Quantum metrology of rotations with mixed spin states
Eduardo Serrano Ensástiga John Martin Chryssomalis Chryssomalakos
 




ed.ensastiga@uliege.be
Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège


B-4000 Liège, Belgium


jmartin@uliege.be
Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège


B-4000 Liège, Belgium


chryss@nucleares.unam.mx
Instituto de Ciencias Nucleares
 Universidad Nacional Autónoma de México PO Box 70-543, 04510, CDMX, México




The efficiency of a quantum metrology protocol can be considerably reduced by the interaction of a quantum system with its environment, resulting in a loss of purity and, consequently, a mixed state for the probing system. In this paper we examine the potential of mixed spin-j states to achieve sensitivity comparable, and even equal, to that of pure states in the measurement of infinitesimal rotations about arbitrary axes. We introduce the concept of mixed optimal quantum rotosensors based on a maximization of the Fisher quantum information and show that it is related to the notion of anticoherence of spin states and its generalization to subspaces. We present several examples of anticoherent subspaces and their associated mixed optimal quantum rotosensors. We also show that the latter maximize negativity for specific bipartitions, reaching the same maximum value as pure states. These results elucidate the interplay between quantum metrology of rotations, anticoherence and entanglement in the framework of mixed spin states.


Quantum metrology of rotations with mixed spin states
    Chryssomalis Chryssomalakos
    April 24, 2024
=====================================================





§ INTRODUCTION



The development of modern measurement techniques is closely tied to advances in all fields of science. The unprecedented manipulation and control of quantum systems is now opening up opportunities for quantum-enhanced metrology, which has become a major field of study. A fundamental example is the precise detection of rotation, first developed for inertial navigation and now used to detect minute magnetic fields through the spin of particles. The study of the measurement of a rotation using a quantum system as a probe reveals the existence of certain states where the estimate can be improved by quantum effects <cit.>. The main tool for estimating the sensitivity of the state of a quantum system is the quantum Cramér-Rao bound  (QCRB) <cit.>. 
If we have no a priori knowledge of the direction of the axis of rotation, we need to find a quantum state that is very sensitive whatever the axis. This corresponds exactly to the notion of an optimal quantum rotosensor (OQR), which has been extensively studied for pure spin-j states in recent years <cit.>. In particular, it was discovered that the OQRs for infinitesimal rotations are anticoherent (AC), a property of quantum states first introduced by Zimba <cit.> and studied in many works, see e.g. <cit.>. An anticoherent state is characterised by the expectation values of the first moments of the angular momentum being homogeneous, i.e. with no preferred direction <cit.>. In this sense, they are the opposite of spin-coherent states <cit.>.

Under real conditions, a quantum system interacts with its environment and its state becomes mixed, i.e. a statistical mixture of pure states. In this scenario, it is not clear that the conditions required for a mixed state to be an OQR are the same as those defining pure OQRs. 
While the characterization of the most sensitive mixed states with a fixed spectrum for an infinitesimal rotation over a known axis, and in general for an infinitesimal transformation given by any generator of the Lie algebra, has been solved in Ref. <cit.>, it remains open for an unknown axis. Additionally, the authors of Ref. <cit.> have characterized the mixed states with the best metrological power via the purification of the state. In this work, precisely, we derive conditions for obtaining the ultimate detection sensitivity of an infinitesimal rotation about an unknown axis with mixed states. We show how to write these conditions in terms of anticoherent subspaces, defined as vector spaces of anticoherent states, which were first introduced and studied by Pereira and Paul-Paddock <cit.>. It turns out that these AC subspaces also have applications in a holonomic quantum computation approach resilient to rotational noise <cit.>.

Multipartite entanglement is a resource to achieve the ultimate sensitivity of measurement with quantum states <cit.>, also known as the Heisenberg limit. For pure states, 
a faithful measure of bipartite entanglement <cit.> with respect to a bipartition of the system is given by the entanglement negativity <cit.>. For mixed states, the negativity is only a witness of entanglement, except for qubit-qubit and qubit-qutrit systems <cit.>. Negativity is easy to compute and is at the core of the notion of distillable entanglement <cit.>. Therefore, in this work we have chosen to study the entanglement of mixed OQRs based on this quantity.

