulations, as stated in the following proposition. This is one of the reasons that it is the mostly widely used bivariate quantum fidelity.

 

Let ρ and σ be quantum states of a system A. Let R be a reference system of the same dimension as A, and let | ψ^ρ⟩ and | ψ^σ⟩ be purifications of ρ and σ, respectively.
 The Uhlmann fidelity F(ρ,σ) between ρ and σ is equal to any one of the following expressions:
  
    F(ρ,σ)    √(ρ)√(σ)_1 
       =sup_U_R ∈𝒰_R | ⟨ψ^ρ | U_R ⊗ I_A | ψ^σ⟩| 
       =inf_(Λ_x)_x ∈ POVM_A∑_x √([Λ_x ρ] [Λ_x σ])
       = 1/2inf_Y_1, Y_2 ≥ 0{[Y_1 ρ] + [Y_2 σ]:  [ Y_1  -I;  -I Y_2 ]≥ 0  }
       =sup_X ∈ℒ_A{ℜ[ [X] ] :   [   ρ   X; X^†   σ ]≥ 0  }.



 Uhlmann's theorem (<ref>) was established in <cit.>. The fact that the fidelity is achieved by a quantum measurement (<ref>) was realized in <cit.>. The SDPs (<ref>) and (<ref>) for bivariate fidelity were established in <cit.>. We refer to <cit.> for proofs of the expressions above (i.e., Theorem 6.8, Theorem 6.12, and Proposition 6.6 therein).
In addition to these expressions, a corollary of <Ref> in <Ref> is a new formulation for bivariate fidelity of two states ρ and σ, which we state in <Ref>.


In the following proposition, we list several properties of the Uhlmann fidelity that we use as a guide for defining multivariate quantum fidelities. 

    The Uhlmann fidelity satisfies the following properties for  states ρ and σ:
    
        
  * Reduction to classical fidelity: If the states ρ and σ commute, then
        
    F(ρ,σ) = ∑_x√(ρ(x) σ(x)),

        where (ρ(x))_x and (σ(x))_x are the spectra of ρ and σ, respectively, in their common eigenbasis.
        
  * Data-processing inequality: For every quantum channel , we have
        
    F(ρ,σ) ≤ F( (ρ), (σ) ).

        
  * Symmetry: Uhlmann fidelity is independent of the order of the quantum states, i.e.,
        
    F(ρ,σ)= F(σ,ρ).

    
  * Faithfulness and orthogonality: Uhlmann fidelity satisfies the inequalities 0≤ F(ρ, σ) ≤ 1. Furthermore, F(ρ,σ)=1 if and only if ρ=σ, and 
         F(ρ,σ)=0 if and only if ρ and σ are orthogonal to each other, i.e., ρσ=0.
         
        
  * Direct-sum property: Let (ρ_x)_x ∈, and (σ_x)_x∈ be tuples of quantum states, where  is a finite alphabet. For the classical-quantum states formed using those tuples, the following equality holds: 
    
    F( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_x, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗σ_x) = ∑_x ∈ p(x)  F(ρ_x, σ_x),
 
    where p is an arbitrary probability distribution on .
    
    
  * Joint concavity: Let (ρ_x)_x ∈, and (σ_x)_x∈ be tuples of quantum states for a finite alphabet , and let (p(x))_x∈ be a probability distribution. Then,
    
    F( ∑_x ∈ p(x) ρ_x, ∑_x ∈ p(x) σ_x) ≥∑_x ∈ p(x)  F(ρ_x, σ_x).

  
    

  We note that the joint concavity of Uhlmann fidelity is a consequence of its properties (b) data-processing inequality under partial trace channel and (e) direct-sum property. We refer the reader to <cit.> for proofs of these properties.[In particular, we refer to Theorem 6.9 for (ii), Theorem 6.7 for (iv), and Theorem 6.11 for (vi) in <cit.>, while (i) and (iii) follow by definition of fidelity.]
  


§ MULTIVARIATE CLASSICAL FIDELITIES
 




The basic requirement that we set for a multivariate classical fidelity is that it should reduce to the bivariate classical fidelity in (<ref>), when only two probability distributions are being considered. Further requirements include satisfying the properties outlined in <ref>, such as the data-processing inequality, symmetry under exchange of the probability distributions, faithfulness, orthogonality, and the direct-sum property. As such, there are many possible functions that satisfy these requirements, and 
the aims of this section are to outline several variants of multivariate classical fidelity and to establish relationships between them.
In subsequent sections, 
we show how quantum generalizations reduce to these definitions for commuting states (states represented using classical distributions), which is one of the desired properties for a quantum generalization of a classical measure. 



 §.§ Matusita multivariate fidelity


Matusita  introduced a generalization of the bivariate classical fidelity of two probability distributions to several probability distributions <cit.>, and therein it was called the affinity of several distributions. Here we call it the Matusita multivariate fidelity, and we recall its definition now.


 
For r∈ℕ, let p_1,…, p_r be probability distributions on a finite set . 
    The Matusita multivariate fidelity is defined as 

    F_r(p_1, …, p_r) ∑_x ∈(p_1(x) ⋯ p_r(x))^1/r.

Furthermore, for a tuple of commuting states (ρ_i)_i=1^r, the Matusita multivariate fidelity is defined as 

    F_r(ρ_1, …, ρ_r) ∑_x( ρ_1(x) ⋯ρ_r(x))^1/r

 where, for i∈ [r], (ρ_i(x))_x is the spectrum of the state ρ_i  in the common eigenbasis of the states.


The Matusita multivariate fidelity satisfies the data-processing inequality, symmetry under exchange of the probability distributions, and the direct-sum property. It also satisfies 0 ≤ F_r(p_1, …, p_r) ≤ 1, and the equality F_r(p_1, …, p_r)=1 holds if and only if p_1=⋯ = p_r.  See Theorem 1 and Corollary 3 of <cit.>. Also, if there exists at least one pair of distributions p_i and p_j with i≠ j ∈ [r] that have disjoint support, then  F_r(p_1, …, p_r)=0. However,  F_r(p_1, …, p_r)=0 does not imply that at least one pair of distributions is disjoint. 
For example, the probability vectors

    p_1=(0,1/2, 1/2),       p_2=(1/2, 0, 1/2),      p_3=(1/2, 1/2, 0)

satisfy F_r(p_1,p_2,p_3)=0, but no two distributions have disjoint support.


The Matusita multivariate fidelity can be written in terms of the classical relative entropy 

    D(p‖ q) ∑_x p(x) ln(p(x)/q(x))     if supp(p)⊆supp(q) 
    
        +∞    else

as follows:

    F_r(p_1, …, p_r) = exp(-inf_w1/r∑_x D(w‖ p_i)),

where the infimum is over every probability distribution w with support contained in the intersection of the supports of p_1, …, p_r (and if no such distribution exists, then the relative entropy is equal to +∞).
We review this representation of the Matusita fidelity in the more general quantum case in <Ref>, 
where we also provide an operational interpretation of it. Since the relative entropy is well known to obey the data-processing inequality, the formulation of the Matusita fidelity in (<ref>) makes it easier to see that it obeys the data-processing inequality.


 

    The Rényi relative entropy between two probability distributions p and q  is defined  for α∈ (0,1) ∪ (1, ∞) as <cit.>
    
    D_α(p‖ q) 1/α-1ln(∑_x p(x) ^α  q(x)^1-α)     if α∈ (0,1) ∨ (α > 1 ∧supp(p)⊆supp(q)) 
    
        +∞    else .

Let α∈ (0,1) be rational such that α=t/r with t<r. Then, for such α we have that 

    D_α(p‖ q) =  1/α-1ln(F_r(p, …,p,q,…,q) ),

where, on the right-hand side, p occurs t times and q occurs r-t times. Thus, the Rényi relative entropy of rational order is a special case of the Matusita fidelity. 

For a tuple of probability distributions p_1, …, p_r and a probability vector s=(s_1,…,s_r), the Hellinger transform is defined as <cit.> (see also <cit.>) 

    H_s(p_1, ..., p_r) ∑_x ∏_i=1^r  p_i(x)^s_i.

If every s_i is rational such that s_i=t_i/t, then we have 

    H_s(p_1, ..., p_r)= F_t(p_1,…,p_1, p_2, …,p_2,…, p_r, …, p_r),

where, on the right-hand side, p_i appears t_i times, for all i ∈ [r]. As such, the Hellinger transform for rational s is a special case of the Matusita fidelity.


The Hellinger distance between two probability distributions p and q is defined as 

    d_H (p, q)  √(p) - √(q)_2 = (∑_x( √(p(x)) - √(q(x)))^2)^1/2 .

Next, we state that the Matusita multivariate fidelity obeys a uniform continuity bound, which quantifies its deviation in terms of the deviation of corresponding probability distributions in two different tuples of probability distributions. 


 

Let (  p_1,…,p_r) and
(  q_1,…,q_r) be two
tuples of probability distributions, and let ε>0 be such that
1/r(  ∑_i=1^r[ d_H (p_i, q_i) ]^2 )  ≤ε, where d_H (·, ·) is defined in (<ref>). Then,

    | F_r(p_1,…,p_r)-F_r(q_1,…,q_r)|≤
    r(ε)  ^1/r.





    See <Ref>.




 §.§ Average pairwise fidelity


The average pairwise Bhattacharyya overlap is defined in <cit.>, which serves as an upper bound on the error probability of an uncoded transmission of classical data through polarized channels. We also call this quantity the  average pairwise fidelity, and note here that it has been considered previously in a similar context for the quantum case <cit.>.

Here we recall the definition of the average pairwise fidelity for classical probability distributions and  for commuting states more generally.

 
Let p_1,…, p_r be probability distributions on a finite set . 
The average pairwise fidelity is defined as 

    F(p_1,…, p_r)    2/r(r-1)∑_i <j F (p_i, p_j) 
        = 2/r(r-1)∑_i <j∑_x ∈√(p_i(x) p_j(x)).

Furthermore, for  commuting states ρ_1, …, ρ_r, the average pairwise fidelity is defined as 
    
    F(ρ_1, …, ρ_r)    2/r(r-1)∑_i <j F(ρ_i,ρ_j) 
       =  2/r(r-1)∑_i <j∑_x √(ρ_i(x) ρ_j(x)) ,

where (ρ_i(x))_x is the tuple of eigenvalues of ρ_i.



The average pairwise fidelity of commuting states obeys a uniform continuity bound, which follows as an immediate consequence of <Ref> below, the latter holding for general quantum states.    





  For a tuple of commuting states (ρ_i)_i=1^r, the following inequality holds, relating the average pairwise fidelity F(ρ_1, …, ρ_r) and the Matusita fidelity F_r(ρ_1, …, ρ_r):
  
    F(ρ_1, …, ρ_r) ≥ F_r(ρ_1, …, ρ_r).




By <ref>, we have

    F(ρ_1, …, ρ_r)      =2/r(r-1)∑_i<j∑_x√(ρ_i(x) ρ_j(x))
        = ∑_x2/r(r-1)∑_i<j√(ρ_i(x) ρ_j(x)),

where (ρ_i(x))_x is the tuple of eigenvalues of ρ_i, for i∈ [r].
For each x, the right-hand side of the above expression is an average of r(r-1)/2 terms.
Applying the inequality of arithmetic and geometric means then gives

    2/r(r-1)∑_i<j√(ρ_i(x) ρ_j(x))   ≥∏_i<j(√(ρ_i(x) ρ_j(x)) )^2/r(r-1)
        = ∏_i<j(ρ_i(x) ρ_j(x))^1/r(r-1).

The desired inequality (<ref>) then follows from the above inequality because each ρ_i(x) appears exactly r-1 times in the  product in (<ref>).



Later on in <Ref>, we generalize the inequality in (<ref>) to the log-Euclidean class of quantum fidelities.
 



 §.§ Average k-wise fidelities




As a generalization of the average pairwise fidelity and to interpolate between this quantity and the Matusita fidelity of order r, we define the average k-wise fidelities of a tuple of r commuting states as follows:
 
    For r∈{3, 4, …}, let ρ_1, …, ρ_r be a tuple of commuting states, where (ρ_i(x))_x is the tuple of eigenvalues of ρ_i. For k ∈{2, …, r}, we define the average k-wise fidelity of ρ_1, …, ρ_r  as
    
    F_k,r(ρ_1, …, ρ_r) 1/rk∑_i_1 < i_2 < ⋯ <i_k F_k(ρ_i_1, ρ_i_2, …, ρ_i_k),

    where F_k(ρ_i_1, ρ_i_2, …, ρ_i_k) = ∑_x (ρ_i_1(x) ⋯ρ_i_k(x))^1/k is the Matusita fidelity of order k.


As a consequence of the properties of Matusita fidelity, all of these quantities obey the data-processing inequality, symmetry under exchange of the states, faithfulness, equal to zero for orthogonal states, and the direct-sum property. They are all also jointly concave, as a consequence of the data-processing inequality and the direct-sum property.

Observe that F(ρ_1, …, ρ_r) = F_2,r(ρ_1, …, ρ_r) and F_r(ρ_1, …, ρ_r) = F_r,r(ρ_1, …, ρ_r), so that both the average pairwise fidelity and the Matusita fidelity of order r are special cases of the average k-wise fidelities.

As a refinement of <Ref>, the average k-wise fidelities are sorted in a descending order, which again follows from an application of the inequality of arithmetic and geometric means, as well as basic combinatorial reasoning:



For r ∈{3,4, …}, let ρ_1, …, ρ_r be a tuple of commuting states. Then

    F_2,r≥ F_3,r≥⋯≥ F_r-1,r≥ F_r,r,

where, for brevity, we have suppressed the dependence of each quantity F_k,r on ρ_1, …, ρ_r.    





    See <Ref>.








§ PROPOSED QUANTUM GENERALIZATIONS
 



In this section, we first introduce several desirable properties of multivariate quantum fidelity, which are generalizations of the properties satisfied by the Uhlmann fidelity in <ref>.
Then we propose four main generalizations of the bivariate quantum fidelities from <Ref>:

    
  * The first generalization is the average pairwise z-fidelity, which is the simplest generalization of the classical average pairwise fidelity in <ref>.

    
  * The next generalization, called the multivariate SDP fidelity, uses the SDP formulation of the Uhlmann fidelity presented in <Ref>.


  * The third generalization, called the secrecy-based multivariate fidelity, is inspired by an existing secrecy measure from <cit.>.


  * The fourth generalization, called the multivariate log-Euclidean fidelity, is defined through the log-Euclidean divergences given in <ref>.

We also show that the second and the third generalizations are  quantum generalizations of the classical average pairwise fidelity introduced in <Ref>. Furthermore, we show that the last one is a quantum generalization of the Matusita multivariate fidelity given in <ref>.  In addition, we define maximal and minimal extensions of multivariate classical fidelities (see <ref> and <ref>). We also show that the proposed multivariate quantum fidelities satisfy several desired properties  presented in <ref>.

 

    For quantum states ρ_1,…, ρ_r, we consider the following properties of a multivariate fidelity quantity 𝐅(ρ_1, …, ρ_r):
     
        
  * Reduction to multivariate classical fidelity: For commuting states ρ_1, …, ρ_r,
        
    𝐅(ρ_1, …, ρ_r)     = 2/r(r-1)∑_i <j F(ρ_i,ρ_j) 
       =  2/r(r-1)∑_i <j∑_x √(ρ_i(x) ρ_j(x)),

or 

    𝐅(ρ_1, …, ρ_r)     =  F_r(ρ_1, …, ρ_r) 
       =∑_x( ρ_1(x) ⋯ρ_r(x))^1/r,

or, for some k∈{3,…, r-1},

    𝐅(ρ_1, …, ρ_r)     =  F_k,r(ρ_1, …, ρ_r) 
       =1/rk∑_i_1 < … < i_k∑_x( ρ_i_1(x) ⋯ρ_i_k(x))^1/k,

where (ρ_i(x))_x is the spectrum of the state ρ_i in the common eigenbasis of all the states.

        
  * Data processing: For a quantum channel ,
        
    𝐅(ρ_1, …, ρ_r) ≤𝐅( (ρ_1), …, (ρ_r) ).

        
  * Symmetry: For every permutation  π of  [r],
        
    𝐅(ρ_1,…, ρ_r)= 𝐅(ρ_π(1), …, ρ_π(r)).

        
  * Faithfulness: 𝐅(ρ_1, …, ρ_r)=1 if and only if all the states are the same, i.e., ρ_i=ρ_j for i,j ∈ [r].
        
        
  * Orthogonality: 𝐅(ρ_1, …, ρ_r)=0 if and only if all states are orthogonal to each other, i.e., ρ_i ρ_j=0 for i ≠ j,  i,j  ∈ [r].[Note that it is only possible for Matusita fidelity to satisfy 𝐅(ρ_1, …, ρ_r)=0 if all states are orthogonal to each other. The example presented in (<ref>) excludes the other implication from holding for Matusita fidelity. The same statements apply to the average k-wise fidelities for r≥ 3 and 3 ≤ k ≤ r.]
        
        
        
  * Direct-sum property: Let (ρ_i^x)_x be tuples of quantum states for all i ∈ [r]. For a probability distribution (p(x))_x∈ where  is a finite alphabet and classical–quantum states formed using those tuples, the following holds: 
    
    𝐅( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x) 
    
             = ∑_x ∈ p(x)  𝐅(ρ_1^x, …, ρ_r^x).
 
    
    


Another desirable property for a multivariate fidelity is joint concavity, stated as follows. Let (ρ_i^x)_x ∈𝒳 be tuples of quantum states for all i ∈ [r], and let (p(x))_x∈𝒳 be a probability distribution. Then
    
    𝐅( ∑_x ∈ p(x) ρ_1^x, …, ∑_x ∈ p(x) ρ_r^x) ≥∑_x ∈ p(x)   𝐅(ρ_1^x, …, ρ_r^x).

This property is an immediate consequence of data processing and the direct-sum property. Indeed, if these latter two properties hold, then one obtains joint concavity by applying (<ref>) and then (<ref>) with the channel 𝒩 as the partial trace over the classical register.





 §.§ Average pairwise fidelities
 



In this subsection, we generalize the  average pairwise classical fidelity in the simplest way possible by using the z-fidelity defined in (<ref>). 


 
    For z ≥ 1/2 and quantum states ρ_1, …, ρ_r, 
   the average pairwise z-fidelity is defined as 
   
    F_z(ρ_1, …, ρ_r) 2/r(r-1)∑_i <j F_z(ρ_i,ρ_j).




    By fixing z=1/2, we obtain the average pairwise Holevo fidelity, denoted by
    
    F_H(  ρ_1,…,ρ_r)   2/r(  r-1)  ∑
    _i<jF_H(ρ_i,ρ_j).

    For z=1, we obtain the average pairwise Uhlmann fidelity, denoted by
    
    F_U(  ρ_1,…,ρ_r)    2/r(  r-1)  ∑
    _i<jF(ρ_i,ρ_j).

    As mentioned previously, the average pairwise Uhlmann fidelity has appeared in <cit.> as a measure of the reliability of a classical-quantum channel.
    Note that in the classical (viz., commuting states) case, the average pairwise z-fidelities are equal for all z ≥ 1/2, and they reduce to the multivariate classical fidelity given in <ref>.




  §.§.§ Uniform continuity bound for average pairwise Uhlmann and Holevo fidelities


Here we prove uniform continuity bounds for the Uhlmann and Holevo average pairwise fidelities.
Before doing so, let us define the Bures <cit.> and Hellinger distances <cit.> and recall that they obey the triangle
inequality:

    d_B(ρ,σ)     √(2[  1-F(ρ,σ)]
    ), 
    
    d_H(ρ,σ)     √(2[  1-F_H(ρ,σ)]
    ).





Let ρ_1, …, ρ_r, σ_1, …, σ_r be quantum states, and let ε>0 be such that 1/r∑_i=1^r d_B(ρ_i,σ_i)≤ε.
Then

    | F_U(  ρ_1,…,ρ_r)-F_U(  σ_1,…,σ_r)|≤ 2√(2)ε.

Similarly, if ε>0 is such that 1/r∑_i=1^r d_H(ρ_i,σ_i
)≤ε,
then

    | F_H(  ρ_1,…,ρ_r)-F_H(  σ_1,…,σ_r)|
    ≤ 2√(2)ε.



    See <Ref>.




  §.§.§ Properties of average pairwise z-fidelities


Here we show that the average pairwise z-fidelities satisfy all desirable properties from <ref>. In addition, we show that it satisfies super-multiplicativity and coarse-graining. 


    For z∈ [1/2,+∞), the average pairwise z-fidelity satisfies all desired properties of a multivariate fidelity, as listed in <ref>.



    Proofs of these properties follow because the z-fidelity (for r=2) satisfies all the properties listed in <ref>. In particular, reduction to the classical setting follows by plugging commuting states into (<ref>) and simplifying.  Faithfulness follows because, if the states are the same, then F_z(ρ, ρ) = ρ^1/4zρ^1/4z_2z^2z= ρ^1/2z_2z^2z = 1. If F_z(ρ, σ) = 1, then due to the fact that the z-fidelities are monotonically decreasing and continuous in z <cit.>, we conclude that F(ρ, σ)=1, from which we conclude that ρ = σ. Orthogonality follows because, if ρσ = 0, then  ρ^1/4zσ^1/4z = 0, from which we conclude that F_z(ρ, σ) = 0. If F_z(ρ, σ) = 0, then ρ^1/4zσ^1/4z = 0 (from positive definiteness of the norm expression in (<ref>)), which implies that ρσ = 0.
    Moreover, data-processing follows from <cit.> and the direct-sum property because
    
    [ ( ( ∑_x p(x) |x⟩⟨ x| ⊗ρ_x)^1/4z( ∑_x  p(x)|x⟩⟨ x| ⊗σ_x)^1/2z( ∑_x p(x)|x⟩⟨ x| ⊗ρ_x)^1/4z)^z] 
    
         = ∑_x p(x) [ (ρ_x^1/4zσ_x^1/2zρ_x^1/4z)^z ],

 thus concluding the proof.
 



    For states 
 
    ρ_1, …, ρ_r and z ≥ 1/2, the average pairwise z-fidelity is supermultiplicative: 
    
    (F_z(ρ_1, …, ρ_r) )^n ≤  F_z(ρ_1^⊗ n, …, ρ_r^⊗ n).




    Using <ref> for the average pairwise z-fidelity, consider the following steps:
    
    F_z(ρ_1^⊗ n, …, ρ_r^⊗ n) 
               = 2/r(r-1)∑_i <j F_z(ρ_i^⊗ n,ρ_j^⊗ n) 
       =  2/r(r-1)∑_i <j(F_z(ρ_i,ρ_j))^n 
       ≥( 2/r(r-1)∑_i <j   F_z(ρ_i,ρ_j)  )^n 
       =  (F_z(ρ_1, …, ρ_r) )^n ,

    where the second equality follows from the multiplicativity of z-fidelity.
    For the inequality, we use convexity of the function x^n for n>1 and apply Jensen's inequality as follows: Let Y be a random variable with a uniform probability distribution over the values y_ij, where y_ij F_z(ρ_i,ρ_j) for all i <j with i,j ∈ [r]. Then Jensen's inequality gives 𝔼[ Y^n] ≥( 𝔼[Y] )^n, where 𝔼 denotes the expectation.


 
 
For states ρ_1,…,ρ_r, …, ρ_r+m,
    the average pairwise z-fidelity satisfies the following inequality for all z ≥ 1/2:
    
    F_z(ρ_1,…, ρ_r) ≤(r+m)(r+m-1)/r(r-1) F_z(ρ_1,…,ρ_r, …, ρ_r+m).




Consider that
    
    ∑_i, j=1 : i ≠ j^r F_z(ρ_i,ρ_j) ≤∑_i, j=1 : i ≠ j^r+m F_z(ρ_i,ρ_j)

    due to 0 ≤ F_z(ρ,σ). Then, by rewriting the above terms by using <ref>, we arrive at the desired inequality.    




 §.§ Multivariate semi-definite programming  fidelity


Here we propose a multivariate generalization of Uhlmann fidelity by generalizing its SDP formulation given  in (<ref>):


    Given quantum states ρ_1, …, ρ_r, we define their multivariate SDP fidelity as 

    F_SDP(ρ_1,…, ρ_r) 
    1/r (r-1)inf_Y_1, …, Y_r ≥ 0 {∑_i=1^r [ Y_i ρ_i] : ∑_i=1^r |i⟩⟨ i| ⊗ Y_i ≥∑_i ≠ j |i⟩⟨ j| ⊗ I }.

    


 <ref> extends to general positive semi-definite (PSD) operators, simply by replacing the tuple of states (ρ_i)_i=1^r with the tuple of PSD operators (A_i)_i=1^r.


In view of the other SDP formulations 
of the Uhlmann fidelity in Proposition <ref>, it is natural to ask if analogous formulations can be constructed for the multivariate SDP fidelity. It is indeed the case, as stated in <ref> and <ref>. 



 
The multivariate SDP fidelity in <ref> has the following dual formulation: 
      
    F_SDP(ρ_1, …, ρ_r)= 
    2/r (r-1)sup_(X_ij)_i ≠ j s.t.
     X_ji =X^†_ij{∑_i <jℜ[ [X_ij]]: ∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ j| ⊗ X_ij≥ 0 }.



    See <Ref>.






  §.§.§ Uniform continuity bound for multivariate SDP fidelity


The goal of this subsection is to prove a uniform continuity bound for the multivariate SDP fidelity. One important step in doing so is to introduce the following quantity, which we show later to be equal to the multivariate SDP fidelity.


    For a tuple (ρ_i)_i=1^r of states, define 

    F_K^⋆(ρ_1, …, ρ_r) 
    1/ r-1sup_K ≥ 0{⟨ψ | K ⊗ I_d | ψ⟩ -1:  K= I_r ⊗ I_d+ ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij},

where

    | ψ⟩1/√(r)∑_i=1^r |i ⟩ | ϕ^ρ_i⟩  ,

with | ϕ^ρ_i⟩ being  a purification of ρ_i for all i ∈ [r] and d the dimension of the Hilbert space of the  states.


We call this the K^⋆-representation, and later on,  Theorem <ref> states that it is equal to the multivariate SDP fidelity:

    F_SDP(ρ_1, …, ρ_r) 
     = F_K^⋆(ρ_1, …, ρ_r)


Let us note that the K^⋆-representation can also be expressed as

    F_K^⋆(ρ_1, …, ρ_r) = 
    1/ r-1sup_K' ∈Herm{⟨ψ | K' ⊗ I_d | ψ⟩ :    K' ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij≤ I_r ⊗ I_d    },

where the optimization is over every Hermitian matrix K'.

Next, we show that the K^⋆-representation  satisfies a uniform continuity bound. This property finds use in the proof of <ref>.


Let ρ_1, …, ρ_r, σ_1, …, σ_r be quantum states, and let ε∈[0,1] be such that

    1/r∑_i=1^rF(ρ_i
    ,σ_i)≥1-ε.

Then,

    | F_K^⋆(ρ_1, …, ρ_r)-F_K^⋆(σ_1, …, σ_r)|≤r/r-1√(ε(  2-ε)  ).




    See <Ref>.



<Ref> below provides another equivalent formulation for the multivariate SDP fidelity, stating that it is equivalent to K^⋆-representation. 
This formulation will be useful in proving the lower bound stated in  <ref> and some consequential properties in <ref>.

 
For 
quantum states ρ_1,…, ρ_r, the multivariate SDP fidelity (<ref>) is equal to the K^⋆-representation: 

    F_SDP(ρ_1, …, ρ_r) = F_K^⋆(ρ_1, …, ρ_r) .



See <Ref>.



By using <ref> and <ref>, we arrive at the following uniform continuity bound for the multivariate SDP fidelity.


Let ρ_1, …, ρ_r, σ_1, …, σ_r be quantum states, and let ε∈[  0,1] be such that

    1/r∑_i=1^rF(ρ_i
    .σ_i)≥1-ε,

Then,

    | F_SDP(ρ_1, …, ρ_r)-F_SDP(σ_1, …, σ_r)|≤r/r-1√(ε(  2-ε)  ).





  §.§.§ Properties of multivariate SDP fidelity


Here we present several properties satisfied by multivariate SDP fidelity. 


   Multivariate SDP fidelity satisfies all the desired properties of multivariate fidelity listed in <ref>. For commuting states, it reduces to the classical average pairwise fidelity.  


    See <Ref>.



  For a tuple of PSD operators (A_i)_i=1^r, the following properties hold.
  For c >0, by applying <Ref> directly, we have the following scaling property: 

    F_SDP( cA_1, … cA_r) = c F_SDP(A_1, …, A_r).

Let (A_i^x)_x be tuples of PSD operators for all i ∈ [r]. Then

    F_SDP( ∑_x ∈ |x⟩⟨ x| ⊗ A_1^x, …, ∑_x ∈  |x⟩⟨ x| ⊗ A_r^x) = ∑_x ∈  F_SDP(A_1^x, …, A_r^x).
 
    A proof of (<ref>) follows similarly to the proof of direct-sum property of multivariate SDP fidelity in <ref>.
    To this end, note that property (vi) in <ref> can be obtained as a special case of (<ref>) by using  (<ref>) and (<ref>).
The other properties (i)-(iii) 
in <ref> also hold by following the same proof arguments, due to the fact that <ref> holds even for PSD operators, among other things.


 
    Let r,m ∈ℕ, and let ρ_1,…, ρ_r+m be quantum states. We have the following inequality: 
    
    F_SDP(ρ_1,…, ρ_r) ≤(r+m)(r+m-1)/r(r-1) F_SDP(ρ_1,…,ρ_r, …, ρ_r+m).
    



    See <Ref>.


 
  Through numerical calculations, we found that there exists a tuple of states for which super-multiplicativity does not hold for the SDP fidelity, i.e., 
  (F_SDP(ρ_1, …, ρ_r) )^n >  F_SDP(ρ_1^⊗ n, …, ρ_r^⊗ n) for some r and (ρ_i)_i=1^r. An example of a tuple of pure states (i.e., ρ_i  | ψ_i⟩⟨ψ_i| for i∈{1,2,3}) for r=3, n=2, and d=3 is as follows (up to numerical approximations of MATLAB): 
  
    |ψ_1 ⟩   =[ -0.8954 + 0.2791 i;     0.2061-0.0805i;    0.2418+ 0.1135i ], 
    
          |ψ_2 ⟩   =[ -0.2422 + 0.2315 i;     0.4386-0.4318i;  -0.6928 - 0.1704i ], 
     
          |ψ_3 ⟩   =[ 0.2560+ 0.5837i;  0.4811-0.3057i; -0.5175- 0.314i ].

For the above example, 

    0.4075=(F_SDP(ρ_1, ρ_2, ρ_3) )^2 >  F_SDP(ρ_1^⊗ 2, ρ_2^⊗ 2, ρ_3^⊗ 2) =0.3820,

which shows a violation of super-multiplicativity.[The MATLAB code used to find the given counterexample is available as arXiv ancillary files along with the arXiv posting of this paper.] 




  §.§.§ Other reductions of multivariate SDP fidelity


 
The development in the proof of <ref> gives insight into how we can view the SDP fidelity as being somewhat like
an average pairwise quantity (see (<ref>)). Indeed, we see that

    F_SDP(ρ_1,…,ρ_r)=
    2/r(  r-1)  sup_K≥0{[            ∑_i<jℜ[  ⟨ψ^ρ_i|K_i,j⊗
    I|ψ^ρ_j⟩]  :; K=∑_i,j=1^r|i⟩⟨ j|⊗ K_i,j, K_i,i=I_d ∀
    i∈[  r] ]}  .





By choosing r=2 in <ref> and applying (<ref>), we obtain yet another formulation for Uhlmann fidelity. 


    For all  quantum states ρ and σ, the Uhlmann fidelity can be expressed as 
    
    F(ρ,σ)=sup_K∈Herm{[ ⟨ϕ|K⊗ I_d|ϕ⟩:
    I_2 ⊗ I_d ≥
    [   0   X; X^†   0 ] K ]}
    ,

    where 
    
    |ϕ⟩1/√(2)(  |0⟩|ψ^ρ⟩
    +|1⟩|ψ^σ⟩)

  with   | ψ^ρ⟩ and | ψ^σ⟩ being purifications of ρ and σ, respectively.


We obtain a formulation for multivariate SDP fidelity of pure states by simplifying the K^⋆-representation and applying  <ref>.

    
For  a tuple of pure states, (  |ψ_1⟩⟨ψ_1|,…,|ψ_r⟩⟨ψ_r|),

    F_SDP(|ψ_1⟩⟨ψ_1|,…,|ψ
    _r⟩⟨ψ_r|)=
    2/r(  r-1)  sup_k_i,j∈ℂ{∑
    _i<jℜ[k_i,j⟨ψ_i|ψ_j⟩]:∑
    _i,j=1^rk_i,j|i⟩⟨ j|≥0, k_i,i=1 ∀ i∈[
    r]  }  .




    See <Ref>.








 §.§ Secrecy-based multivariate fidelity
 


In this subsection, we introduce another multivariate quantum fidelity inspired by an existing secrecy measure in (<ref>) and analyse its properties.

Recall the following secrecy measure from <cit.>, defined in terms of the average Uhlmann fidelity of a tuple of states to a reference state:

    S(ρ_1, …, ρ_r) sup_σ∈𝒟1/r∑_i=1^rF(  ρ_i, σ) =sup_σ∈𝒟 F(ρ_XA, ρ_X ⊗σ),

where ρ_XA∑_i=1^r1/r|i⟩⟨ i|_X⊗ρ_i. Observe that ρ_X = I/r. The square of this  secrecy measure was given a direct operational interpretation in <cit.> as the maximum success probability, in a quantum interactive proof, that a prover could pass a test for the states being similar. See <cit.> for a comprehensive study on the optimization problem involved with this secrecy measure. In particular, by putting together Eqs. (22)–(23), Definition 3, Lemma 4, and Theorems 5 and 6 of <cit.> and performing some other simplifications, we arrive at the following primal and dual SDP formulations of S(ρ_1, …, ρ_r):

    S(ρ_1,…,ρ_r)  
        =1/rsup_X_i,j=X_j,i^†∈ℒ,
    ∀ i,j∈{0,1, …, r}  ,
    σ≥0{[                                ∑_i=1^rℜ[Tr[X_i,0
    ]] : Tr[σ]=1,; |0⟩⟨
    0|⊗σ
    +∑_i=1^r|i⟩⟨ i|⊗ρ_i
    +∑_i,j=0: i≠ j^r|i⟩⟨ j|⊗ X_i,j≥0 ]}
        =1/2rinf_Y_1,…,Y_r≥0,
    μ≥0{[                                                  μ+∑_i=1^rTr[Y_iρ_i]:; |0⟩⟨
    0|⊗μ I_d +
    ∑_i=1^r|i⟩⟨ i|⊗ Y_i
    ≥∑_i=1^r(  |i⟩⟨ 0|+|0⟩⟨ i|)
    ⊗ I_d ]} .



Using the secrecy measure, we define another multivariate fidelity as follows.
 
Let ρ_1, …, ρ_r be quantum states.
We define the multivariate secrecy   fidelity as 

    F_S(ρ_1, …, ρ_r) 1/r-1( r [ S(ρ_1, …, ρ_r)]^2 -1 ),

   where S(ρ_1,…,ρ_r) is defined in (<ref>).





    By applying <cit.>,  the following equality holds:

    [ S(ρ_1, …, ρ_r)]^2 = sup_P_T'RF → T^”F'∈𝒰1/r^2∑_x,y=1^r ⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA,

 where 
    
    P^x_R → F'⟨ x|_T^” P_T'RF → T^”F'|x ⟩_T' |0 ⟩_F ,

    with P_T'RF → T^”F' being a unitary, and | ϕ^ρ_x⟩ is a purification of state ρ_x for all x ∈ [r].
    Using this expression, the secrecy-based multivariate fidelity can be written somewhat similar to an average pairwise fidelity as follows:
    
    F_S(ρ_1, …, ρ_r)     = sup_P_T'RF → T^”F'∈𝒰{2/r(r-1)∑_x<yℜ[⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA] 
            -
            1/r(r-1)∑_x=1^r ⟨ϕ^ρ_x|_RA(I_R - (P^x_R → F')^†  P^x_R → F') | ϕ^ρ_x⟩_RA}
       ≤sup_P_T'RF → T^”F'∈𝒰2/r(r-1)∑_x<yℜ[⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA],

    where the inequality follows because,  for all x,  P^x_R → F' is a contraction.








  §.§.§ Uniform continuity bound for secrecy-based multivariate fidelity


 
Let ρ_1, …, ρ_r, σ_1, …, σ_r be quantum states, and let ε∈[  0,1] be such that

    1/r∑_i=1^r d_B(ρ_i,σ_i)≤ε,

where the Bures distance d_B is defined in (<ref>).
Then,

    | F_S(ρ_1, …, ρ_r) - F_S(σ_1, …, σ_2)  | ≤ 2 √(2)( r  /r-1)ε.



    See <Ref>.








  §.§.§ Properties of secrecy-based multivariate fidelity


Next we prove that secrecy-based multivariate fidelity also satisfies the desirable properties of a multivariate fidelity from <ref>.

 
    The secrecy-based multivariate fidelity satisfies all the desired properties of a multivariate fidelity listed in <ref>. For commuting states, it reduces to the classical average pairwise fidelity.



  See <ref>.



Similar to the average pairwise fidelity and multivariate SDP fidelity, F_S also satisfies the coarse-graining property with an extra additive factor. 
 
   Let r,m ∈ℕ, and let ρ_1,…, ρ_r+m be quantum states. We have the following inequality: 
    
    F_S(ρ_1,…, ρ_r) ≤(r+m)(r+m-1)/r(r-1) F_S(ρ_1,…,ρ_r, …, ρ_r+m) +m/r(r-1).
  


    See <ref>.




    Through numerical calculations, we found that there exists a tuple of states for which super-multiplicativity does not hold for the SDP fidelity, i.e., 
  (F_S(ρ_1, …, ρ_r) )^n >  F_S(ρ_1^⊗ n, …, ρ_r^⊗ n) for some r and (ρ_i)_i=1^r. The same example that violates super-multiplicativity of SDP multivariate fidelity in <ref>, also violates the super-multiplicativity of secrecy-based multivariate fidelity. 






 §.§ Relations between proposed generalizations


In this section, we present some inequalities relating the average pairwise Holevo fidelity, the secrecy-based multivariate fidelity, the multivariate SDP fidelity, and the average pairwise Uhlmann fidelity.
In particular, we show that the multivariate SDP fidelity and secrecy-based multivariate fidelity are sandwiched between the average pairwise Holevo fidelity and the average pairwise Uhlmann fidelity, from below, and above, respectively.



 
For a tuple (ρ_i)_i=1^r of states,
the multivariate SDP fidelity and secrecy-based multivariate fidelity lie between the average pairwise Holevo fidelity and the 
average pairwise Uhlmann fidelity:

    F_H(ρ_1, …, ρ_r) ≤ F_S(ρ_1, …, ρ_r) ≤ F_SDP(ρ_1, …, ρ_r) ≤ F_U(ρ_1, …, ρ_r) ≤√(F_H(ρ_1, …, ρ_r)).
  


    See <Ref>. 





In <ref> we showed that multivariate SDP fidelity is  bounded from above by the average pairwise Uhlmann fidelity. Next, we show that it is also  bounded from below by a function of pairwise Uhlmann fidelities (note that it is not an average).

    The multivariate SDP fidelity is bounded from below by a multiple of the maximum of sums of pairwise fidelities of the given states. More precisely, given quantum states ρ_1,…, ρ_r, we have
  
    2/r(r-1)sup_π∈ S_r∑_i=1^⌊r/2⌋ F(ρ_π(i), ρ_π(i+⌊ r/2 ⌋))  
        ≤ F_SDP(ρ_1, …, ρ_r).