For an individual spin-j-system, the notion of entanglement has no obvious meaning. It is only by interpreting spin-j states as symmetric N=2j-qubit states that the notion becomes relevant and meaningful. 
In this correspondence, the quantum property of anticoherence is related to negativity since pure AC states maximise negativity for a certain bipartition of the N=2j-qubit Hilbert space <cit.>. In this paper, we also study the possible connections between negativity and anticoherence for mixed states. 

The paper is organized as follows: After reviewing the basic concepts in Sec. <ref>, we derive the conditions for pure and mixed states to be OQR in Sec. <ref>. We study the question of the existence of mixed OQRs for small spin and present illustrative examples in Secs. <ref>–<ref>, respectively. The discussion concerning the negativity of mixed OQR states is presented in Sec. <ref>. Finally, we summarize our results in Sec. <ref>.





§ CONCEPTS AND MATHEMATICAL FRAMEWORK





 §.§ Fidelity and Quantum Fisher Information for pure and mixed states


Let us consider a general quantum state |Ψ⟩ in a Hilbert space  where the elements of the rotation group ∈SO(3) are represented by (𝐧 , η  ) = e^-iη 𝐧·𝐉, where 𝐧 = (n_x , n_y , n_z) and η are the axis-angle parameters, and 𝐉 = (J_x , J_y , J_z) is an angular momentum operator. The Hilbert space  has a unique decomposition into subspaces that transform independently by a SU(2) transformation (the double cover of SO(3)) through one of its irreducible representations (irreps), labeled by j=0,1/2,1, …

    = ⊕_j ^(j)  ,


where the superindex of each subspace ^(j) indicates the (2j+1)-dimensional irrep (spin j). For simplicity, we restrict our search to states living in a single subspace ^(j), later interpreted as the symmetric sector of the Hilbert space of N=2j qubits. As we explain in the next subsection, the sensitivity of a state to detect infinitesimal rotations increases with a factor proportional to j(j+1), suggesting that it is convenient to work with states belonging to the ^(j) subspace associated with the  the highest-dimensional irrep. A similar result has been proved for symmetric multiqudit pure-states <cit.>. The fidelity between a state |Ψ⟩∈^(j) and its image under the rotation = ^(j) ( 𝐧, η ) reads

    F ( |Ψ⟩, |Ψ⟩) ≡ 
        |⟨Ψ||Ψ⟩ |^2   .

Our main goal is to find the states that minimize F(|Ψ⟩ , |Ψ⟩) averaged over all possible axes of rotation

    F (η, |Ψ⟩) ≡1/4π∫ 
        F(|Ψ⟩ ,  (𝐧 ,η ) |Ψ⟩) 
        d 𝐧  ,

for an infinitesimal angle of rotation η≪ 1. For mixed states ρ in the Hilbert-Schmidt space of operators (^(j)), we use the Uhlmann-Jozsa fidelity F(ρ, σ) <cit.>, which can be related to the Bures distance <cit.>,

    F(ρ ,  ) ≡√(√(ρ) √(ρ))  .

Like any well-defined fidelity function for mixed states <cit.>, the Uhlmann-Jozsa fidelity coincides with the fidelity (<ref>) when the states are pure. For infinitesimal angles η≪ 1 about an axis of rotation 𝐧, one must calculate the Quantum Fisher information (QFI) I(𝐧, ρ), given by the Bures distance between ρ and the infinitesimally close state ρ_= (𝐧,η) ρ (𝐧,η)^†, which reads <cit.>

    I(𝐧 , ρ )  ≡. 1/4∂^2 F(ρ,ρ_)/∂η^2|_η=0 = lim_η→ 01- F(ρ , ρ_)/2η^2.