    See <Ref>.





 §.§ Minimal and maximal extensions of multivariate classical fidelities


One way to construct a multivariate quantum fidelity is by optimizing classical multivariate fidelities, and there are at least two ways of doing so. The first way is to perform a measurement of the r quantum states, resulting in r probability distributions, and then to minimize a classical multivariate fidelity of the r distributions over all possible measurements. The measurement is also called a quantum-to-classical transformation. The second way is to begin with r probability distributions and act on them with a preparation channel that realizes the r quantum states, and then to maximize a classical multivariate fidelity over all possible preparation channels. A preparation channel is also known as a classical-to-quantum transformation.
The former approach leads to a quantity that we call a maximal extension of multivariate fidelity (or measured multivariate fidelity), and the latter leads to a minimal extension of multivariate fidelity (or preparation multivariate fidelity).

These concepts have been introduced and studied extensively in prior work. The measured relative entropy was defined in <cit.> and its extension to Petz-Rényi divergences  in <cit.> (see also <cit.>). The preparation relative entropy was defined in <cit.> and shown therein to be equal to the Belavkin–Staszewski relative entropy from <cit.>. 
The preparation fidelity and more general f-divergences were defined in <cit.> and studied further in <cit.>.
See also <cit.> for maximal and minimal extensions of resource measures, including generalized fidelity for sub-normalized states. Most recently, maximal and minimal extensions of multivariate Chernoff divergence were defined in <cit.>.



We  define the maximal and minimal extensions of classical multivariate fidelity as follows.




We define a maximal extension of multivariate fidelity (also called a measured multivariate fidelity) as 

    𝐅(ρ_1, …, ρ_r) inf_( Λ_x)_x𝐅(Λ(ρ_1), …, Λ(ρ_r))

where 𝐅 is one of the multivariate fidelities listed in (i) of <Ref> and Λ(ρ_i)∑_x Tr[Λ_x ρ]|x⟩⟨ x| is the measurement channel associated with the POVM ( Λ_x)_x (see also (<ref>)).   




    We define a minimal extension of multivariate fidelity (also called a preparation multivariate fidelity) as 
    
    𝐅(ρ_1, …, ρ_r) sup_∈CPTP,  ω_i ∈𝒟_c ∀ i ∈ [r]{𝐅( ω_1, …, ω_r ) : (ω_i) =ρ_i  ∀ i ∈ [r]},

where 𝐅 is one of the multivariate fidelities listed in (i) of <Ref> and  is a classical-quantum channel satisfying (ω_i) =ρ_i for all i∈ [r] and 𝒟_c denotes the set of all states diagonal in the standard basis (so that 𝒟_c is a set of commuting states).


Note that, by the above definitions and data processing of multivariate fidelity (<ref>), we have the following inequalities:

    𝐅(ρ_1, …, ρ_r) ≤𝐅(ρ_1,…, ρ_r) ≤𝐅(ρ_1, …, ρ_r).

Proofs for these inequalities follow the same approach used in <cit.>.
Both the maximal and minimal extensions also satisfy the data-processing inequality.


In the bivariate setting, the maximal extension of the classical fidelity is equal to the Uhlmann fidelity, and the minimal extension is equal to the geometric fidelity, as defined in (<ref>):

By <cit.> we have 

    𝐅(ρ,σ) ≤𝐅(ρ, σ) ≤𝐅(ρ,σ)= F(ρ,σ),

where the equality follows from (<ref>).
Furthermore, <cit.> shows that 

    𝐅(ρ,σ)= inf_ε >0 [ ρ(ε) #σ(ε)],

where A # B denotes the geometric mean of the positive definite operators A and B:

    A # B  A^1/2( A^-1/2 B A^-1/2)^1/2 A^1/2,

ρ(ε) ρ + ε I,
and σ(ε) σ + ε I.
Due to this formulation, the minimal extension of the classical bivariate fidelity is also called the geometric fidelity. In <cit.>, the geometric fidelity was called the Matsumoto fidelity, and several of its properties were established by means of semi-definite programming.




  §.§.§ Measured multivariate fidelity


In this section, we consider various properties satisfied by the measured multivariate fidelity defined in <ref>.




If the underlying classical multivariate fidelity satisfies data processing, symmetry, faithfulness, orthogonality, and the direct-sum property, then the maximal extension satisfies data processing, symmetry, faithfulness, orthogonality, and the direct-sum property.



    See <ref>.




By choosing the underlying classical multivariate fidelity to be the average pairwise fidelity, here we define measured average pairwise fidelity.

 
    For states ρ_1, …, ρ_r, the measured average pairwise fidelity is defined as 

    F_M(ρ_1, …, ρ_r)inf_( Λ_x)_x2/r(r-1)∑_i <j F( Λ(ρ_i), Λ(ρ_j) )

where Λ(ρ_i)∑_x Tr[Λ_x ρ]|x⟩⟨ x| is the measurement channel induced by applying the POVM ( Λ_x)_x on the state ρ_i (see also (<ref>)).


Note that <ref> is the largest quantum generalization of average  pairwise classical fidelity that satisfies data processing.


When 𝐅 is chosen to be a multivariate fidelity that generalizes average pairwise classical fidelity, we have that

    𝐅(ρ_1, …, ρ_r) = F_M(ρ_1,…, ρ_r).

Similarly, when 𝐅 is chosen to be a multivariate fidelity that generalizes Matusita multivariate fidelity, we have that 

    𝐅(ρ_1, …, ρ_r) = inf_(Λ_x)_x F_r( Λ(ρ_1), …, Λ(ρ_r) ).

The next proposition generalizes <ref> to the case of measured multivariate fidelities. 
 

The following inequality holds: 
  
    F_M(ρ_1,…, ρ_r) ≥inf_(Λ_x)_x F_r( Λ(ρ_1), …, Λ(ρ_r) ),
  
where the Matusita multivariate fidelity F_r is defined in (<ref>).


    The proof of the first statement follows from applying <ref> for each measurement channel Λ and then taking the infimum over all measurement channels. 






 

    Let ρ_1, …, ρ_r, σ_1, …, σ_r be quantum states, and let ε>0 be such that 1/r∑_i=1^r d_B(ρ_i,σ_i)≤ε.
Then,

    | F_M(  ρ_1,…,ρ_r)-F_M(  σ_1,…,σ_r)|≤ 2√(2)ε.

Furthermore, if ε>0 is such that 1/r∑_i=1^rF(ρ_i
,σ_i)≥1-ε,
then

    | F_M(  ρ_1,…,ρ_r)-F_M(  σ_1,…,σ_r)|≤r/r-1√(ε(  2-ε)  ).



    For F_M(  ρ_1,…,ρ_r),
    the underlying classical multivariate fidelity is the average pairwise fidelity. Considering the reduction to the classical case for the quantum generalizations we have proposed, the following equalities hold for every measurement channel Λ:
    
    F( Λ(ρ_1), …, Λ(ρ_r))=  F_U( Λ(ρ_1), …, Λ(ρ_r))=  F_SDP( Λ(ρ_1), …, Λ(ρ_r)).

Since classical multivariate fidelity satisfies data processing we also have the following implications:

    1/r∑_i=1^r d_B(ρ_i,σ_i)≤ε1/r∑_i=1^r d_B( Λ(ρ_i),Λ(σ_i) )≤ε, 
       1/r∑_i=1^rF(ρ_i
    ,σ_i)≥1-ε1/r∑_i=1^rF(Λ(ρ_i)
    ,Λ(σ_i) )≥1-ε.

Now consider that

    F_M(  ρ_1,…,ρ_r) -  F_M(  σ_1,…,σ_r) 
        =  inf_( Λ_x)_x   F( Λ(ρ_1), …, Λ(ρ_r)) -    inf_( Λ'_x)_x F( Λ'(σ_1), …, Λ'(σ_r)) 
        =  sup_( Λ'_x)_x( inf_( Λ_x)_x   F( Λ(ρ_1), …, Λ(ρ_r)) -  F( Λ'(σ_1), …, Λ'(σ_r))  ) 
       ≤sup_( Λ'_x)_x(    F( Λ'(ρ_1), …, Λ'(ρ_r)) -  F( Λ'(σ_1), …, Λ'(σ_r))  ).

At this point, we conclude the two inequalities by applying <ref> and <ref> along with (<ref>).


 

    Using a similar proof technique as given in the proof of <ref> and using uniform continuity of the Matusita multivariate fidelity in <ref>, it is possible to obtain uniform continuity bound for measured Matusita multivariate fidelity as well under the assumption that 1/r( ∑_i=1^r [ d_B(ρ_i,σ_i)]^2) ≤ε.
 

The following proposition establishes inequalities between the measured average pairwise fidelity and average pairwise Uhlmann fidelity. 
 
The following inequality holds for every tuple (ρ_1,…,ρ_r) of states:

    F_M(ρ_1, …, ρ_r) ≥  F_U(ρ_1, …, ρ_r).
   





    Consider the following:

    F_M(ρ_1, …, ρ_r) 
           =inf_( Λ_x)_x2/r(r-1)∑_i <j F( Λ(ρ_i), Λ(ρ_j) ) 
       ≥2/r(r-1)∑_i <jinf_( Λ_x)_x F( Λ(ρ_i), Λ(ρ_j) ) 
       = 2/r(r-1)∑_i <j F(ρ_i,ρ_j),

where the first inequality follows from placing the infimum inside the sum, the second equality follows from the Fuchs–Caves characterization of Uhlmann fidelity in terms of the measured bivariate fidelity (see (<ref>)). 







    When r=2, the Uhlmann fidelity is equal to the measured fidelity, as stated in (<ref>). However, when r>2, through numerical evaluations, it follows that there exists a tuple of states satisfying 
    F_M(ρ_1, …, ρ_r) > F_SDP(ρ_1,…, ρ_r). This is numerically validated by showing that F_U ≥ F_SDP in (<ref>) is a strict inequality for some tuples of states and then combining with (<ref>). One such tuple of states (up to numerical approximations of MATLAB)  
    is the following:
    
    ρ_1    = [  0.3465+0.0000i -0.2036+0.2643i; -0.2036-0.2643i  0.6535+0.0000i ], 
    ρ_2    = [  0.6546+0.0000i -0.3308-0.2297i; -0.3308+0.2297i  0.3454+0.0000i ],
    ρ_3    =[ 0.6169+0.0000i 0.0327-0.0321i; 0.0327+0.0321i 0.3831+0.0000i ].

The MATLAB code used to find the given counterexample is available as arXiv ancillary files along with the arXiv posting of this paper.

Furthermore, it is interesting to determine whether there exists a tuple of states for which F_M(ρ_1, …, ρ_r) > F_U(ρ_1,…, ρ_r)—we leave this as an open question.








  §.§.§ Preparation multivariate fidelity


The preparation multivariate fidelity, as outlined in <ref>, is also known as the minimal extension of multivariate fidelity. Next, we discuss various properties satisfied by the preparation multivariate fidelity.

  

The preparation multivariate fidelity in <ref> satisfies data processing, symmetry, faithfulness, and the direct-sum property, as listed  in <ref>.

Furthermore, when the states are orthogonal to each other, we have 𝐅(ρ_1, …, ρ_r) =0 (weak orthogonality).


    See <ref>.



The minimal extension of the multivariate fidelity does not satisfy uniform continuity in general. To see that consider the following counterexample: Fix r=2  and choose 

    ρ_1 = ρ_2= σ_1= |0 ⟩⟨ 0|    and  σ_2= |ψ⟩⟨ψ|,

where |ψ⟩√(1-ε) |0 ⟩  + √(ε) |1⟩ for ε>0.
Then, we obtain the following by using the properties of Uhlmann fidelity from <ref>:

    F(ρ_1,σ_1)     =1 ,
    
        F(ρ_2, σ_2)    = √(1-ε) ,
    𝐅(ρ_1, ρ_2)    =1 ,
    𝐅(σ_1, σ_2)    = 0,

where the third equality follows by the faithfulness of preparation fidelity and the last equality by the fact that the minimal extension of the fidelity between two pure states  evaluates to zero <cit.> (see also <cit.>).
With the above evaluations we see that  1/2( F(ρ_1,σ_1) + F(ρ_2,σ_2) ) ≥ 1- ε and yet 

    | 𝐅(ρ_1, ρ_2) - 𝐅(σ_1, σ_2) |=1,

implying that the preparation multivariate fidelity does not satisfy uniform continuity in general. 








 §.§ Multivariate log-Euclidean fidelity
 



In this section, we recall the definition of multivariate log-Euclidean divergences <cit.>, we show how one possible quantum Matusita fidelity is a special case, and we give operational interpretations of them in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis, specifically making use of the main result of <cit.>. 



    Let ρ_1,…,ρ_r be positive definite states and let s=(  s_1,…,s_r) be a probability distribution.
    The multivariate
log-Euclidean divergence is defined as

    H_s(ρ_1,…,ρ_r) -lnTr[  exp(
    ∑_i=1^rs_ilnρ_i)  ]  .

For general states, the multivariate log-Euclidean divergence is defined as

    H_s(ρ_1,…,ρ_r) inf_ε > 0  H_s(ρ_1(ε),…,ρ_r(ε)) = lim_ε→ 0^+  H_s(ρ_1(ε),…,ρ_r(ε)),

where ρ_i(ε) ρ_i + ε I for all i∈[r] and the rightmost equality follows from operator monotonicity of the logarithm and monotonicity of (·)→Tr[exp(·)]. The argument of the trace in (<ref>) is known as the log-Euclidean Frechet mean <cit.>.


For simplicity, in most of what follows we focus on positive definite states and note that all of the statements can be extended as in (<ref>).

Now, we define the multivariate log-Euclidean fidelity that extends (<ref>) for commuting states to the general quantum setting.
 
Let (ρ_i)_i=1^r be a tuple of states. We define the multivariate log-Euclidean fidelity for positive definite states as
    
    F^♭_r(ρ_1,…,ρ_r)Tr[  exp(
    1/r∑_i=1^rlnρ_i)  ]  ,

and for the general case as

    F^♭_r(ρ_1,…,ρ_r)inf_ε >0  F^♭_r(ρ_1(ε),…,ρ_r(ε)) = lim_ε→ 0^+  F^♭_r(ρ_1(ε),…,ρ_r(ε)),

where ρ_i(ε)ρ_i + ε I.


Note that <ref> reduces to (<ref>) for the bivariate setting.


 
   When s is uniform (i.e., s_i=1/r for all i ∈ [r]),  the multivariate log-Euclidean divergence is equal to the negative logarithm of the multivariate log-Euclidean fidelity in <ref>:

    -ln(F^♭_r(ρ_1,…,ρ_r) ) = -lnTr[  exp(  ∑_i=1^r1/rlnρ_i
    )  ].

For the special case of commuting states, the multivariate log-Euclidean divergence reduces to the negative logarithm of the Hellinger transform from (<ref>):

    H_s(ρ_1,…,ρ_r)= -lnTr[  ∏_i=1^rρ_i^s_i].



 
     The multivariate log-Euclidean fidelity in <ref> satisfies several desirable properties of a multivariate fidelity listed in <ref>, including reduction to classical Matusita multivariate fidelity in (<ref>), data processing, symmetry, faithfulness, direct-sum property, and uniform continuity.

     If all states are orthogonal to each other, i.e., ρ_i ρ_j=0 for i ≠ j,  i,j  ∈ [r], then F^♭_r (ρ_1, …, ρ_r)=0 (weak orthogonality). However, F^♭_r (ρ_1, …, ρ_r)=0 does not imply that the states are orthogonal to each other.



    See <ref> for a proof of the properties.
    As a counterexample for one direction of the orthogonality property mentioned above, consider the diagonal states formed by the probability vectors given in (<ref>). 





Next, we establish a quantum generalization of the inequality from <ref>.

 
Let (ρ_i)_i=1^r be a tuple of states. Then, we have

    F^♭_r(ρ_1, …, ρ_r) ≤2/r(r-1)∑_i<j F_♭(ρ_i,ρ_j).
   


For positive definite states ρ_1, …, ρ_r, consider that

    2/r(  r-1)  ∑_i<jF_♭(ρ_i,ρ
    _j)
        =2/r(  r-1)  ∑_i<jexp(  -inf_ω∈𝒟[  1/2D(ω‖ρ_i)+1/2
    D(ω‖ρ_j)]  )  
       ≥exp(  -2/r(  r-1)  ∑_i<jinf_ω∈𝒟[  1/2D(ω‖ρ_i)+1/2
    D(ω‖ρ_j)]  )  
       ≥exp(  -inf_ω∈𝒟2/r(  r-1)
    ∑_i<j[  1/2D(ω‖ρ_i)+1/2D(ω‖ρ_j)]  )  
        =exp(  -inf_ω∈𝒟1/r∑_i=1^r
    D(ω‖ρ_i))  
        =F^♭_r(ρ_1,…,ρ_r),

where the first equality follows from (<ref>); the first inequality  from convexity of the exponential
function; the second inequality follows because

    inf_ω∈𝒟2/r(  r-1)  ∑_i<j[
    1/2D(ω‖ρ_i)+1/2D(ω‖ρ_j)]  
    ≥2/r(  r-1)  ∑_i<jinf_ω∈𝒟
    [  1/2D(ω‖ρ_i)+1/2D(ω‖ρ
    _j)];

the penultimate equality is from a counting argument, and the final equality
is again from (<ref>). The general case follows by a limiting argument.







  §.§.§ Connection to oveloh information

The generalized Holevo information of a tuple of states (
ρ_i)  _i with uniform prior distribution is defined as

    inf_σ_A∈𝒟D(ρ_XA‖ρ_X⊗σ_A),

where D is a generalized divergence that satisfies data processing under quantum channels <cit.> and

    ρ_XA1/r∑_i=1^r|i⟩⟨ i|_X⊗ρ_i.


Fix α∈ (0,1) ∪ (1, ∞). The Petz–Rényi relative entropy of a state ρ and a PSD operator σ is defined as
<cit.>

    D_α(ρ‖σ) 1/α-1ln[ ρ^ασ^1- α],    α∈ (0,1) 
    (α > 1 (ρ)⊆(σ))
    
    +∞,

It is a generalized divergence for α∈ [0,1) ∪ (1,2] <cit.>.
Taking the limit α→ 1 gives the quantum relative entropy defined as  <cit.>

    D(ρσ) [ ρ ( logρ -logσ)],    supp(ρ) ⊆supp(σ), 
    
                +∞,    otherwise.



In the spirit of the definition of lautum information (reverse of mutual information) <cit.>, we define the oveloh information (reverse of Holevo information): 

    𝒪(X;A)_ρinf_σ∈𝒟D(ρ_X⊗σ‖ρ_XA) =inf_σ∈𝒟1/r∑_i=1^rD(σ‖ρ
    _i),

where ρ_XA is given in (<ref>).
Also note that oveloh information is a special case of log-Euclidean divergence with the choice of s being a uniform distribution (see (<ref>)). 
To this end, 
exp[- 𝒪(X;A)_ρ] is precisely the  multivariate log-Euclidean fidelity in <ref>: 

    F^♭_r (ρ_1, …, ρ_r)= exp[- 𝒪(X;A)_ρ] =exp(-inf_σ∈𝒟1/r∑_i=1^rD(σ‖ρ
    _i)).


With this connection, we establish a relationship between Holevo and oveloh information as given below.

 
    Let (ρ_i)_i be a tuple of 
    states. Then the following inequality holds: 
    
    exp[ - inf_σ∈𝒟D_1/2(ρ_XA‖ρ
    _X⊗σ) ] ≥1/r + r-1/rexp[  -𝒪(X;A)_ρ],

    where ρ_XB is given in (<ref>) and 
    D_1/2(·· ) is the Petz–Rényi relative entropy of order 1/2 in (<ref>). 



By using (<ref>) together with (<ref>), we have 

    exp[  -inf_σ∈𝒟D(ρ_X⊗σ‖ρ
    _XA)]   
        = F^♭_r(ρ_1, …, ρ_r) 
       ≤2/r(r-1)∑_i<j F_♭(ρ_i, ρ_j),

where the inequality follows from <ref>.
Then, by recalling that z-fidelities are antimonotone  for z ≥ 1/2 together with (<ref>), we have F_♭(ρ,σ) ≤ F_H(ρ,σ) leading to 

    exp[  -inf_σ∈𝒟D(ρ_X⊗σ‖ρ
    _XA)]    ≤2/r(r-1)∑_i<j F_H(ρ_i, ρ_j).

Also, using (<ref>), we obtain the following: 

    exp[  -inf_σ∈𝒟D_1/2(ρ_XA‖ρ_X ⊗σ
    )]   = 1/r +r-1/r[ 2/r(r-1)∑_i<j F_H(ρ_i, ρ_j)].

Combining (<ref>) and the previous equality 
completes the proof. 




Note that the inequality stated above is tighter than that which one would obtain by using the facts that 

    inf_σ∈𝒟D_1/2(ρ_XA‖ρ
    _X⊗σ)    = inf_σ∈𝒟D_1/2(ρ
    _X⊗σ‖ρ_XA) , 
    inf_σ∈𝒟D_1/2(ρ
    _X⊗σ‖ρ_XA)    ≤inf_σ∈𝒟D(ρ
    _X⊗σ‖ρ_XA).




  §.§.§ Operational interpretation of multivariate log-Euclidean divergences
 
The following formula is known (see <cit.>, and <cit.>, as well as <cit.> for a related observation):

    H_s(ρ_1,…,ρ_r)=inf_ω∈𝒟∑_i=1^r
    s_iD(ω‖ρ_i)=inf_ω∈𝒟D(π(s)⊗ω‖ρ(s)),

where 𝒟 is the set of density matrices and the quantum relative entropy is given in (<ref>).
The second equality
follows from the direct-sum property of quantum relative entropy <cit.>, along with
the following definitions:

    π(s)   ∑_i=1^rs_i|i⟩⟨ i|,    ρ(s)   ∑_i=1^rs_i|i⟩⟨ i|⊗ρ_i.



Now we present the hypothesis testing setting and result that provides an operational interpretation of the multivariate log-Euclidean divergence. The setting was considered in a general way in <cit.> and is  known as quantum hypothesis
testing with arbitrarily varying null hypothesis and simple alternative hypothesis. 
The null hypothesis for our case is that a state  is chosen adversarially from the following set:

    Ω^(n)_s{⊗_i=1^n(  π(s)⊗ω_i)  :ω_i∈𝒟 ∀ i∈[  n]
    }  .

This set is an example of what is called an arbitrarily varying source in <cit.>.
The alternative hypothesis is that the state ρ(s)^⊗ n is chosen.
The receiver of the unknown state is allowed to perform a measurement {  M^(n),I^⊗
n-M^(n)} acting on all n systems in order to figure out which
is the case, such that 0≤ M^(n)≤ I^⊗ n. The outcome
M^(n) is associated with choosing the null hypothesis, and the outcome
I^⊗ n-M^(n) is associated with choosing the alternative
hypothesis. Formally, the type I and type II error probabilities are
respectively defined in this setting as

    α(M^(n))     sup_σ^(n)∈Ω^(n)_sTr
    [  (  I^⊗ n-M^(n))  σ^(n)]  ,
    β(M^(n))     Tr[M^(n)ρ(s)^⊗ n].

The fact that the type I error probability is maximized over arbitrary
σ^(n)∈Ω^(n)_s is what makes the
hypothesis testing scenario adversarial, as the state σ^(n) is
selected from Ω^(n)_s in such a way, for a given measurement operator
M^(n), to make the type I error probability as large as possible. For
ε∈[  0,1] and n ∈ℕ, the  optimal non-asymptotic type II error exponent is
defined as

    1/nD_H^ε(Ω^(n)_s‖ρ(s)^⊗ n
    ) -1/nln(  inf_M^(n)∈ℳ^(n){β(M^(n)):α(M^(n))≤ε})  ,

where ℳ^(n) denotes the set of all measurement operators:

    ℳ^(n){ M^(n): 0≤ M^(n)≤ I^⊗ n}.



 
Let s be a probability distribution on [r], and let ρ_1, …, ρ_r be  states. For all ε∈(  0,1), the following equality holds:

    lim_n→∞1/nD_H^ε(Ω^(n)_s‖ρ(s)^⊗ n)=H_s(ρ_1,…,ρ_r),

where 1/n D_H^ε(Ω^(n)_s‖ρ(s)^⊗ n) is defined in (<ref>) and the multivariate log-Euclidean divergence in <ref>.



    The proof follows by applying <cit.> and the identity in
(<ref>).



Thus, we obtain a fundamental operational meaning for H_s(ρ_1
,…,ρ_r) in this hypothesis testing scenario, as the optimal asymptotic type II error exponent. By picking s to be
uniform, we obtain an operational meaning for the negative logarithm of the  multivariate log-Euclidean fidelity in (<ref>).



 §.§ Average k-wise log-Euclidean fidelities
 

As a generalization of the ideas presented in <Ref> for the classical case, we define the average
k-wise log-Euclidean fidelities as follows:


For r∈{  3,4,…}, let ρ_1,…,ρ_r be quantum states. For k∈{
2,3,…,r}, define the average k-wise log-Euclidean fidelity as
follows:

    F_k,r^♭(ρ_1,…,ρ_r):=1/rk∑
    _i_1<⋯<i_kF_k^♭(ρ_i_1,…,ρ_i_k).




We can establish an ordering relationship between them, generalizing that
found in <Ref> for the classical case. The proof below is similar to previous ones, combining
elements of the proofs of <Ref> and <Ref>:



For r∈{  3,4,…}, let ρ_1,…,ρ_r be
quantum states. Then the following inequalities hold:

    F_2,r^♭≥ F_3,r^♭≥⋯≥ F_r-1,r^♭≥
    F_r,r^♭,

where, for brevity, we have suppressed the dependence of F_k,r^♭ on
ρ_1,…,ρ_r.



See <Ref>.


















§ CONCLUDING REMARKS AND FUTURE DIRECTIONS
 
 
In this work, we extended bivariate fidelities to multivariate fidelities, which are measures characterizing the similarity between multiple quantum states. In the classical case, we analysed the Matusita multivariate fidelity <cit.>, average pairwise fidelity, and average k-wise fidelities. We proposed three variants for the quantum case that reduce to the average pairwise fidelity for commuting states, namely, average pairwise z-fidelities for z ≥ 1/2, the multivariate SDP fidelity, inspired by the SDP of the Uhlmann fidelity, and the secrecy-based multivariate fidelity inspired by the secrecy measure proposed in  <cit.>. We analysed their mathematical properties, such as data processing, faithfulness, uniform continuity bounds, and connections between these three variants. To this end, we proved that multivariate SDP fidelity and secrecy-based multivariate fidelity are sandwiched between the average pairwise z-fidelities at z=1/2 and z=1. 
We also explored a quantum extension of Matusita multivariate fidelity, which we called as multivariate log-Euclidean fidelity while analysing its properties and providing an operational interpretation to this in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis.


Open questions for future work include the following (in no particular order):

    
  * What are similarity measures  for a tuple of channels, which generalize multivariate fidelity of states? One approach towards this is to extend the SDP of channel fidelity <cit.> from two channels to multiple channels, by using the same approach we followed here when defining multivariate SDP fidelity  (<ref>). In particular, we can define multivariate channel SDP fidelity for a tuple of channels ( _i )_i=1^r (each one of those is a CPTP map from ℒ_A to ℒ_B) as follows:

    F_SDP(_1, …, _r ) 
       inf_ρ_RA∈𝒟_RA  F_SDP( _1(ρ_RA), …, _r(ρ_RA) )  
        = 1/r (r-1)inf_Y_RB^i ≥ 0  ∀  i ∈ [r], 
    ρ_R ∈𝒟{[                        ∑_i=1^r [ Y_RB^i   Γ_RB^_i] :; ∑_i=1^r |i⟩⟨ i| ⊗ Y_RB^i ≥∑_i ≠ j |i⟩⟨ j| ⊗ρ_R ⊗ I_B ]},

where Γ_RB^_i is the Choi operator of channel _i for all i∈[r] (i.e., Γ_RB^_i∑_k,ℓ |k⟩⟨ℓ| ⊗𝒩_i(|k⟩⟨ℓ|)). The proof of the equality in the last line follows from the same reasoning used to prove <cit.>.

Furthermore, similar to average pairwise z-fidelity for states in <ref>, we can define average pairwise channel z-fidelity as follows for z ≥ 1/2:

    F_z(_1, …, _r) 
          inf_ρ_RA∈𝒟_RA  F_z( _1(ρ_RA), …, _r(ρ_RA) ) 
        = inf_ρ_RA∈𝒟_RA2/r(r-1)∑_i <j F_z (_i(ρ_RA),_j(ρ_RA)).



  * Another interesting future direction is to investigate  multivariate geometric fidelities based on average pairwise fidelities built from (<ref>) or by generalizing the following SDP formulation of geometric fidelity from <cit.> and <cit.>:

    𝐅(ρ,σ)=sup_W ∈Herm{[W] 
         :   [ ρ W; W σ ]≥ 0}.

Indeed a multivariate generalization of the SDP above is as follows:

    2/r (r-1)sup_(X_ij)_i ≠ j s.t.
     X_ji =X_ij∈Herm{∑_i <j[X_ij]: ∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ j| ⊗ X_ij≥ 0 },

and one can verify that it satisfies several basic requirements for a multivariate fidelity listed in <ref>, by following our proof approaches for <ref>.


  * Are there information-theoretic tasks that provide operational interpretations to quantum generalizations of multivariate fidelity, including average pairwise z-fidelities and multivariate SDP fidelity? 


  * Is √(2(1-F_z(ρ,σ))) a distance measure for z ∈ (1/2,1)∪(1,∞)?  Knowing this in turn would provide uniform continuity bounds for average pairwise z-fidelities, as  <ref> does for the special cases z ∈{1/2,1}. 


  * Another question that we pose is related to multivariate generalizations of the Holevo fidelity. It was shown in <cit.> that the Holevo fidelity has the following SDP characterization:

    F_H(ρ,σ) = sup_C≥ 0{⟨Γ | C |Γ⟩: [   ρ⊗ I      C;      C I ⊗σ^T ]≥ 0},

where |Γ⟩∑_i |i⟩|i⟩.
This characterization follows because an optimal choice for C is the matrix geometric mean of the commuting operators ρ⊗ I and I ⊗σ^T (see, e.g., <cit.>), which is simply (ρ⊗σ^T)^1/2, and because ⟨Γ | (ρ⊗σ^T)^1/2 |Γ⟩ = Tr[ρ^1/2σ^1/2]. The dual of the SDP in (<ref>) is given by

    inf_Y,Z≥0, 
     W∈ℒ{Tr[Y(  ρ⊗ I)  ]+Tr[Z(
    I⊗σ^T)  ]:
    W+W^†≥|Γ⟩⟨Γ|,
    [   Y   W; W^†   Z ]≥0
    } .

However, it is unclear how to generalize the constructions in (<ref>) and (<ref>) to multiple states, given that it makes use of properties specific to bipartite systems. Interestingly, the constructions in (<ref>) and (<ref>) do not encounter this difficulty. 


  * Is there a possibility of
finding analytical or alternative variational expressions for minimal and maximal extensions of multivariate classical fidelity in <ref> and <ref>, respectively?






§ ACKNOWLEDGEMENTS


We thank Komal Malik for helpful discussions. TN  acknowledges support from the NSF under grant no. 2329662. HKM  acknowledges support from the NSF under grant no. 2304816 and AFRL under agreement no. FA8750-23-2-0031.
FL acknowledges support from the Department of Mathematics at UIUC for a research visit to Cornell University, as well as the hospitality of Mark Wilde's research group during that stay.
MMW acknowledges support from the NSF under grants 2329662, 2315398, 1907615, 2304816. 

This material is based on research
sponsored by Air Force Research Laboratory under agreement number
FA8750-23-2-0031. The U.S. Government is authorized to reproduce and
distribute reprints for Governmental purposes notwithstanding any copyright
notation thereon. The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of Air Force Research
Laboratory or the U.S. Government. 

alpha
 





§ PROOFS




In the following appendices, we present technical proofs of the results that are stated in the main text of the paper. 



 §.§ Proof of Equation (<ref>) (Log-Euclidean fidelity as limit of z-fidelity)




For ε>0, define ρ(ε)ρ+ε I and σ(ε)σ+ε I.
Observe that 

    F_z(ρ,σ) = lim_ε→ 0^+ F_z(ρ(ε),σ(ε)) = inf_ε >0 F_z(ρ(ε),σ(ε)),

which follows from the operator monotonicity of the function x^p for p∈(0,1] and monotonicity of (·) →Tr[(·)^z] for all z≥ 1/2. Now recalling that the z-fidelities are antimonotone for z≥ 1/2 <cit.> and continuous in z, we conclude that

    lim_z→∞ F_z(ρ,σ)     = 
        inf_z ≥1/2 F_z(ρ,σ) 
        = inf_z ≥1/2inf_ε >0 F_z(ρ(ε),σ(ε)) 
        = inf_ε >0inf_z ≥1/2  F_z(ρ(ε),σ(ε)) 
        = inf_ε >0lim_z →∞  F_z(ρ(ε),σ(ε)) 
        = inf_ε >0Tr[exp(1/2 (lnρ(ε) + lnσ(ε))].

The last equality follows from an application of the Lie–Trotter product formula. Similar observations were made in <cit.> using the fact that for operators A and B, lim_z →∞exp(A/z) exp(B/z) = exp(A+B) with the choice A= 1/2lnρ(ε) and B=1/2lnσ(ε).

The last equality in (<ref>) follows from operator monotonicity of the logarithm function and monotonicity of (·) →Tr[exp(·)].



 §.§ Proof of Proposition <ref> (Uniform continuity of Matusita fidelity)




Consider that

    | F_r(p_1,…,p_r)-F_r(q_1,…,q_r)|
         =|∑_x(  p_1(x)⋯ p_r(x))  ^1/r
    -∑_x(  q_1(x)⋯ q_r(x))  ^1/r|
         =|∑_x[  (  p_1(x)⋯ p_r(x))
    ^1/r-(  q_1(x)⋯ q_r(x))  ^1/r]
    |
         =|∑_i=1^r∑_x[  (  p_1(x)⋯
    p_i-1(x))  ^1/r(  p_i(x)^1/r-q_i
    (x)^1/r)  (  q_i+1(x)⋯ q_r(x))  ^1/r]  |
       ≤∑_i=1^r|∑_x[  (  p_1(x)⋯
    p_i-1(x))  ^1/r(  p_i(x)^1/r-q_i
    (x)^1/r)  (  q_i+1(x)⋯ q_r(x))  ^1/r]  |
       ≤∑_i=1^r‖(  p_1⋯ p_i-1q_i+1⋯
    q_r)  ^1/r‖ _r/r-1‖ p_i
    ^1/r-q_i^1/r‖ _r
         =∑_i=1^rF_r-1(p_1,…,p_i-1,q_i+1,…,q_r
    )^r-1/r‖ p_i^1/r-q_i^1/r
    ‖ _r
       ≤∑_i=1^r‖ p_i^1/r-q_i^1/r
    ‖ _r
         =r(  1/r∑_i=1^r(  ‖ p_i^1/r
    -q_i^1/r‖ _r^r)  ^1/r)  
       ≤ r(  1/r∑_i=1^r‖ p_i^1/r
    -q_i^1/r‖ _r^r)  ^1/r
       ≤ r(  1/r∑_i=1^r‖√(p_i)-√(q_i)‖
    _2^2)  ^1/r
       ≤ r( ε)  ^1/r.

The third equality follows from rewriting as a telescoping sum. The first
inequality follows from the triangle inequality. The second inequality follows
from Hölder's inequality with parameters r,s≥1 satisfying 1/r+1/s=1, so that s=r/r-1. The fourth equality follows from
applying definitions, and the third inequality follows because F_r≤1 for
all probability distributions and for every integer r≥2.
The fourth inequality follows from the concavity of function x^1/r for r≥ 2 for all x ≥ 0. To see the penultimate
inequality, consider 

that the following inequality holds for all distributions p and q by using <cit.>:

    ‖ p^1/r-q^1/r‖ _r^r=∑
    _x| p(x)^1/r-q(x)^1/r| ^r≤∑_x( √(p(x))-√(q(x)))^2 =‖√(p)-√(q)‖ _2^2.

Finally, the last inequality follows from the the assumption 1/r(  ∑_i=1^r[ d_H (p_i, q_i) ]^2 )  ≤ε, along with (<ref>).







 §.§ Proof of Proposition <ref> (Inequalities relating classical average k-wise fidelities)




It suffices to prove the following inequality:

    F_k-1,r(ρ_1,…,ρ_r)≥ F_k,r(ρ_1,…,ρ_r),

for all k∈{  3,…,r}. So we prove this one, using a generalization of the approach used in the proof of <Ref>.
Let S_k-1(i_1,…,i_k) denote all size-(k-1) subsets of
i_1,…,i_k, of which there are k of them. For example, if r=7,
k=4, and i_1=2, i_2=4, i_3=5, i_4=7, then

    S_k-1(i_1,i_2,i_3,i_4)=S_3(2,4,5,7)={{
    2,4,5}  ,{  2,4,7}  ,{  2,5,7}
    ,{4,5,7}}  .

Note that each symbol i_j appears k-1 times in each subset of
S_k-1(i_1,…,i_k). Then consider that

    F_k(ρ_i_1,…,ρ_i_k)      =∑_x(  ρ_i_1(x)⋯ρ_i_k(x))  ^1/k
         =∑_x(  ∏_t∈ S_k-1(i_1,…,i_k)[
    ρ_t(1)(x)⋯ρ_t(k-1)(x)]  ^1/k-1)  ^1/k
       ≤∑_x1/k∑_t∈ S_k-1(i_1,…,i_k)[
    ρ_t(1)(x)⋯ρ_t(k-1)(x)]  ^1/k-1
         =1/k∑_t∈ S_k-1(i_1,…,i_k)∑_x[
    ρ_t(1)(x)⋯ρ_t(k-1)(x)]  ^1/k-1
         =1/k∑_t∈ S_k-1(i_1,…,i_k)F_k-1(ρ_t(1)
    ,…,ρ_t(k-1)),

where the inequality follows as a consequence of the inequality of arithmetic
and geometric means. Now consider that

    F_k,r(ρ_1, …, ρ_r)     =  1/rk∑_i_1<⋯<i_kF_k(ρ_i_1
    ,…,ρ_i_k)
       ≤1/rk∑_i_1<⋯<i_k1/k∑_t∈
    S_k-1(i_1,…,i_k)F_k-1(ρ_t(1),…,ρ_t(k-1)) 
         =k-1!r-k!/r!∑_i_1<⋯<i_k∑_t∈ S_k-1
    (i_1,…,i_k)F_k-1(ρ_t(1),…,ρ_t(k-1))
         =k-1!r-k!/r!(  r-k+1)  ∑_i_1<⋯<i_k-1
    F_k-1(ρ_i_1,…,ρ_i_k-1)
         =1/rk-1∑_i_1<⋯<i_k-1F_k-1(ρ_i_1
    ,…,ρ_i_k-1)
        = F_k-1,r(ρ_1, …, ρ_r).

The second equality follows because the sum in the third line includes each
distinct term r-k+1 times. Indeed, the sum in the third line arises by
picking size-k subsets from [  r], and from each of these
size-k subsets, picking size-(  k-1) subsets, thus leading to
a total number of subsets given by rkkk-1. However,
picking subsets in this way leads to rk-1 distinct subsets, each
occurring with multiplicity rkkk-1/rk-1
=r-k+1.