 §.§ Multipole operators



The multipole (or polarization) operators T_LM, with L=0, …, 2j and M=-L, … ,L <cit.>, form an orthonormal basis with respect to the Hilbert-Schmidt inner product
( T_L_1 M_1^† T_L_2 M_2 ) = δ_L_1 L_2δ_M_1 M_2. Hence, a mixed state ρ expanded in this basis reads

    ρ = ∑_L=0^2j∑_M=-L^L ρ_LM T_LM = ρ_0 + ∑_L=1^2j∑_M=-L^L ρ_LM T_LM  ,

where ρ_LM =  (ρ T^†_LM) and ρ_0 = (2j+1)^-11 is the maximally mixed state in ^(j). Since T_L M^† = (-1)^M T_L  -M <cit.>, the hermiticity of the density matrix enforces that ρ^*_LM = (-1)^M ρ_L -M. Additionally, the T_LM's are traceless operators except for T_00 = 1/√(2j+1). The multipole operators can be constructed as products of the angular momentum operators J_± = J_x ± i J_y and J_z (see e.g. Eq. (4), p. 44 of Ref. <cit.>). For example, 

    T_10∝ J_z ,    
      T_20∝𝐉^2 - 3 J_z^2 ,
    
           T_1± 1∝ J_± ,     T_2± 1∝ J_z J_± + J_± J_z ,    
    T_2± 2∝ J_±^2 .

More details on multipole operators can be found in Refs. <cit.>. 




 §.§ Anticoherent states and subspaces



We begin with the definition of anticoherent states as given in Ref. <cit.>, extended here to mixed states.


  
*Definition. A state ρ∈ (^(j)), either pure or mixed, is t-anticoherent if 
(ρ J_𝐧^K) is independent of 𝐧 for K=1 , 2, …, t, where J_𝐧≡𝐧·𝐉. 
For the sake of brevity in this paper, we will use the abbreviation t-AC for t-anticoherent.

The condition of t-anticoherence is equivalent to a zero expectation value of the multipole operators, i.e. ⟨ T_L M⟩ =  (ρ   T_L M) = 0, for L=1, … ,t and M= -L , … , L <cit.>. In particular, the 1-anticoherence condition for pure states reduces to ⟨ψ| J_a |ψ⟩ = 0 for a=x,y,z <cit.>. The t-anticoherence of a state ρ can be quantified with the functions [The functions (<ref>) define measures of anticoherence for pure states, see Ref. <cit.> for more details.]

    𝒜_t (ρ) = t+1/t[ 1-  (ρ_t^2) ]

for integer t⩾ 1, where ρ_t= _N-t(ρ) is the reduced density matrix obtained after tracing out N-t spin-1/2 among the N making up the N/2=j spin. These functions range from 0 for spin-coherent states to 1 for t-AC states <cit.>. In addition, a state ρ is t-AC if and only if 𝒜_K(ρ)=1 for K=1,2,…, t <cit.>. By definition, t-anticoherence implies t'-anticoherence for t' < t. While for pure states there are no t-AC states for certain values of t ⩽ 2j <cit.>, one can always find, at least sufficiently close to the maximally mixed state, states which are AC to any order t. We give a proof of the last statement in Appendix <ref>. 

The notion of an anticoherent state was generalized in Ref. <cit.> to subspaces as follows.


  
*Definition. A k-dimensional subspace V of ^(j)  
is t-AC if any state |ψ⟩∈ V is t-AC. Such a subspace will be referred to as (j,k,t) AC subspace.

A necessary, but not sufficient, condition for a basis ℬ= {|ψ_m⟩}_m to span a t-AC subspace V is that its vectors |ψ_m⟩ are t-AC states. Orthonormal bases made of t-AC states for the whole Hilbert space have been studied in Ref. <cit.>.

There are equivalent characterizations for (j,k,t) AC subspaces as specified in the following lemma.
 

     Let V be a subspace of dimension k of ^(j). The following statements are equivalent:
     
         
  * V is a (j,k,t) AC subspace.
         
  * A (non-necessarily orthogonal) basis ℬ={|ψ_m⟩}_m=1^k of V satisfies the following conditions:
         
    ⟨ψ_m_1| T_L M|ψ_m_2⟩ = 0   ,

         for any
         m_1,   m_2 = 1, … , k, L=1,… , t and M= -L , … , L.
         
  * Any pair of states |ψ_1⟩, |ψ_2⟩∈ V satisfies the following conditions:
         
    ⟨ψ_1| T_L M|ψ_2⟩ = 0   ,

         for any L=1,… , t and M= -L , … , L.
     