 §.§ Proof of Proposition <ref> (Uniform continuity of average pairwise fidelities)





Define the shorthand 𝒮_ρ (ρ_1,…, ρ_r) and 𝒮_σ (σ_1,…, σ_r).
Consider that

    F_U(𝒮_ρ)-F_U(𝒮_σ)      =2/r(
    r-1)  ∑_i<jF(ρ_i,ρ_j)-2/r(  r-1)
    ∑_i<jF(σ_i,σ_j)
         =2/r(  r-1)  ∑_i<j[  F(ρ_i,ρ
    _j)-F(σ_i,σ_j)].

We have

    2[  F(ρ_i,ρ_j)-F(σ_i,σ_j)]
    
         =2[  1-F(σ_i,σ_j)-(  1-F(ρ_i,ρ
    _j))  ]  
         =2[  √(1-F(σ_i,σ_j))√(1-F(σ
    _i,σ_j))-√(1-F(ρ_i,ρ_j))√(1-F(ρ_i
    ,ρ_j))]  
         =d_B(σ_i,σ_j)d_B(σ_i,σ_j)-d_B(ρ
    _i,ρ_j)d_B(ρ_i,ρ_j)
         =d_B(σ_i,σ_j)d_B(σ_i,σ_j)-d_B(σ
    _i,σ_j)d_B(ρ_i,ρ_j)
           +d_B(σ_i,σ_j)d_B(ρ_i,ρ_j)-d_B(ρ
    _i,ρ_j)d_B(ρ_i,ρ_j)
         =[  d_B(σ_i,σ_j)+d_B(ρ_i,ρ_j)]
    [  d_B(σ_i,σ_j)-d_B(ρ_i,ρ_j)]  
       ≤[  d_B(σ_i,σ_j)+d_B(ρ_i,ρ_j)]×
           [  d_B(σ_i,ρ_i)+d_B(ρ_i,ρ_j)+d_B(ρ
    _j,σ_j)-d_B(ρ_i,ρ_j)]  
         =[  d_B(σ_i,σ_j)+d_B(ρ_i,ρ_j)]
    [  d_B(σ_i,ρ_i)+d_B(ρ_j,σ_j)]  
       ≤ 2√(2)[  d_B(σ_i,ρ_i)+d_B(ρ_j,σ_j)] 
        = 2√(2)[  d_B(ρ_i,σ_i)+d_B(ρ_j,σ_j)],

where the first inequality follows from the triangular inequality of the Bures distance and the second inequality because d_B(ω,τ) ≤√(2) for all quantum states ω and τ.
This implies that

    F(ρ_i,ρ_j)-F(σ_i,σ_j)≤√(2)[  d_B(ρ_i,σ_i)+d_B(ρ_j,σ_j)].

Substituting (<ref>) into (<ref>) and using the fact that

    ∑_i<j[  d_B(ρ_i,σ_i)+d_B(ρ_j,σ_j)] = (r-1) ∑_i=1^r d_B(ρ_i,σ_i)

gives

    F_U(𝒮_ρ)-F_U(𝒮_σ)  
        ≤√(2)(2/r(  r-1)  ) ∑_i<j[  d_B(ρ_i,σ_i)+d_B(ρ_j,σ_j)] 
        = 2√(2)(1/r∑_i=1^r d_B(ρ_i,σ_i)) 
       ≤ 2√(2)ε,

 where the last inequality follows from the assumption that 1/r∑_i=1^r d_B(ρ_i,σ_i)≤ε.
 This concludes the proof of the inequality in (<ref>).

A similar proof works for the inequality in 
(<ref>) by replacing all instances of Uhlmann fidelity and Bures distance  with Holevo fidelity and quantum Hellinger distance, respectively.




 §.§ Proof of Proposition <ref> (Dual of multivariate SDP fidelity)
 


Recall that a standard primal SDP and its dual are given as follows (cf. <cit.>): 

    inf_Y≥0{Tr[BY]:Φ^†(Y)≥ A}, 
       sup_X≥0{Tr[AX]:Φ(X)≤ B},

where A and B are Hermitian matrices and Φ is a Hermiticity-preserving superoperator. It is always true that ≥, and  strong duality is said to hold in the case of equality. A sufficient condition for strong duality to hold is that there exists a feasible point for the  and a strictly feasible point for the ; i.e., the latter meaning that there exists a Y such that Y>0 and Φ^†(Y) > A. This is known as Slater's condition.
In what follows, we identify the A, B, and Φ^† from the multivariate SDP formulation (<ref>) to derive its dual, and we then argue that strong duality holds using Slater's theorem.

Comparing the primal SDP (<ref>) to the multivariate SDP fidelity (<ref>) without considering the normalization term r(r-1), 
we make the following choices of A, B, and Φ^†: given Y=∑_i,j=1^r |i⟩⟨ j| ⊗ Y_ij, define

    A    ∑_i,j∈ [r]: i ≠ j |i⟩⟨ j| ⊗ I , 
    
        B    ∑_i=1^r |i⟩⟨ i| ⊗ρ_i, 
    Φ^†(Y)    ∑_i=1^r |i⟩⟨ i| ⊗ Y_ii.

Thus, given any Hermitian matrix X=∑_i,j=1^r |i⟩⟨ j| ⊗ X_ij and using the relation [ Y Φ(X)  ] = [ Φ^†(Y) X   ], we obtain

    Φ(X)= ∑_i=1^r |i⟩⟨ i| ⊗ X_ii.

Also, we have 

    [AX]= ∑_i ≠ j[X_ij]= 2 ∑_i <jℜ[ [X_ij]].

The  condition Φ(X) ≤ B is given by

    ∑_i=1^r |i⟩⟨ i| ⊗ X_ii≤∑_i=1^r |i⟩⟨ i| ⊗ρ_i,

which, by adding ∑_i ≠ j |i⟩⟨ i| ⊗ X_ij to both sides,  evaluates to

    0 ≤  X≤∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ i| ⊗ X_ij.

Using the relations (<ref>) and (<ref>) in the  form and the fact that X is Hermitian gives the desired dual formulation (<ref>).


We use Slater's condition to show that strong duality holds, which requires finding a strictly feasible point in the SDP formulation (<ref>) and a feasible point in the dual formulation (<ref>). The positive definite matrix Y=∑_i=1^r |i⟩⟨ i| ⊗ r I is a strictly feasible point because Φ^†(Y)-A>0. To see this, consider that

    Φ^†(Y)-A    = ∑_i=1^r |i⟩⟨ i| ⊗ rI - ∑_i ≠ j |i⟩⟨ j| ⊗ I  
       = ∑_i=1^r (r+1) |i⟩⟨ i| ⊗ I - ∑_i, j=1^r |i⟩⟨ j| ⊗ I 
       =((r+1) I - uu^T ) ⊗ I,

where u is the r-column vector of all ones. The matrix (r+1) I - uu^T is positive definite because its (distinct) eigenvalues are 1 and r+1. We thus conclude that Φ^†(Y)-A>0. A feasible point for (<ref>) consists of simply picking X_ij= 0 for all i,j. 



 §.§ Proof of Theorem <ref> (Uniform continuity of K^⋆-representation)





Recall by definition that 

    F_K^⋆(ρ_1, …, ρ_r)=1/r-1sup_K≥0{[                 ⟨ψ^ρ|K⊗ I|ψ^ρ⟩-1:; K=I_r⊗ I_d+∑_i≠ j|i⟩⟨ j|⊗ K_ij≥0, ]}  ,

where |ψ^ρ⟩=1/√(r)∑_i=1^r|i⟩|ϕ^ρ_i
⟩ and |ϕ^ρ_i⟩ is an arbitrary purification of ρ_i.
Fix

    K=I_r⊗ I_d+∑_i≠ j|i⟩⟨ j|⊗ K_ij
    ≥0.

Also fix

    |ψ^ρ⟩    =1/√(r)∑_i=1^r|i⟩|ϕ
    ^ρ_i⟩,
    
    |ψ^σ⟩    =1/√(r)∑_i=1^r|i⟩
    |ϕ^σ_i⟩,

where these purifications are those that achieve the (root) fidelity, so that
⟨ϕ^σ_i|ϕ^ρ_i⟩=F(ρ_i,σ_i).
Hölder's inequality implies that

    |⟨ψ^ρ|K⊗ I|ψ^ρ⟩-⟨ψ^σ|K⊗ I|ψ^σ⟩|
        =|Tr[(  |ψ^ρ⟩⟨ψ^ρ|-|ψ^σ⟩⟨ψ^σ|)  K⊗ I]|
       ≤‖ |ψ^ρ⟩⟨ψ^ρ|-|ψ^σ
    ⟩⟨ψ^σ|‖ _1‖ K⊗ I‖
    _∞
        =‖ |ψ^ρ⟩⟨ψ^ρ|-|ψ^σ
    ⟩⟨ψ^σ|‖ _1‖ K‖ _∞
       ≤‖ |ψ^ρ⟩⟨ψ^ρ|-|ψ^σ
    ⟩⟨ψ^σ|‖ _1‖ r(  I_r⊗
    I_d)  ‖ _∞
        =r‖ |ψ^ρ⟩⟨ψ^ρ|-|ψ^σ
    ⟩⟨ψ^σ|‖ _1
        =r√(1-|⟨ψ^σ|ψ^ρ⟩|
    ^2)
        =r√(1-|1/r∑_i=1^r⟨ϕ^σ_i
    |ϕ^ρ_i⟩| ^2)
        =r√(1-(  1/r∑_i=1^rF(ρ_i,σ_i))
    ^2)
       ≤ r√(1-(  1-ε)  ^2)
        =r√(ε(  2-ε)  ).

The second to last inequality follows because 

    (  I_r⊗ I_d)   = ∑
    _x(  |x⟩⟨ x|⊗ I)  K(  |x⟩⟨
    x|⊗ I)  
        =1/r∑_k=0^r-1(  Z(k)⊗ I)  K(
    Z(k)⊗ I)  ^†
       ≥1/rK,

where Z(k)∑_ℓ=0^r-1e^2π iℓ k/r|ℓ⟩⟨ℓ| is
the Heisenberg phase shift operator. The last inequality follows from
antimonotonicity of the function √(1-x^2) on the interval x∈[
0,1] and the fact that 1/r∑_i=1^rF(ρ_i
,σ_i)≥1-ε by assumption (see (<ref>)).
From (<ref>) we thus get

    1/r-1(  ⟨ψ^ρ|K⊗ I|ψ^ρ⟩
    -1)    
       ≤1/r-1(  ⟨ψ^σ|K⊗
    I|ψ^σ⟩-1)  +r/r-1√(ε(
    2-ε)  )
       ≤ F_K^⋆(σ_1, …, σ_r)+r/r-1√(ε(
    2-ε)  ),

where the last inequality follows from the expression in
(<ref>). Since the inequality holds for all K satisfying
the condition, we conclude that

    F_K^⋆(ρ_1, …, ρ_r)≤ F_K^⋆(σ_1, …, σ_r)+r/r-1√(ε(  2-ε)  ),

after applying (<ref>) once again. 
The other inequality
follows from the same reasoning, but instead swapping ρ_1, …, ρ_r
and σ_1, …, σ_r.




 §.§ Monotonicity of multivariate SDP fidelity


For every tuple of states ρ_1, … , ρ_r (not necessarily invertible) and for all ε > 0, all states of the following form are invertible:

    ρ_i^(ε)11+ε(ρ_i +  ε/d I ).

As stated below, the multivariate SDP fidelity F_SDP(ρ_1^(ε), …, ρ_r^(ε)) is monotonic as a function of ε, up to a multiplicative factor.

    Let ρ_1,…, ρ_r be  quantum states, and fix ε_1 and ε_2 such that 0≤ε_1 ≤ε_2. Then 
    
    (1+ε_1)F_SDP(ρ_1^(ε_1), …, ρ_r^(ε_1))  ≤ (1+ε_2)F_SDP(ρ_1^(ε_2), …, ρ_r^(ε_2)).




     Let (Y_i)_i=1^r be a candidate tuple for the SDP for F_SDP(ρ_1^(ε_2), …, ρ_r^(ε_2)).
Then

    1/r(r-1)∑_i=1^n [ Y_i ρ_i^(ε_2)] 
           = 1/r(r-1)(1+ε_2)∑_i=1^n [ Y_i (ρ_i + ε_2 I/d) ]  
       ≥1/r(r-1)(1+ε_2) ∑_i=1^n [ Y_i (ρ_i + ε_1 I/d) ] 
        =(1+ ε_1)/r(r-1)(1+ ε_2)∑_i=1^n [ Y_i ρ_i^(ε_1)] 
       ≥(1+ ε_1)/(1+ ε_2)  F(ρ_1^(ε_1), …, ρ_r^(ε_1)).

The first inequality holds because ε_1 ≤ε_2, and the last inequality follows from  <ref>. 
By optimizing the left-hand side of the above over all (Y_i)_i=1^r satisfying the constraints in the SDP for F_SDP(ρ_1^(ε_2), …, ρ_r^(ε_2)), and then rearranging the terms, we arrive at the desired inequality (<ref>).




 §.§ Proof of Proposition <ref> (Alternative formulation of multivariate SDP fidelity)
 



The  proof of <Ref> is structured in four steps as follows: 
In the first step we show an equivalent formulation for the K^⋆-representation in (<ref>).
This helps us in the second step to prove that F_K^⋆(ρ_1, …, ρ_r) ≤ F_SDP(ρ_1, …, ρ_r) for general states. 
In the third step, we show that F_K^⋆(ρ_1, …, ρ_r) = F_SDP(ρ_1, …, ρ_r) for invertible states. 
Finally, in the fourth step, with the assistance of <ref> and <ref> among other things, we establish the equivalence (<ref>) for the general setting. 


Step 1: Equivalent formulation for K^⋆-representation


Define 

    F_Int(ρ_1, …, ρ_r)  
            
    2/r(r-1)sup_K_ji=K_ij^†{∑_i<jℜ[ [ ρ_i^1/2K_ijρ_j^1/2]]: I_r ⊗ I_d+ ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij≥ 0 } .
 
We first show that F_Int(ρ_1, …, ρ_r) = F_K^⋆(ρ_1, …, ρ_r). 
For all i∈ [r], define the canonical purification of ρ_i as 

    | ϕ^ρ_i⟩ =(ρ_i^1/2⊗ I_d) | Γ⟩,

where |Γ⟩∑_i=1^d |i⟩⊗ |i⟩ is the unnormalized maximally entangled vector and d denotes the dimension of the Hilbert space on which each ρ_i is defined.
With this, we can write each term in the objective function as 

    ℜ[ [ ρ_i^1/2K_ijρ_j^1/2]]= ℜ[ ⟨ϕ^ρ_i| K_ij⊗ I_d | ϕ^ρ_j⟩].

Recalling the definition of K from (<ref>)

    K  I_r ⊗ I_d+ ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij ,

and considering that K≥ 0,
we also obtain that

    1/√(r)( ∑_k=1^r ⟨ k| ⟨ϕ^ρ_k| )(   I_r ⊗ I_d ⊗ I_d+ ∑_i ≠ j=1^r  |i ⟩⟨ j| ⊗ K_ij⊗ I_d) 1/√(r)( ∑_ℓ=1^r |ℓ⟩  |ϕ^ρ_ℓ⟩) ≥ 0

and 

    1/√(r)( ∑_k=1^r ⟨ k| ⟨ϕ^ρ_k| )(   I_r ⊗ I_d ⊗ I_d+ ∑_i ≠ j=1^r |i ⟩⟨ j| ⊗ K_ij⊗ I_d) 1/√(r)( ∑_ℓ=1^r |ℓ⟩  |ϕ^ρ_ℓ⟩)  
       = 1 + 1/r(∑_i ≠ j=1^r ⟨ϕ^ρ_i| K_ij⊗ I_d | ϕ^ρ_j⟩) 
       = 1 + 2/r(∑_i < jℜ[⟨ϕ^ρ_i| K_ij⊗ I_d | ϕ^ρ_j⟩]  ).

The above evaluates to the following equality: 

    ∑_i < jℜ[⟨ϕ^ρ_i| K_ij⊗ I_d | ϕ^ρ_j⟩] = r/2( ⟨ψ | K ⊗ I_d | ψ⟩ -1),

where 

    | ψ⟩ = 1/√(r)∑_i=1^r |i ⟩ | ϕ^ρ_i⟩.

Then, for the canonical purification of each ρ_i, we deduce that the objective function of the optimizations in both F_Int(ρ_1, …, ρ_r) and F_K^⋆(ρ_1, …, ρ_r) are equivalent. 

For any other purification, recall that all purifications are related by an isometry. 
Let 

    |Φ^ρ_i⟩ = ( U_i ⊗ I_d )|ϕ^ρ_i⟩,

where U_i is an isometry for all i ∈ [r].
Then, the objective function inside the SDP (i.e., without considering the constants) evaluates to 

    ∑_i < jℜ[⟨Φ^ρ_i| K_ij⊗ I_d | Φ^ρ_j⟩] = ∑_i < jℜ[⟨ϕ^ρ_i| U_i^† K_ij U_j⊗ I_d | ϕ^ρ_j⟩].

Also note that the matrix formed by replacing K_ij in (<ref>) with U_i^† K_ij U_j also satisfies the constraint due to: 

    ∑_i=1^r |i ⟩⟨ i| ⊗ U_i^†(I_r ⊗ I_d+ ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij) ∑_k=1^r |k ⟩⟨ k| ⊗ U_k ≥ 0,

which follows from the fact that U_i^† U_i =I.
Then, the previous arguments in the canonical purification follow for all purifications of the considered states. We then conclude that 

    F_K^⋆(ρ_1, …, ρ_r)=  F_Int(ρ_1, …, ρ_r)  
       =
     2/r(r-1)sup_K_ji=K_ij^†{∑_i<jℜ[ [ ρ_i^1/2K_ijρ_j^1/2]]: I_r ⊗ I_d+ ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij≥ 0 }.




Step 2: F_K^⋆(ρ_1,…, ρ_r)≤ F_SDP(ρ_1, …, ρ_r) for general states 


Consider the constraint: I_r ⊗ I_d+ ∑_i ≠ j |i⟩⟨ j| ⊗ K_ij≥ 0 in (<ref>). 
By left and right multiplying the above inequality by 
∑_i=1^r |i⟩⟨ i| ⊗ρ_i^1/2, we obtain 

    ∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ j| ⊗ρ_i ^1/2 K_ijρ_j^1/2≥ 0.

Then referring to the dual SDP of multivariate SDP fidelity in <ref>, we see that a feasible point in the optimization (<ref>) is given by X_ij= ρ_i ^1/2 K_ijρ_j^1/2 for i≠ j.
With that we have:

    2/r(r-1)∑_i<jℜ[ [ ρ_i^1/2K_ijρ_j^1/2]] ≤ F_SDP(ρ_1,…, ρ_r).

Then, optimizing over all K satisfying the required constraint, we arrive at the claim:

    F_K^⋆(ρ_1,…,ρ_r) ≤ F_SDP(ρ_1,…, ρ_r).


Step 3: F_K^⋆(ρ_1,…,ρ_r) =F_SDP(ρ_1,…, ρ_r) for invertible states


Consider an arbitrary feasible point in the dual SDP (<ref>) of multivariate SDP fidelity:

    ∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ j| ⊗ X_ij≥ 0.

Using the assumption that ρ_i  is invertible for all i ∈ [ r], by left and right multiplying the above inequality by the positive definite matrix
∑_i=1^r |i⟩⟨ i| ⊗ρ_i^-1/2, we obtain 

    ∑_i=1^r |i⟩⟨ i| ⊗ I_d + ∑_i ≠ j |i⟩⟨ j| ⊗ρ_i^-1/2 X_ijρ_j^-1/2≥ 0.

This implies that the matrices K_ij=ρ_i^-1/2 X_ijρ_j^-1/2 for i ≠ j form a feasible point for the optimization in (<ref>).
With this choice of a feasible point, we get 

    2/r(r-1)∑_i<jℜ[ [ X_ij]] ≤ F_K^⋆(ρ_1,…, ρ_r).

Since the above inequality holds for an arbitrary feasible point in (<ref>), it follows by the definition that

    F_SDP(ρ_1,…, ρ_r) ≤   F_K^⋆(ρ_1,…,ρ_r).

Combining (<ref>) with (<ref>), we get

    F_SDP(ρ_1,…, ρ_r) =   F_K^⋆(ρ_1,…,ρ_r)

for invertibles states.

Step 4: Extending to non-invertible states


For all i∈ [r] and ε > 0,

    ρ_i^(ε)11+ε(ρ_i +  ε/d I )

is an invertible state. Then, we have that 

    1/2ρ_i - ρ_i^(ε)_1 = ε/2(1+ε)ρ_i - I/d_1 ≤ε/1+εε',

leading to F(ρ_i, ρ_i^(ε)) ≥ 1- ε'. 
Then by <ref>, we get

    | F_K^⋆(ρ_1, …, ρ_r) -F_K^⋆( ρ_1^(ε), …, ρ_r^(ε))   | ≤r/r-1√(ε'(2-ε')).


Using <ref> with the choice ε_1=0 and ε_2=ε >0, we have that 

    F_SDP(ρ_1, …, ρ_r )     ≤ (1+ε)F_SDP(ρ_1^(ε), …, ρ_r^(ε))
       ≤ F_SDP(ρ_1^(ε), …, ρ_r^(ε)) + ε,

where the last inequality holds because F_SDP(ρ_1^(ε), …, ρ_r^(ε)) ≤ 1.
Then consider that

    F_SDP(ρ_1^(ε), …, ρ_r^(ε))   = F_K^⋆(ρ_1^(ε), …, ρ_r^(ε))
       ≤ F_K^⋆(ρ_1, …, ρ_r) +r/r-1√(ε'(2-ε'))
       ≤ F_SDP(ρ_1, …, ρ_r ) + r/r-1√(ε'(2-ε')),

where the first equality follows from (<ref>); the second inequality from (<ref>); and finally the third inequality from (<ref>) for general states. 

Combining (<ref>) and (<ref>), we arrive at 

    | F_SDP(ρ_1, …, ρ_r) -F_SDP( ρ_1^(ε), …, ρ_r^(ε))   |    ≤max{ε, r/r-1√(ε'(2-ε'))}
       ≤ 2 √(ε'(2-ε'))
        = 2 √(ε/1+ε(2-ε/1+ε)).


With the established machinery, consider 

    | F_SDP(ρ_1, …, ρ_r) -  F_K^⋆(ρ_1, …, ρ_r) |     ≤| F_SDP(ρ_1, …, ρ_r) -F_SDP( ρ_1^(ε), …, ρ_r^(ε))   |     
            + |F_SDP( ρ_1^(ε), …, ρ_r^(ε))  -F_K^⋆( ρ_1^(ε), …, ρ_r^(ε))   | 
             + | F_K^⋆(ρ_1, …, ρ_r) -F_K^⋆( ρ_1^(ε), …, ρ_r^(ε))   |
       ≤  2 √(ε/1+ε(2-ε/1+ε)) + 0 + r/r-1√(ε/1+ε(2-ε/1+ε)),

where the first inequality follows from triangular inequality; the second inequality from using (<ref>) and (<ref>) along with the fact that F_SDP( ρ_1^(ε), …, ρ_r^(ε)) =F_K^⋆( ρ_1^(ε), …, ρ_r^(ε)) due to (<ref>).

Finally, by taking limit ε→ 0 in the above inequality, we conclude that 

    | F_SDP(ρ_1, …, ρ_r) -  F_K^⋆(ρ_1, …, ρ_r) | =0,

which completes the proof for the general case.






 §.§ Proof of Theorem <ref> (Properties of multivariate SDP fidelity)
 



Note that multivariate SDP fidelity satisfies reduction to classical average pairwise fidelity, faithfulness, and orthogonality by <ref> and the fact that average pairwise fidelities satisfy these properties as stated in <ref>.




Data processing:
Let : ℒ(_A) →ℒ(_B) be a channel, where the dimension of _A and _B be d_A and d_B, respectively. 
    For i ∈ [r], let Y_i  be such that 
    ∑_i=1^r |i⟩⟨ i| ⊗ Y_i ≥∑_i ≠ j |i⟩⟨ j| ⊗ I_d_B. 
    Consider that
    
    1/r(r-1)∑_i=1^r [ Y_i (ρ_i) ] = 1/r(r-1)∑_i=1^r [ ^†(Y_i) ρ_i ].

Then, due to  being completely positive, along with  being trace-preserving (so that ^†(I_d_B) =I_d_A),  we have that

    0 ≤∑_i=1^r |i⟩⟨ i| ⊗^†(Y_i) - ∑_i ≠ j |i⟩⟨ j| ⊗^†(I_d_B) = ∑_i=1^r |i⟩⟨ i| ⊗^†(Y_i) - ∑_i ≠ j |i⟩⟨ j| ⊗ I_d_A   .

With that, the tuple (^†(Y_i) )_i=1^r is a candidate for F_SDP(ρ_1, …, ρ_r). Thus, 

    F_SDP(ρ_1, …, ρ_r) ≤1/r(r-1)∑_i=1^r [ Y_i (ρ_i) ].

This holds for all (Y_i)_i=1^r satisfying ∑_i=1^r |i⟩⟨ i| ⊗ Y_i ≥∑_i ≠ j |i⟩⟨ j| ⊗ I_d_A. Finally, taking the infimum over all such (Y_i)_i=1^r concludes the proof.


    We note that the proof for data processing given above holds not just for completely positive, trace-preserving maps, but more generally for r-positive, trace-preserving maps.




Symmetry: This follows directly from the definition of the SDP fidelity.




Direct-sum property:
    Let ( Y_i^x)_i=1^r for all x ∈ satisfy 
    ∑_i=1^r |i⟩⟨ i| ⊗ Y^x_i - ∑_i ≠ j |i⟩⟨ j| ⊗ I ≥ 0.
    Consider that
    
    1/r (r-1)∑_x ∈ p(x)  ∑_i=1^r [ Y_i^x ρ_i^x] 
        =  1/r(r-1)∑_i=1^r  [∑_x ∈ p(x)  Y_i^x ρ_i^x ]  
       = 1/r(r-1)∑_i=1^r  [(∑_x' ∈ |x' ⟩⟨ x'| ⊗ Y_i^x')  ∑_x ∈ p(x)   |x ⟩⟨ x | ⊗ρ_i^x ].

    From the conditions satisfied by (Y_i^x)_i=1^r, we also get 
    
    ∑_i=1^r |i⟩⟨ i|  ⊗ |x ⟩⟨ x| ⊗ Y^x_i  - ∑_i ≠ j |i⟩⟨ j| ⊗ |x ⟩⟨ x|⊗ I_d  ≥ 0.

    By summing over x ∈, we get
     
    ∑_i=1^r |i⟩⟨ i| ⊗∑_x ∈ |x ⟩⟨ x| ⊗ Y^x_i   - ∑_i ≠ j |i⟩⟨ j| ⊗  I_|𝒳|⊗ I_d ≥ 0.

    Then, ( ∑_x ∈ |x ⟩⟨ x| ⊗ Y^x_i )_i=1^r forms a feasible point for the optimization in F_SDP( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x).
    Using the relation (<ref>), this leads to

    F_SDP( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x) 
    ≤1/r (r-1)∑_x ∈ p(x)  ∑_i=1^r [ Y_i^x ρ_i^x].

Since the last inequality holds for all (Y_i^x)_i=1^r and for all x ∈, this also holds for  the optimal ones achieving F(ρ_1^x, …, ρ_r^x).
We thus get 
 
    F_SDP( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x) ≤∑_x ∈ p(x)  F_SDP(ρ_1^x, …, ρ_r^x).


To prove the reverse direction, 
let ( Z_i)_i=1^r satisfy 
    ∑_i=1^r |i⟩⟨ i| ⊗ Z_i - ∑_i ≠ j |i⟩⟨ j|  ⊗ I_|𝒳|⊗ I_d ≥ 0. Then
   
    1/r(r-1)∑_i=1^r  [ Z_i  ∑_x ∈ p(x)   |x ⟩⟨ x | ⊗ρ_i^x ] 
     = ∑_x ∈ p(x)  1/r(r-1)∑_i=1^r  [ (⟨ x| ⊗ I_d  ) Z_i ( | x⟩⊗ I_d ) ρ_i^x ].

From the conditions satisfied by  ( Z_i)_i=1^r, we also get for all x ∈ that
    
    ∑_i=1^r |i⟩⟨ i|  ⊗ (⟨ x| ⊗ I_d  ) Z_i ( | x⟩⊗ I_d )   - ∑_i ≠ j |i⟩⟨ j| ⊗⟨ x|x⟩⊗ I_d  ≥ 0 
    ⟹    ∑_i=1^r |i⟩⟨ i|  ⊗ (⟨ x| ⊗ I_d  ) Z_i ( | x⟩⊗ I_d )   - ∑_i ≠ j |i⟩⟨ j| ⊗   I_d  ≥ 0.

Then ( (⟨ x| ⊗ I_d  ) Z_i ( | x⟩⊗ I_d ))_i=1^r is a candidate for F_SDP(ρ_1^x, …, ρ_r^x), leading to 

    F_SDP(ρ_1^x, …, ρ_r^x) ≤1/r(r-1)∑_i=1^r  [  (⟨ x| ⊗ I_d  ) Z_i ( | x⟩⊗ I_d )ρ_i^x ].

Summing over all x ∈ together with (<ref>) leads to

    ∑_x ∈ p(x) F_SDP(ρ_1^x, …, ρ_r^x) ≤1/r(r-1)∑_i=1^r  [ Z_i  ∑_x ∈ p(x)   |x ⟩⟨ x | ⊗ρ_i^x ].

Since the above inequality also holds for an optimal choice of (Z_i)_i=1^r, we arrive at
 
    ∑_x ∈ p(x)  F_SDP(ρ_1^x, …, ρ_r^x) ≤  F_SDP( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x)  .

  Lastly, by combining (<ref>)  and (<ref>), we conclude the proof of (<ref>). 

Joint concavity:
Using the direct-sum property (property 6 in <ref>), we have
    
    F_SDP( ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x) = ∑_x ∈ p(x)  F_SDP(ρ_1^x, …, ρ_r^x).

    Then, using the above and applying data processing  with respect to the partial trace channel (i.e., taking a partial trace over the classical system) to the left-hand side concludes the proof. 





   





 §.§ Proof of Proposition <ref> (Coarse-graining property of SDP fidelity)
 
    Let (Y_i)_i=1^r+m be such that 
    
    ∑_i=1^r +m|i⟩⟨ i| ⊗ Y_i ≥∑_i ≠ j |i⟩⟨ j| ⊗ I.

    Then that choice is a possible candidates for the SDP of F_SDP(ρ_1,…,ρ_r, …, ρ_r+m) with the extension of (<ref>) to r+m states. Consider 
    
    ∑_i=1^r+m[Y_i ρ_i]    ≥∑_i=1^r[Y_i ρ_i]  
       ≥ r(r-1)  F_SDP(ρ_1, …, ρ_r)

    where the first inequality follows from Y_i ≥ 0 for all i ∈ [r+m], and the last inequality from the fact that (Y_i)_i=1^r ( r elements from the tuple (Y_i)_i=1^r+m) forms a possible candidate for the SDP of F_SDP(ρ_1,…, ρ_r) in (<ref>) due to 
    
    ∑_i=1^r |i⟩⟨ i| ⊗ Y_i ≥∑_i ≠ j |i⟩⟨ j| ⊗ I.
 We conclude the proof by infimizing the LHS above over all possible candidates (Y_i)_i=1^r+m satisfying the constraints, which renders 
    
    (r+m) (r+m-1)  F_SDP(ρ_1,…,ρ_r, …, ρ_r+m) ≥ r(r-1)  F_SDP(ρ_1, …, ρ_r).




 §.§ Proof of Proposition <ref> (Multivariate SDP fidelity for pure states)




Consider the general formula for arbitrary states (  ρ_1
,…,ρ_r) from <ref>:

    F_SDP(ρ_1,…,ρ_r)=
    1/r-1sup_K≥0{⟨ψ|K⊗ I_d|ψ⟩-1:K=∑_i,j=1^r|i⟩⟨ j|⊗ K_i,j, K_i,i
    =I_d ∀ i∈[  r]  }  ,

where |ψ⟩=1/√(r)∑_i=1^r|i⟩|ψ^ρ_i
⟩ and |ψ^ρ_i⟩ is a purification of ρ_i. In
the case that each ρ_i is pure, the purification system for each state
can be taken to be a trivial one-dimensional system. So then each matrix
K_i,j reduces to a scalar k_i,j∈ℂ, and the constraints in
(<ref>) reduce to those in (<ref>).
Then

    ⟨ψ|K⊗ I_d|ψ⟩
        =(  1/√(r)∑_i^''=1^r⟨
    i^''|⟨ψ_i^''|)  (  ∑
    _i,j=1^rk_i,j|i⟩⟨ j|⊗ I_d)  (  1/√(r)∑_i^'=1^r|i^'⟩|ψ_i^'
    ⟩)  
        =1/r∑_i^'',i,j,i^'=1^rk_i,j⟨
    i^''|i⟩⟨ j|i^'⟩ ⟨ψ
    _i^''|ψ_i^'⟩
        =1/r∑_i,j=1^rk_i,j⟨ψ_i|ψ_j⟩
        =1/r∑_i=1^rk_i,i⟨ψ_i|ψ_i⟩+1/r∑_i,j=1:i≠ j^rk_i,j⟨ψ_i|ψ_j⟩
        =1+2/r∑_i<jℜ[k_i,j⟨ψ_i|ψ
    _j⟩].

Substituting (<ref>) into the objective function in (<ref>) and
simplifying then gives (<ref>).



 §.§ Proof of Proposition <ref> (Uniform continuity of secrecy-based multivariate fidelity)
 
    
    By <ref>,
    
    | F_S(ρ_1, …, ρ_r) - F_S(σ_1, …, σ_2)  | 
       = r/r-1| [ S(ρ_1, …, ρ_r )]^2 -  [ S(σ_1, …, σ_r )]^2  |
           
       =  r/r-1[ S(ρ_1, …, ρ_r ) + S(ρ_1, …, ρ_r )] | S(ρ_1, …, ρ_r )- S(ρ_1, …, ρ_r ) |
       ≤2r/r-1| S(ρ_1, …, ρ_r )- S(ρ_1, …, ρ_r ) |,

    where the last inequality follows because S(ρ_1, …,ρ_r) ≤ 1 for every tuple (ρ_i)_i=1^r of states.

    By using (<ref>), we have that
    
    | S(ρ_1, …, ρ_r )- S(σ_1, …, σ_r ) |    ≤sup_ω∈𝒟|  1/r∑_i=1^r ( F(ρ_i, ω) - F(σ_i, ω) )| 
       ≤sup_ω∈𝒟1/r∑_i=1^r  |  F(ρ_i, ω) - F(σ_i, ω)|

For an arbitrary ω∈𝒟, consider that

    |F(ρ_i,ω) - F(σ_i,ω) |
            = | [ 1-F(σ_i,ω)] - [ 1-F(ρ_i,ω)]  | 
       =1/2|[d_B(σ_i, ω)]^2 - [d_B(ρ_i, ω)]^2 | 
       =1/2[ d_B(σ_i, ω)+ d_B(ρ_i, ω) ] |d_B(σ_i, ω) -d_B(ρ_i, ω) | 
       ≤1/2[ d_B(σ_i, ω)+ d_B(ρ_i, ω) ] d_B(σ_i, ρ_i) 
       ≤√(2) d_B(σ_i, ρ_i),

  where the penultimate inequality follows from triangular inequality of the Bures distance and the last inequality because the Bures distance is bounded by √(2).
Together with that, we have 

    | S(ρ_1, …, ρ_r )- S(σ_1, …, σ_r ) | ≤√(2)/r∑_i=1^r d_B(σ_i, ρ_i) ≤√(2)ε,

  where the last inequality follows from the assumption 1/r∑_i=1^r d_B(ρ_i,σ_i) ≤ε. Finally combining (<ref>)–(<ref>) and (<ref>), we conclude that 
  
    |F_S(ρ_1, …, ρ_r) - F_S(σ_1, …, σ_2)|  ≤r/r-1 2 √(2)ε,

  completing the proof.










 §.§ Proof of Theorem <ref> (Properties of secrecy-based multivariate fidelity)



Note that the secrecy-based multivariate fidelity satisfies reduction to classical average pairwise fidelity, faithfulness, and orthogonality by <ref> and the fact that average pairwise fidelities satisfy these properties as stated in <ref>.

Symmetry: This follows by the definition of secrecy measure in (<ref>). 



Data processing:
    Consider 
    
    sup_σ∈𝒟1/r∑_i=1^rF(  ρ_i, σ)    ≤sup_σ∈𝒟1/r∑_i=1^rF( ( ρ_i), (σ) ) 
       ≤sup_σ'  ∈𝒟1/r∑_i=1^rF( ( ρ_i), σ' ),

    where the first inequality follows from data processing of the Uhlmann fidelity (<ref>, Property 2); and the last inequality by supremizing over a larger set. 
    With that we arrive at 
     
    S(ρ_1, …, ρ_r) ≤ S( (ρ_1), …, (ρ_r) ).

By recalling <ref>, we conclude that 
 
    F_S(ρ_1, …, ρ_r) ≤ F_S( (ρ_1), …, (ρ_r) ).


Direct-sum property:
 A purification of the state ∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_i^x is 

    |Ψ_i⟩∑_x √(p(x)) |x⟩⊗ | Φ^ρ_i^x⟩,

    where | Φ^ρ_i^x⟩ is a purification of state ρ_i^x. 
Let an optimal unitary of the prover achieving the optimum value of [S(ρ_1^x, …, ρ_r^x )]^2 related to the tuple of states ( ρ_i^x )_i in (<ref>) be Z^x_T'RF → T^”F' for all x ∈. i.e., 

    [S(ρ_1^x, …, ρ_r^x )]^2= 1/r^2∑_i,j=1^r  ⟨Φ^ρ_i^x|_RXA(Z^x,i_R → F')^†  Z^y,j_R → F'| Φ^ρ_j^x⟩_RXA,

where

    Z^x,y_R → F'⟨ y|_T^” Z^x_T'RF → T^”F'|y ⟩_T' |0 ⟩_F

Then, choose the controlled unitary 

    Z ∑_x  |x⟩⟨ x| ⊗ Z^x.

and the resulting contraction

    Z^y_R → XF'∑_x  |x⟩⟨ x| ⊗  Z^x,y_R → F'


So, now with the notations developed, consider

    =1/r^2∑_i,j=1^r ⟨Ψ^i|_RXA(Z^i_R → XF')^† Z^j_R → XF'| Ψ^j⟩_RXA
       = 1/r^2∑_i,j=1^r (∑_x √(p(x)) |x⟩⊗⟨Φ^ρ_i^x|) ( ∑_x  |x⟩⟨ x| ⊗(Z^x,i_R → F')^† Z^y,j_R → F')  ×
                                               (∑_x √(p(x)) |x⟩⊗ | Φ^ρ_i^x⟩)   
       = 1/r^2∑_i,j=1^r ∑_x p(x) ⟨Φ^ρ_i^x|_RXA(Z^x,i_R → F')^†  Z^y,j_R → F'| Φ^ρ_j^x⟩_RXA
       =∑_x p(x)1/r^2∑_i,j=1^r  ⟨Φ^ρ_i^x|_RXA(Z^x,i_R → F')^†  Z^y,j_R → F'| Φ^ρ_j^x⟩_RXA
       = ∑_x p(x) [S(ρ_1^x, …, ρ_r^x)]^2,

where the first inequality follows by substituting the definitions in (<ref>) and (<ref>), and the last equality from (<ref>).

This shows that controlled unitary in (<ref>) is a possible candidate for the unitary in the optimization of [S(∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x)]^2. 
With that, we conclude the following inequality:

    [S(∑_x ∈ p(x) |x⟩⟨ x| ⊗ρ_1^x, …, ∑_x ∈ p(x)  |x⟩⟨ x| ⊗ρ_r^x)]^2 
    ≥∑_x ∈ p(x) [S(ρ_1^x, …, ρ_r^x)]^2.