 
In particular, a (j,k,1) AC subspace fulfills

    ⟨ψ_1| J_a |ψ_2⟩ = 0

for any pair of states |ψ_1⟩, |ψ_2⟩∈ V and a=x,y,z (note that |ψ_1⟩=|ψ_2⟩ is not excluded). One method of searching for AC subspaces, which is well suited to numerical approaches, is to define an objective function which takes as arguments vectors and which is zero if and only if these vectors span a (j,k,t) AC subspace for certain values of j,k,t. To this end, we first introduce, for integers L,M, the functions

    G_L M(ℬ) = ∑_1 = m_1 ≤ m_2^k | ⟨ψ_m_1| T_L M|ψ_m_2⟩ |^2   ,

where ℬ = {|ψ_m⟩}_m=1^k is now assumed to be an orthonormal basis of some subspace V of dimension k. The last equation can be re-written as

    
    G_L M(Π) = ( Π  T_L M Π T_L M^†)

    with

    Π = ∑_|ψ_m⟩∈ℬ|ψ_m⟩⟨ψ_m|.

The positive function G_LM(Π) is zero if and only if all the terms in the sum (<ref>), which are all positive, vanish simultaneously. Therefore, the  objective function 
    G_t(Π) = ∑_L=1^t ∑_M=-L^L G_L M(Π)   .
   
is zero if and only if ⟨ψ_m_1| T_LM|ψ_m_2⟩ = 0 for any L=1,… , t and M=-L, … , L and any m_1 , m_2 = 1, … , k, which corresponds to the second equivalent statement of a subspace being a (j,k,t) AC subspace mentioned in Lemma <ref>. In this work, most of the AC subspaces presented in Sections <ref>-<ref> were found by searching for zeros of the objective function G_t(Π) by numerical optimization.




 §.§ Negativity of pure AC states




The negativity of a state ρ of a bipartite system AB with Hilbert space ^A ⊗^B is defined as <cit.>

    (ρ) = 
     ∑_k |_k| -_k/2  ,


    |ψ⟩ = 1/√(t+1)∑_α=1^t+1|ϕ^A_α⟩⊗|ϕ^B_α⟩  ,

with {|ϕ_α^A⟩} an orthonormal basis of ^A, and {|ϕ_α^B⟩} an orthonormal subset of ^B. An immediate consequence of (<ref>) is that _t of a pure t-AC state |ψ_t-AC⟩ is maximal and equal to <cit.>

    _t(|ψ_t-AC⟩⟨ψ_t-AC|) = t/2  .

This result is no longer valid for mixed states, as shown by the example of the maximally mixed state ρ_0, which is t-AC to any order t but for which _t(ρ_0)=0 because ρ_0 is fully separable. Consequently, for arbitrary mixed states, there is a priori no relationship between negativity and anticoherence. However, in Sec. <ref>, we will uncover a connection between t-AC subspaces and mixed states with maximal negativity over certain bipartitions.




§ OPTIMAL QUANTUM ROTOSENSORS FOR INFINITESIMAL ROTATIONS



Now, let us consider the problem of optimal quantum rotosensors (OQR) <cit.> for infinitesimal rotations, i.e. for rotation angles η≪ 1. 
The best possible estimate of η, i.e., the minimum variance Var(η̂) of the estimator η̂ of the parameter η, is given by the QCRB <cit.>, M Var [η̂] ⩾ 1/I( 𝐧 , ρ ), which is written here in terms of the QFI (<ref>) and achievable by optimizing over all Positive Operator Valued Measures. Here, M is the number of times the measurement is repeated. As the axis of rotation is unknown, we seek to minimize the QCRB averaged over all rotation axes

    M Var [η̂]  = 
     M/4π∫Var [η̂]   d 𝐧⩾I(𝐧,ρ)^-1  .

The  of the inequality is lower bounded by Jensen's inequality <cit.>

    I(𝐧,ρ)^-1⩾1/I(𝐧,ρ)  ,
 
which is tight when the function I(𝐧 , ρ) is independent of the variable of integration 𝐧. Therefore, our goal is to find a state ρ that minimizes and saturates the QCRB, which is the case when ρ satisfies the following two conditions:

       1. ρ maximizes  I(𝐧 , ρ)  

       2. I(𝐧 , ρ) is independent of 𝐧 


As we will show in the following, these two conditions lead naturally to the notion of AC states and subspaces.