Now we prove the opposite inequality. Let Δ(·)∑_x
|x⟩⟨ x|(·)|x⟩⟨ x| denote the dephasing channel.
Recall that

    S^2(  ∑_xp(x)|x⟩⟨ x|⊗ρ_1^x,…
    ,∑_xp(x)|x⟩⟨ x|⊗ρ_r^x)  
    
    =[  sup_σ_XA1/r∑_i=1^rF(  ∑
    _xp(x)|x⟩⟨ x|⊗ρ_i^x,σ_XA)  ]
    ^2.

Let σ_XA be an arbitrary state, and define the probability
distribution q and states (  σ^x)  _x in terms of

    (  Δ⊗id)  (  σ_XA)
    =∑_xq(x)|x⟩⟨ x|⊗σ^x,

where id is the Identity channel. 
Then

    1/r∑_i=1^rF(  ∑_xp(x)|x⟩⟨ x|⊗ρ_i^x,σ_XA)  
       ≤1/r∑_i=1^rF(  (  Δ⊗id
    )  (  ∑_xp(x)|x⟩⟨ x|⊗ρ_i^x)
    ,(  Δ⊗id)  (  σ_XA)
    )  
        =1/r∑_i=1^rF(  ∑_xp(x)|x⟩⟨
    x|⊗ρ_i^x,∑_xq(x)|x⟩⟨ x|⊗σ^x)
    
        =1/r∑_i=1^r∑_x√(p(x)q(x))F(  ρ_i
    ^x,σ^x)  
        =∑_x√(p(x)q(x))1/r∑_i=1^rF(  ρ_i
    ^x,σ^x)  .

Now applying the Cauchy–Schwarz inequality, we conclude that

    ∑_x√(p(x)q(x))1/r∑_i=1^rF(  ρ_i^x
    ,σ^x) 
       ≤√(∑_xq(x))√(∑_xp(x)[  1/r∑_i=1
    ^rF(  ρ_i^x,σ^x)  ]  ^2)
        =√(∑_xp(x)[  1/r∑_i=1^rF(  ρ_i
    ^x,σ^x)  ]  ^2)
       ≤√(∑_xp(x)S^2(ρ_1^x,…,ρ_r^x)).

We have thus shown that the following inequality holds for every state σ_XA:

    [  1/r∑_i=1^rF(  ∑_xp(x)|x⟩⟨
    x|⊗ρ_i^x,σ_XA)  ]  ^2≤∑_x
    p(x)S^2(ρ_1^x,…,ρ_r^x).

By taking the optimization over every state σ_XA, we conclude that

    S^2(  ∑_xp(x)|x⟩⟨ x|⊗ρ_1^x,…
    ,∑_xp(x)|x⟩⟨ x|⊗ρ_r^x)  ≤∑
    _xp(x)S^2(ρ_1^x,…,ρ_r^x).

Note that equality is achieved by picking σ_XA = ∑_x p(x) |x⟩⟨ x| ⊗σ^x, where σ^x is an optimal choice for S^2(ρ_1^x,…,ρ_r^x).
This completes the proof.



 §.§ Proof of Proposition <ref> (Coarse-graining property of secrecy-based fidelity)
 
    Let μ≥ 0 and ( Y_i )_i=1^r+m be a tuple of PSD matrices that satisfy |0⟩⟨
0|⊗μ I_d +
∑_i=1^r+m|i⟩⟨ i|⊗ Y_i≥∑_i=1^r+m(  |i⟩⟨ 0|+|0⟩⟨ i|)
⊗ I_d and consider the SDP formulation of the secrecy measure in (<ref>). 
Given that Y_i for all i ∈[r+m] is PSD, we get 

    μ + ∑_i=1^r+m[Y_i ρ_i]  ≥μ + ∑_i=1^r[Y_i ρ_i].

Also note that by choosing the first r elements of the tuple ( Y_i )_i=1^r+m, ( Y_i )_i=1^r, those also satisfy 

    |0⟩⟨
    0|⊗μ I_d +
    ∑_i=1^r|i⟩⟨ i|⊗ Y_i≥∑_i=1^r(  |i⟩⟨ 0|+|0⟩⟨ i|).

With that, μ and ( Y_i )_i=1^r are possible candidates to the SDP of S(ρ_1,…, ρ_r). 
Then, we arrive at 

    1/2r(μ + ∑_i=1^r+m[Y_i ρ_i]   ) ≥ S(ρ_1,…, ρ_r).

By optimizing over all μ≥ 0 and ( Y_i )_i=1^r+m that satisfy the required constraints, we arrive at 

    (r+m)/r S(ρ_1,…, ρ_r+m) ≥ S(ρ_1,…, ρ_r).

Note that by rearranging the terms in <ref>, we have the equality 

    (r-1) F_S(ρ_1,…,ρ_r) +1  = r [ S(ρ_1,…, ρ_r)]^2.

Then, by combining above-mentioned relations, we get 

    r+m/r(   (r+m -1) F_S(ρ_1,…,ρ_r+m) +1 ) ≥  (r-1) F_S(ρ_1,…,ρ_r) +1.

Finally, we arrive at the desired result by algebraic simplifications.



 §.§ Proof of Theorem <ref> (Relations between multivariate fidelities)
 



We prove the chain of inequalities in the theorem statement from left to right. 


F_H(ρ_1, …,ρ_r) ≤ F_S(ρ_1, …, ρ_r):


Recall the Petz <cit.> and sandwiched <cit.>
Rényi relative entropies, defined for α∈ (0,1)∪(1,∞) respectively as follows:

    D_α(ρ‖σ)     1/α-1lnTr
    [ρ^ασ^1-α],
    D_α(ρ‖σ)     1/α-1lnTr[(  σ^(  1-α)  /2αρσ^(  1-α)  /2α)  ^α].

Also note that D_1/2(ρ‖σ) = -2 ln F(ρ,σ) and D_1/2(ρ‖σ) = -2 ln F_H(ρ,σ).
The following inequality is known for α∈(  0,1) from <cit.>:

    D_α(ρ‖σ)≤ D_α(ρ‖σ).

Let

    ρ_XA1/r∑_i=1^r|i⟩⟨ i|_X⊗ρ_i.

Plugging into (<ref>), taking  infima with respect to a state
σ_A, and setting α=1/2 then gives

    inf_σ_AD_1/2(ρ
    _XA‖ρ_X⊗σ_A)≤inf_σ_AD_1/2
    (ρ_XA‖ρ_X⊗σ_A).

Consider that

    inf_σ_AD_1/2(ρ_XA‖ρ_X
    ⊗σ_A)      =inf_σ_A[  -2ln F(  ρ_XA
    ,ρ_X⊗σ_A)  ]  
        =-2lnsup_σ_AF(  ρ_XA,ρ_X⊗σ_A)
    
        =-2lnsup_σ1/r∑_i=1^rF(  ρ_i,σ) 
        = -2 ln S(ρ_1, …, ρ_r)
    
    .

For the penultimate line, we used the direct-sum property of the fidelity in (<ref>).

Now applying  <cit.> (which holds for α∈(0,1)), consider that

    inf_σ_AD_1/2(ρ_XA‖ρ_X⊗σ
    _A)
        =-lnTr[  (  Tr_X[ρ_X
    ^1/2ρ_XA^1/2])  ^2]  
        =-lnTr[  (  Tr_X[  (
    ∑_i1/r|i⟩⟨ i|⊗ I)  ^1/2(  ∑
    _j1/r|j⟩⟨ j|⊗ρ_j)  ^1/2]
    )  ^2]  
        =-lnTr[  (  Tr_X[  ∑
    _i1/r|i⟩⟨ i|⊗ρ_i^1/2]  )
    ^2]  
        =-lnTr[  (  ∑_i1/rρ_i
    ^1/2)  ^2]  
        =-lnTr[  (  ∑_i1/rρ_i
    ^1/2)  (  ∑_j1/rρ_j^1/2)  ]  
        =-lnTr[  1/r^2∑_iρ_i+1/r^2∑_i≠ jρ_i^1/2ρ_j^1/2]  
        =-ln(  1/r+1/r^2∑_i≠ jTr
    [  ρ_i^1/2ρ_j^1/2]  )  
        =-ln(  1/r+2/r^2∑_i<jF_H(ρ_i,ρ
    _j))  
        =-ln(  1/r+r-1/r[  2/r(  r-1)
    ∑_i<jF_H(ρ_i,ρ_j)]  )  
        =-ln(  1/r+r-1/rF_H(ρ_1, …, ρ_r ))  .

Combining (<ref>), (<ref>), and (<ref>), we get

    -2ln S(ρ_1, …, ρ_r)  
       ≤-ln(  1/r+r-1/rF_H(ρ_1, …, ρ_r))  ,

which is the same as

    [  S(ρ_1, …, ρ_r)  ]  ^2≥1/r+r-1/rF_H(ρ_1, …, ρ_r).

 Then, by rearranging the terms in the above inequality, we arrive at the required bound.

F_S(ρ_1, …,ρ_r) ≤ F_SDP(ρ_1, …, ρ_r):




By using  (<ref>), we have 

    [ S(ρ_1, …, ρ_r)]^2    = sup_P_T'RF → T^”F'∈𝒰1/r^2∑_x,y=1^r ⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA
        = sup_P_T'RF → T^”F'∈𝒰1/r^2∑_x=1^r ⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^x_R → F'| ϕ^ρ_x⟩_RA +            2/r^2∑_x< y=1 ^r ℜ[  ⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA],

    where 
    
    P^x_R → F'⟨ x|_T^” P_T'RF → T^”F'|x ⟩_T' |0 ⟩_F

    with P_T'RF → T^”F' being a unitary, 
    and | ϕ^ρ_x⟩ is a purification of state ρ_x for all x ∈ [r].
With P^x_R → F' being a contraction, we have (P^x_R → F')^† P^x_R → F'≤ I_d (also see the proof of <cit.>). Applying that, we have 

    [ S(ρ_1, …, ρ_r)]^2  ≤1/r +  sup_P_T'RF → T^”F'∈𝒰2/r^2∑_x< y=1 ^r ℜ[  ⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA].

Now, observe that 

    ∑_x,y |x⟩⟨ y| ⊗(P^x_R → F')^† P^y_R → F' = (∑_x |x⟩⊗(P^x_R → F')^†)(∑_y ⟨ y| ⊗ P^y_R → F')≥0.

Again applying (P^x_R → F')^† P^x_R → F'≤ I_d, we see that 

    I_r ⊗ I_d+ ∑_x ≠ y |x⟩⟨ y| ⊗(P^x_R → F')^† P^y_R → F'≥∑_x,y |x⟩⟨ y| ⊗(P^x_R → F')^† P^y_R → F'≥ 0.

Then the choice K_xy=(P^x_R → F')^† P^y_R → F' for x≠ y ∈ [r] and K_xx=I_d for x ∈ [r] is a possible candidate to the F_SDP(ρ_1,…, ρ_r) in (<ref>). To this end, we have 

    ∑_x< y=1 ^r ℜ[  ⟨ϕ^ρ_x|_RA(P^x_R → F')^† P^y_R → F'| ϕ^ρ_y⟩_RA] ≤r (r-1)/2 F_SDP(ρ_1,…, ρ_r).

Supremizing over P_T'RF → T^”F'∈𝒰 and 
incorporating (<ref>) , we have 

    [ S(ρ_1, …, ρ_r)]^2  ≤1/r +  (r-1)/r F_SDP(ρ_1,…, ρ_r).

Rearranging the terms and recalling <ref>, we conclude by arriving at 

    F_S(ρ_1, …, ρ_r) ≤   F_SDP(ρ_1,…, ρ_r).




F_SDP(ρ_1, …,ρ_r) ≤ F_U(ρ_1, …, ρ_r): 



  Let X_ij satisfy X_ji=X_ij^† along with the following relation:
    
    ∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ j| ⊗ X_ij≥ 0.

 Then for each pair of states ρ_i,ρ_j with i < j, we also have 
 
    [    ρ_i   X_ij; X_ij^†    ρ_j ]≥ 0.

 Using this together with the dual SDP of Uhlmann fidelity given in (<ref>), we get 
 
    2 ∑_i <jℜ[ [X_ij]] ≤ 2 ∑_i <j  F(ρ_i, ρ_j).

 Taking the supremum over all (X_ij)_ij satisfying the relation
 ∑_i=1^r |i⟩⟨ i| ⊗ρ_i + ∑_i ≠ j |i⟩⟨ j| ⊗ X_ij≥ 0, we obtain 
 
    r(r-1) F_SDP(ρ_1, …, ρ_r) ≤  2 ∑_i <j^r  F(ρ_i, ρ_j),

 which implies the desired inequality after rearrangement of terms.

 F_U(ρ_1,…, ρ_r) ≤√(F_H(ρ_1,…,ρ_r)):

Notice that for states ρ and σ

    F(ρ,σ) ≤√(F_H(ρ,σ)),

which follows by adapting <cit.> with the choice s=1/2, A=ρ, and B=σ. 
This gives 

    F_U(ρ_1,…, ρ_r) 
           = 2/r(  r-1)  ∑_i<jF(ρ_i,ρ_j) 
       ≤2/r(  r-1)  ∑_i<j√(F_H(ρ_i,ρ_j))
       ≤√(2/r(  r-1)  ∑_i<jF_H(ρ_i,ρ_j) )
       = √(F_H(ρ_1,…,ρ_r)),

where the last inequality holds due to the concavity of the square root function. 






 §.§ Proof of Proposition <ref> (Lower bound on multivariate SDP fidelity)
 
 Fix an arbitrary permutation π∈ S_r. For all 1 ≤ i ≤⌊ r/2 ⌋, we have from (<ref>) that
    
    F(ρ_π(i), ρ_π(i+⌊ r/2 ⌋))    = sup_Z_i{ℜ[ [Z_i]]: [         ρ_π(i)            Z_i;          Z_i^† ρ_π(i+⌊ r/2 ⌋) ]≥ 0 }.

    Consider arbitrary operators Z_i satisfying the constraint of (<ref>):
    
    [         ρ_π(i)            Z_i;          Z_i^† ρ_π(i+⌊ r/2 ⌋) ]≥ 0,      1 ≤ i ≤⌊ r/2 ⌋.

    Define operators X_ij for 1 ≤ i < j ≤ r as
    
    X_ij = 
            
                Z_i    if  1≤ i ≤⌊ r/2 ⌋ and  j=i+⌊ r/2 ⌋, 
    
                0    otherwise,

    where 0 denotes the zero operator. The positive semi-definiteness of the matrices in (<ref>) implies that 
    
    ∑_i=1^r |i⟩⟨ i| ⊗ρ_π(i) + ∑_i ≠ j |i⟩⟨ j| ⊗ X_ij≥ 0.

    Hence, the SDP dual (<ref>) and the permutation symmetry of the multivariable SDP fidelity yield
    
    F_SDP(ρ_1, …, ρ_r) 
                   ≥2/ r(r-1)∑_1 ≤ i <j ≤ rℜ[ [X_ij]] 
       =2/ r(r-1)∑_i=1^⌊r/2⌋ℜ[[Z_i] ].

    Combining (<ref>) and (<ref>), we get 

    F_SDP(ρ_1, …, ρ_r) ≥2/ r(r-1)∑_i=1^⌊r/2⌋ F(ρ_π(i), ρ_π(i)+⌊r/2⌋).

Since the above inequality holds for arbitrary π∈ S_r, the desired inequality (<ref>) follows.



 §.§ Proof of Proposition <ref> (Properties of measured multivariate fidelity)

Symmetry follows by the symmetry followed by the underlying classical multivariate fidelity. 

Data processing: 
Let Λ' be a measurement channel and  be a channel. Then
consider 

    𝐅( Λ'((ρ_1)), …, Λ'((ρ_r))) ≥inf_( Λ_x)_x𝐅(Λ(ρ_1), …, Λ(ρ_r))

due to the composed measurement channel Λ' ∘ being one of the choices of measurement channels.
Infimizing over all measurement channels  Λ', we arrive at 

    inf_( Λ'_x)_x𝐅( Λ'((ρ_1)), …, Λ'((ρ_r))) ≥inf_( Λ_x)_x𝐅(Λ(ρ_1), …, Λ(ρ_r)),

proving the data processing property. 

Faithfulness:
When ρ_i=ρ_j, by the faithfulness of multivariate fidelity we have 

    𝐅(ρ_1, …, ρ_r) ≥𝐅(ρ_1, …, ρ_r)=1,

Since the underlying classical multivariate fidelity is less than or equal to one, we also have 
𝐅(ρ_1, …, ρ_r) =1.

To prove the reverse implication, assume 𝐅(ρ_1, …, ρ_r) =1. Again due to the fact that the underlying classical multivariate fidelity is less than or equal to one, under all measurement channels we have 
𝐅(Λ(ρ_1), …, Λ(ρ_r))=1. This includes tomographically complete measurements, which lead to distributions that are in one-to-one correspondence with the underlying density operators (i.e., there is a linear invertible map relating the resulting distributions and the original density operators).
From the faithfulness of the underlying classical multivariate fidelities, the distributions are the same, and due to the aforementioned bijection, the states are also the same. i.e., {M_x}_x=1^r with M_x >0, ∑_x=1^r M_x =I and [M_x(ρ_i-ρ_j)] implies that ρ_i=ρ_j.

Orthogonality: Let  𝐅(ρ_1, …, ρ_r) =0. Then we have 

    0= 𝐅(ρ_1, …, ρ_r) ≥𝐅(ρ_1, …, ρ_r).

Then, by orthogonality of multivariate fidelity that generalizes the underlying classical multivariate fidelity (e.g., average pairwise z-fidelity that generalizes average pairwise fidelities), we have the condition that for ρ_i ρ_j =0 for all i≠ j ∈ [r].

To prove the reverse, assume that ρ_i ρ_j =0 for all i≠ j ∈ [r]. Then there exists a measurement to distinguish them perfectly, leading to orthogonal distributions. Then the underlying multivariate classical fidelity is equal to zero.


Direct-sum property:
For all i∈[  r], define

    ρ_XA^i∑_xp(x)|x⟩⟨ x|⊗ρ_i^x,

where (  p(x))  _x is a probability distribution, {
|x⟩}  _x is an orthonormal basis, and each ρ_i^x is a
state.


Defining the measurement channels

    Λ(ω_XA)     ∑_yTr[Λ_yω
    _XA]|y⟩⟨ y|,
    Λ^x(τ_A)     ∑_yTr[Λ_y^xτ
    _A]|y⟩⟨ y|,
    Λ(ω_XA)     ∑_x,yTr[
    (  |x⟩⟨ x|⊗Λ_y^x)  ω_XA]
    |x⟩⟨ x|⊗|y⟩⟨ y|,

consider that

    𝐅(ρ_XA^1,…,ρ_XA^r)
        =inf_Λ𝐅(Λ(ρ_XA^1),…,Λ(ρ_XA^r))
       ≤inf_Λ𝐅(Λ(ρ_XA^1
    ),…,Λ(ρ_XA^r))
        =inf_Λ𝐅(  ∑_xp(x)|x⟩⟨
    x|⊗Λ^x(ρ_1^x),…,∑_xp(x)|x⟩⟨
    x|⊗Λ^x(ρ_r^x))  
        =inf_Λ∑_xp(x)𝐅(  Λ^x(ρ_1
    ^x),…,Λ^x(ρ_r^x))  
        =∑_xp(x)inf_Λ^x𝐅(  Λ^x(ρ_1^x
    ),…,Λ^x(ρ_r^x))  
        =∑_xp(x)𝐅(ρ_1^x,…,ρ_r^x) .

The first inequality follows because Λ is a special kind
of measurement. The third equality follows from the direct-sum property of the
underlying classical multivariate fidelity.