 §.§ Pure states



For a pure state ρ_ψ≡|ψ⟩⟨ψ|, the Uhlmann-Jozsa fidelity (<ref>) reduces to (<ref>), and the QFI (<ref>) is just the variance of the angular momentum component along the 𝐧 direction

    I(𝐧 , ρ )  = ⟨ J_𝐧^2 ⟩ - ⟨ J_𝐧⟩^2    .

The averaged QFI in all directions is then equal to

    I(𝐧 , ρ) 
     = 
     1/3∑_a=x,y,z( ⟨ J_a^2 ⟩ - ⟨ J_a ⟩^2 )
     = 1/3(
    j(j+1) - ⟨𝐉⟩^2
    )

where we used the identity

    ∫_S^2 n_a n_b   d 𝐧 = 4π/3δ_ab

to evaluate the integral defining the average. The conditions (<ref>)–(<ref>) for a pure state |ψ⟩ are fulfilled whenever ⟨ J_𝐧⟩=0 and ⟨ J_𝐧^2 ⟩ = j(j+1)/3 for any 𝐧, that is when |ψ⟩ is a 2-AC state <cit.>. In this case, the optimal lower bound for the averaged QCRB reads

    Var [η̂] ⩾3/Mj(j+1)  ,

that is inversely proportional to j(j+1). This means that the higher the spin quantum number, the better the optimal estimate that can be obtained.





 §.§ Mixed states



Now, we consider mixed states of a spin-j, ρ∈ (^(j)). Let {_m , |ψ_m⟩}_m=1^d denote the eigenvalues and the corresponding eigenvectors of ρ with d=2j+1 the dimension of ^(j). Without loss of generality, we label by m=1, …, k the eigenvectors with the non-zero eigenvalues, which span the image (ρ) of ρ, with ( (ρ))=k. When the state ρ is not full rank (k< d), the states |ψ_m⟩ with m=k+1,…, d span the kernel of ρ, (ρ). Both (ρ) and (ρ) are complementary subspaces in (^(j)). We denote by P_ρ and P_ρ^⊥ the projectors onto (ρ) and (ρ) =  (ρ)^⊥, respectively,

    P_ρ = ∑_m=1^k |ψ_m⟩⟨ψ_m|  ,    
        P_ρ^⊥ = 1 - P_ρ = ∑_m=k+1^d |ψ_m⟩⟨ψ_m|  .

The QFI (<ref>), in terms of the Uhlmann-Jozsa fidelity (<ref>), is now given for an infinitesimal rotation about the axis 𝐧 by <cit.>

    I(𝐧,ρ) = 1/2∑_l,m=1^d p_lm |⟨ψ_l| J_𝐧|ψ_m⟩|^2

with

    p_lm = {[                     0           if _m=_l =0; ( _m - _l )^2/_m + _l             otherwise ]. .
 
We now rewrite Eq. (<ref>) as follows

    I (𝐧, ρ) =    1/2∑_l,m =1^k p_lm  |⟨ψ_l| J_𝐧|ψ_m⟩|^2 +   (ρ J_𝐧 P_ρ^⊥ J_𝐧) 
        
    
        =    1/2∑_l,m =1^k p_lm  |⟨ψ_l| J_𝐧|ψ_m⟩|^2 + ( ρ J_𝐧^2 )
        
       
        - 1/2∑_l,m =1^k ( _l + _m )  |⟨ψ_l| J_𝐧|ψ_m⟩|^2 
        
    
        =    ( ρ J_𝐧^2 )
        -2 ∑_l,m=1^k _l  _m/_l + _m |⟨ψ_l| J_𝐧|ψ_m⟩|^2
          .

It is easy to observe from the last equation that Eq. (<ref>) reduces to Eq. (<ref>) for the case of pure states. The QFI for mixed state (<ref>) averaged over all directions is then given by 

    I(𝐧 ,ρ) = 1/3( 
        j(j+1) - 2 ∑_l,m=1^k _l  _m/_l + _m |⟨ψ_l|𝐉|ψ_m⟩|^2
        )   .