For the opposite inequality, consider that

    𝐅(ρ_XA^1,…,ρ_XA^r)
        =inf_Λ𝐅(Λ(ρ_XA^1),…,Λ(ρ_XA^r))
        =inf_Λ𝐅(  ∑_xp(x)Λ(  |x⟩⟨
    x|⊗ρ_1^x)  ,…,∑_xp(x)Λ(
    |x⟩⟨ x|⊗ρ_r^x)  )  
       ≥inf_Λ∑_xp(x)𝐅(  Λ(  |x⟩⟨
    x|⊗ρ_1^x)  ,…,Λ(  |x⟩⟨
    x|⊗ρ_r^x)  )  
       ≥∑_xp(x) 𝐅(ρ_1^x,…,ρ_r^x).

The first equality follows from linearity of the measurement channel Λ. The first inequality follows because the underlying classical fidelities
are jointly concave (as a consequence of the data-processing inequality and the direct-sum property). The final inequality follows because tensoring in the
state |x⟩⟨ x| and performing the measurement Λ is a
particular kind of measurement channel for the tuple (  ρ_1
^x,…,ρ_r^x), and so the resulting fidelity cannot be
smaller than 𝐅(ρ_1^x,…,ρ_r^x), which is defined as
the infimum over all measurement channels.

By combining (<ref>) and (<ref>), we conclude the proof. 






 §.§ Proof of Proposition <ref> (Properties of minimal extension of multivariate fidelity)




Recall 
    𝐅(ρ_1, …, ρ_r) sup_∈CPTP,  ω_i ∈𝒟_c ∀ i ∈ [r]{𝐅( ω_1, …, ω_r ) : (ω_i) =ρ_i  ∀ i ∈ [r]},

where  is a classical-quantum channel with (ω_i) =ρ_i.

Symmetry follows by the symmetry of the underlying classical multivariate fidelity. 
 

Data processing:
Let  be a quantum channel. 
Let ω'_i ∈𝒟_c such that '(ω'_i)= ρ_i. This also leads to 
('(ω'_i))= (ρ_i), providing a feasible preparation channel for (ρ_i) resulted from ∘'.
So we have 

    𝐅( ω'_i, …, ω'_r )  ≤𝐅((ρ_1), …, (ρ_r) )

Then, supremizing over all  and ω_i, we obtain that 

    𝐅(ρ_1, …, ρ_r ) ≤𝐅((ρ_1), …, (ρ_r) ).


Faithfulness:
Let 𝐅(ρ_1, …, ρ_r ) =1. Then, we also have that 𝐅(ρ_1, …, ρ_r ) =1. With the faithfulness property of multivariate fidelity (e.g., average z-pairwise fidelity), we find that ρ_i=ρ_j for all i,j ∈ [r]. 

To prove the reverse implication, assume that ρ_i=ρ_j =ρ for all i,j ∈ [r].  Then a simple preparation channel is trace and replace the deterministic state ω_i  |0 ⟩⟨ 0| with ρ. The fidelity of the common state is equal to one, due to the faithfulness of the classical multivariate fidelity. Since the prep fidelity involves a supremum over all preparations and the underlying fidelity cannot exceed one, this proves that 𝐅(ρ_1, …, ρ_r ) =1.

Direct-sum property:
For all i∈[  r], define

    ρ_XA^i∑_xp(x)|x⟩⟨ x|⊗ρ_i^x,

where (  p(x))  _x is a probability distribution, {
|x⟩}  _x is an orthonormal basis, and each ρ_i^x is a
state. 


Define the preparation channel 𝒫^x and the state ω_i
^x∈𝒟_c, such that

    𝒫^x(ω_i^x)=ρ_i^x ∀ i∈[  r]  .

Defining

    𝒫^'(τ_XS)    ∑_x|x⟩⟨
    x|⊗𝒫^x(⟨ x|_Xτ_XS|x⟩_X),
    ω_i^'   ∑_xp(x)|x⟩⟨ x|⊗ω_i^x

we then find that, for all i∈[  r],

    𝒫^'(  ω_i^')  =∑_x
    p(x)|x⟩⟨ x|⊗𝒫^x(ω_i^x)=ρ_XA^i.

Then consider that

    𝐅(ρ_XA^1,…,ρ_XA^r)
         =sup_𝒫,
    ω_1,…,ω_r∈𝒟_c{𝐅(ω_1,…,ω_r
    ):𝒫(ω_i)=ρ_XA^i ∀ i∈[  r]
    }
       ≥sup_𝒫^',
    ω_1^'
    ,…,ω_r^'∈𝒟_c{𝐅(ω
    _1^',…,ω_r^'):𝒫^'(ω
    _i^')=ρ_XA^i ∀ i∈[  r]  }
         =sup_𝒫^',
    ω_1^',…
    ,ω_r^'∈𝒟_c{[ 𝐅(  ∑_xp(x)|x⟩⟨ x|⊗ω_i
    ^x,…,∑_xp(x)|x⟩⟨ x|⊗ω_i^x)  :;                           𝒫^'(ω_i^')=ρ_XA^i ∀ i∈[
    r] ]}
         =sup_𝒫^',
    ω_1^',…
    ,ω_r^'∈𝒟_c{∑_xp(x)𝐅(
    ω_i^x,…,ω_i^x)  :𝒫^'(ω
    _i^')=ρ_XA^i ∀ i∈[  r]  }
         =∑_xp(x)sup_𝒫^x,
    ω_1^x,…
    ,ω_r^x∈𝒟_c{𝐅(  ω_i
    ^x,…,ω_i^x)  :𝒫^x(ω_i^x)=ρ
    _i^x ∀ i∈[  r]  }
         =∑_xp(x)𝐅(ρ_1^x,…,ρ_r^x).



Now we prove the following inequality

    𝐅(ρ_XA^1,…,ρ_XA^r)≤∑
    _xp(x)𝐅(ρ_1^x,…,ρ_r^x
    ).

For a probability distribution (  q_i(y))  _y, define the
state

    ω_i∑_yq_i(y)|y⟩⟨ y|,

for all i∈[  r]. Then consider that the action of a general
preparation channel 𝒫 is as follows

    𝒫(ω_i)=ρ_XA^i  ∀ i∈[  r]  .

Now define

    𝒫(|y⟩⟨ y|)=σ_XA^y,

so that

    ∑_yq_i(y)σ_XA^y=ρ_XA^i  ∀ i∈[  r]
    .

Then from the action of the completely dephasing channel Δ, define the
probability distribution (  r(x|y))  _x and tuple (
σ_A^x,y)  _x of states as

    (  Δ⊗id)  (σ_XA^y)=∑
    _xr(x|y)|x⟩⟨ x|⊗σ_A^x,y,

and consider that

    ρ_XA^i     =(  Δ⊗id)  (
    ρ_XA^i)  
         =∑_yq_i(y)(  Δ⊗id)
    (σ_XA^y)
         =∑_x,yq_i(y)r(x|y)|x⟩⟨ x|⊗σ_A^x,y.

Now observe that

    ∑_xp(x)|x⟩⟨ x|=Tr_A[ρ_XA^i
    ]=∑_x(  ∑_yq_i(y)r(x|y))  |x⟩⟨ x|,

proving that the marginal over y of the joint distribution q_i(y)r(x|y)
is p(x). Now let us define the conditional distribution s_i(y|x) by

    s_i(y|x)p(x)=q_i(y)r(x|y),

and we note that

    ρ_XA^i     =∑_x,ys_i(y|x)p(x)|x⟩⟨ x|⊗σ_A^x,y
         =∑_xp(x)|x⟩⟨ x|⊗∑_ys_i(y|x)σ_A
    ^x,y
         =∑_xp(x)|x⟩⟨ x|⊗ρ_i^x.

Thus, conditioned on x, generating y at random according to s_i(y|x)
and preparing the state σ_A^x,y is a particular way of preparing
ρ_i^x. That is, we can define the preparation channel 𝒫
^x as

    𝒫^x(|y⟩⟨ y|)=σ_A^x,y,

so that, for all i∈[  r],

    𝒫^x(  ∑_ys_i(y|x)|y⟩⟨ y|)
    =∑_ys_i(y|x)𝒫^x(  |y⟩⟨ y|)
    =∑_ys_i(y|x)σ_A^x,y=ρ_i^x.

Putting everything together, let 𝒫 and ω_1,…
,ω_r be a particular way of preparing ρ_XA^1,…,ρ
_XA^r, respectively. Then

    𝐅(ω_1,…,ω_r)
         =𝐅(  ∑_yq_1(y)|y⟩⟨ y|,…,∑
    _yq_r(y)|y⟩⟨ y|)  
       ≤𝐅(  ∑_x,yq_1(y)r(x|y)|x⟩⟨
    x|⊗|y⟩⟨ y|,…,∑_x,yq_r(y)r(x|y)|x⟩⟨ x|⊗|y⟩⟨ y|)  
         =𝐅(  ∑_x,ys_1(y|x)p(x)|x⟩⟨
    x|⊗|y⟩⟨ y|,…,∑_x,ys_1(y|x)p(x)|x⟩⟨ x|⊗|y⟩⟨ y|)  
         =∑_xp(x)𝐅(  ∑_ys_1(y|x)|y⟩⟨
    y|,…,∑_x,ys_1(y|x)|y⟩⟨ y|)  
       ≤∑_xp(x)sup_𝒫^x,
    ω_1^x
    ,…,ω_r^x∈𝒟_c{𝐅(
    ω_i^x,…,ω_i^x)  :𝒫^x(ω_i
    ^x)=ρ_i^x ∀ i∈[  r]  }
         =∑_xp(x)𝐅(ρ_1^x,…,ρ_r^x).

The first inequality follows from data processing. The second equality follows
from the identity in (<ref>). The third equality follows from the
direct-sum property for the underlying classical multivariate fidelity. The
last inequality follows because, from 𝒫 and ω_1
,…,ω_r, we have constructed a particular method of preparing
ρ_1^x,…,ρ_r^x from the channel 𝒫^x and the
commuting states ∑_ys_1(y|x)|y⟩⟨ y|,…,∑_y
s_r(y|x)|y⟩⟨ y|, as given in (<ref>). The
proof of the inequality in (<ref>) is complete after
noticing that we have proven the inequality 𝐅(ω_1
,…,ω_r)≤∑_xp(x)𝐅(ρ_1^x
,…,ρ_r^x) for all possible preparations of the states ρ
_XA^1,…,ρ_XA^r, and so we can take a supremum over all such preparations.


Weak orthogonality: Suppose that ρ_iρ_j=0 for all i≠ j ∈ [r]. Then, by the orthogonality of multivariate fidelity that generalizes the underlying classical multivariate fidelity (e.g., average pairwise z-fidelity that generalizes average pairwise fidelity and multivariate log-Euclidean fidelity that generalizes Matusita multivariate fidelity), we have 

    𝐅(ρ_1, …, ρ_r) ≤𝐅(ρ_1, …, ρ_r) =0,

which implies that 𝐅(ρ_1, …, ρ_r)=0 concluding the proof.





 §.§ Proof of Theorem <ref> (Properties of multivariate log-Euclidean fidelity)
 



  Symmetry follows by the definition of oveloh information in (<ref>). 

Reduction to classical Matusita multivariate fidelity:
By (<ref>), we have 
    
    inf_σ∈𝒟1/r∑_i=1^rD(σ‖ρ
    _i)=-lnTr[  exp(  ∑_i=1^r1/r
    lnρ_i)  ]  .

Then, when the states are commuting, we also have the following identity: 

    exp(  ∑_i=1
    ^r1/rlnρ_i)  =∏_i=1^rρ_i^1/r.

Combining the above two identitites, we arrive at 

    exp[  -inf_σ∈𝒟D(ρ_X⊗σ‖ρ
    _XA)]        =exp[  -inf_σ∈𝒟1/r
    ∑_i=1^rD(σ‖ρ_i)]  
         =Tr[  ∏_i=1^rρ_i^1/r
    ]  .

Observing that the last expression is precisely the Matusita multivariate fidelity for commuting states then establishes the claim.

  Data processing:
  Consider an arbitrary σ∈𝒟, due to the data processing of quantum relative entropy, we have 
  
    D(ρ_X ⊗σ‖ρ_XA) ≥ D(ρ_X ⊗(σ) ‖id⊗( ρ_XA)),

  where id is the identity channel and : ℒ(_A) →ℒ(_B) is a quantum channel.
  Note that 
  
    (id⊗)( ρ_XA)= 1/r∑_i=1^r |i⟩⟨ i|_X⊗(ρ_i).

  So, we arrive at 
  
    inf_σ∈𝒟1/r∑_i=1^rD(σ‖ρ
    _i)     ≥inf_σ∈𝒟1/r∑_i=1^rD((σ) ‖(ρ
    _i)) 
       ≥inf_σ' ∈𝒟1/r∑_i=1^rD(σ' ‖(ρ
    _i)),

where the last inequality follows from optimization over a larger set of states.
This then leads to 

    exp( - inf_σ∈𝒟1/r∑_i=1^rD(σ‖ρ
    _i) ) ≤exp(-  inf_σ' ∈𝒟1/r∑_i=1^rD(σ' ‖(ρ
    _i)) ),

due to the monotonicity of the exponential function. 

Faithfulness: If ρ_i= ρ for all i ∈ [r], then for arbitrary σ∈𝒟

    D(ρ_X ⊗σ‖ρ_XA)=  D(ρ_X ⊗σ‖ρ_X ⊗ρ) ≥ 0.

Then D(ρ_X ⊗σ‖ρ_XA)=0 by the choice σ= ρ. With that choice, infimum is achieved leading to exp[- 𝒪(X;A)_ρ]= exp(0) =1.

To prove the reverse direction, assume that exp[- 𝒪(X;A)_ρ]=1, so that we have

    inf_σ∈𝒟1/r∑_i=1^rD(σ‖ρ
    _i) =0

Since quantum relative entropy is non-negative, there exists σ∈𝒟 such that D(σ‖ρ_i) =0 for all i ∈ [r]. Then,  faithfulness of quantum relative entropy <cit.> implies that σ= ρ_i for all i, which is the desired implication. 

Direct-sum property:
Let 

    ρ_i ∑_x p(x) |x⟩⟨ x|_X ⊗ρ_i^x

for all i ∈ [r].
Then, by the spectral theorem, we have 

    lnρ_i = ∑_x  |x⟩⟨ x|_X ⊗ln(p(x) ρ_i^x ) = ∑_x  |x⟩⟨ x|_X ⊗ln(p(x) ) I + ∑_x  |x⟩⟨ x|_X ⊗ln( ρ_i^x ).

Summing over all i ∈ [r] and averaging, we get 

    ∑_i=1^r 1/rlnρ_i = ∑_x  |x⟩⟨ x|_X ⊗ln(p(x) ) I +  ∑_x  |x⟩⟨ x|_X ⊗∑_i=1^r 1/rln( ρ_i^x ).

Using the fact that the two summands on the right above commute, that [  exp(A+B)]=[  exp(A) exp(B)] for commuting operators A and B, and again applying the spectral theorem,
we arrive at 

    Tr[  exp(  ∑_i=1^r1/r
    lnρ_i)  ] 
       = [ exp(∑_x  |x⟩⟨ x|_X ⊗ln(p(x)) I ) exp(∑_x'  |x'⟩⟨ x'|_X ⊗∑_i=1^r 1/rln( ρ_i^x'))  ] 
       = [ (∑_x  |x⟩⟨ x|_X ⊗ p(x) I ) (∑_x'  |x'⟩⟨ x'|_X ⊗exp( ∑_i=1^r 1/rln( ρ_i^x') ))  ] 
        = [ ∑_x |x⟩⟨ x|_X ⊗ p(x) exp( ∑_i=1^r 1/rln( ρ_i^x) )] 
       = ∑_x p(x) [exp( ∑_i=1^r 1/rln( ρ_i^x) )].

Recalling (<ref>) together with <ref>, we conclude the proof. 

Weak orthogonality: Let ρ_iρ_j=0 for all i≠ j ∈ [r]. Then, the channel mapping ρ_i → |i ⟩⟨ i| for all i ∈ [r] is a reversible mapping. Then, due to the data processing property of log-Euclidean multivariate fidelity, we have 

    F^♭_r (ρ_1,…, ρ_r) =F^♭_r( |1⟩⟨ 1|, …, |r⟩⟨ r|).

For inf_σ1/r∑_i=1^r D(σ‖ |i⟩⟨ i| ) < ∞, there should be σ∈𝒟 such that D(σ‖ |i⟩⟨ i| ) < ∞ for all i ∈ [r]. This indeed happens only if supp(σ) ⊆supp(|i⟩⟨ i|) for all i. Since, {|i⟩⟨ i|} is a collection of orthonormal projections, we cannot find σ such that it satisfies the finiteness condition for all i ∈ [r]. This implies that 

    inf_σ1/r∑_i=1^r D(σ‖ |i⟩⟨ i| )= ∞.

Then, by recalling <ref>, we conclude that 
F^♭_r(ρ_1, …, ρ_r) =0.

Continuity: 
The multivariate log-Euclidean fidelity F_♭ given in Definition <ref> is continuous at tuples (ρ_1,…, ρ_r) of invertible states.
This follows from the fact that the matrix exponential and logarithmic functions are continuous at positive definite operators, which implies that the multivariate log-Euclidean fidelity is a composition of continuous functions. 




 §.§ Proof of Proposition <ref> (Inequalities relating average k-wise log-Euclidean fidelities)




Let S_k-1(i_1,…,i_k) denote all size-(k-1) subsets of
i_1,…,i_k, of which there are k of them. Note that each symbol
i_j appears k-1 times in each subset of S_k-1(i_1,…,i_k).
Then consider that

    1/k∑_t∈ S_k-1(i_1,…,i_k)F_k-1^♭
    (ρ_t(1),…,ρ_t(k-1)) 
         =1/k∑_t∈ S_k-1(i_1,…,i_k)exp(
    -inf_ω∈𝒟1/k-1∑_j∈[  k-1]
    D(ω‖ρ_t(j)))  
       ≥exp(  -1/k∑_t∈ S_k-1(i_1,…,i_k)
    inf_ω∈𝒟1/k-1∑_j∈[  k-1]
    D(ω‖ρ_t(j)))  
       ≥exp(  -inf_ω∈𝒟1/k(  k-1)
    ∑_t∈ S_k-1(i_1,…,i_k)∑_j∈[  k-1]
    D(ω‖ρ_t(j)))  
         =exp(  -inf_ω∈𝒟1/k∑_ℓ∈{i_1,…,i_k}D(ω‖ρ_ℓ))  
         =F_k^♭(ρ_i_1,…,ρ_i_k),

where the first inequality follows from the convexity of the exponential
function and the second from the fact that

    1/k∑_t∈ S_k-1(i_1,…,i_k)inf_ω∈𝒟
    1/k-1∑_j∈[  k-1]  D(ω‖ρ_t(j))≤inf_ω∈𝒟1/k(  k-1)  ∑_t∈
    S_k-1(i_1,…,i_k)∑_j∈[  k-1]  D(ω‖ρ_t(j)).

The penultimate equality follows because

    1/k(  k-1)
    ∑_t∈ S_k-1(i_1,…,i_k)∑_j∈[  k-1]
    D(ω‖ρ_t(j))  
      =1/k∑_ℓ∈{i_1,…,i_k}D(ω‖ρ_ℓ),

as a generalization of the justification behind (<ref>).
Now consider that

    1/rk∑_i_1<⋯<i_kF_k^♭(ρ_i_1
    ,…,ρ_i_k)
       ≤1/rk∑_i_1<⋯<i_k1/k∑_t∈
    S_k-1(i_1,…,i_k)F_k-1^♭(ρ_t(1),…,ρ
    _t(k-1))
         =k-1!r-k!/r!∑_i_1<⋯<i_k∑_t∈ S_k-1
    (i_1,…,i_k)F_k-1^♭(ρ_t(1),…,ρ_t(k-1))
         =k-1!r-k!/r!(  r-k+1)  ∑_i_1<⋯<i_k-1
    F_k-1^♭(ρ_i_1,…,ρ_i_k-1)
         =1/rk-1∑_i_1<⋯<i_k-1F_k-1^♭
    (ρ_i_1,…,ρ_i_k-1).

The reasoning for these steps is precisely the same as that given for (<ref>)–(<ref>).