Hence, the first condition (<ref>) defining a mixed OQR reduces to satisfying ⟨ψ_l| J_a |ψ_m⟩=0 for a=x,y,z and any pair of states in (ρ), which is equivalent to (ρ) being a 1-AC subspace. This condition implies that ρ is a 1-AC state, but it is stronger. Once the first condition (<ref>) is met, the second condition (<ref>) reduces to (ρ J_𝐧^2) being independent of 𝐧, , ρ being a 2-AC state. Remarkably, the mixed states satisfying these two conditions, which we now call mixed OQRs, are as good estimators as the pure OQRs because they give the same lower bound on the averaged QCRB (<ref>). We can therefore conclude that a less pure spin state is not necessarily a worse estimator. 

We summarize the conditions we have found for a state to be OQR in Table <ref>, both for the pure case and for the mixed case. The condition (<ref>) can always be fulfilled for any value of the spin quantum number by sufficiently mixing a state with the maximally mixed state and thus decreasing its purity P= (ρ^2) (see proof in Appendix <ref>). On the other hand, the first condition is all the more difficult to fulfill the higher the rank of the state ρ is. There is therefore a certain tension between the two conditions. Since (ρ) is of dimension equal to the rank k of ρ, condition (<ref>) will be harder to meet for highly mixed states. The rank k of a state is in fact directly related to the minimal purity it can reach, since (ρ^2) = ∑_m=1^k λ_m^2⩾ 1/ k. For a given spin quantum number j, the maximum dimension k_max of a 1-AC subspace sets in turn a lower bound on the purity attainable by mixed OQRs. We have determined the value of k_max by numerically searching for zeros of the objective function (<ref>) for j⩽ 17/2 (see discussion in Sec. <ref> and Fig. <ref>, where we show the dependence of k_max on j). 

A general approach to identify mixed OQRs, which we will pursue in the next section, is as follows. The first step is to find 1-AC subspaces V which can be used to construct mixed states satisfying the condition (<ref>), followed by the search for an appropriate mixture of states within V that will satisfy the condition (<ref>). Another, more direct approach is to find 2-AC subspaces from which a mixture of basis states will automatically define a mixed OQR. In both cases, it is essential to search for and identify AC subspaces.   







§ MIXED OQRS FOR SMALL SPIN




While pure spin states defining OQRs have been extensively studied in the literature <cit.>, we present here several examples of mixed OQRs, first performing a systematic search for small spin quantum numbers.



 §.§ Spin 1


The only 1-AC spin-1 states are given by ^(1)|1,0⟩ <cit.>, where ^(1) = ^(1) (α,β, γ) is the rotation matrix in the j=1-irrep parametrized with the Euler angles. First, let us search for a (1,2,1) AC subspace spanned by an orthonormal basis {|ψ_1⟩, |ψ_2⟩}. The two states |ψ_a⟩ must be obtained from the state |1,0⟩ by rotations, since they must be 1-AC. Moreover, since the order of anticoherence is independent of the orientation of the subspace, we can reorient the basis so that

    |ψ_1⟩=    |1,0⟩ = (0,1,0)   , 
        
    |ψ_2⟩ =    ^(1)(α, β, γ) |1,0⟩ = ( -e^-i αsinβ√(2),cosβ ,e^i αsinβ√(2))

where the vectors indicate the components of the states in the |j,m⟩ basis, with m=j,…, -j. By direct calculation, we obtain ⟨ψ_2 | ψ_1 ⟩ = cosβ and ⟨ψ_2| J_+|ψ_1⟩ = e^-i αsinβ, which cannot be made simultaneously zero. 
Consequently, there are no (1,k,1) AC subspaces with k>1. On the other hand, the only 2-AC state of spin-1 is the maximally mixed state ρ_0 = 1/3. Therefore, no state, pure or mixed, of a spin 1 can be an OQR. 



 §.§ Spin 3/2


As for spin 1, there is only one 1-AC state, up to rotation, given by the GHZ state |ψ_GHZ⟩= (1,0,0,1 )/√(2) <cit.>.  We can start searching for a (3/2,2,1) AC subspace spanned by a reoriented orthonormal basis such that

    |ψ_1⟩=   |ψ_GHZ⟩  ,   |ψ_2⟩ =  ^(3/2) (α , β , γ)|ψ_1⟩  .

The orthogonality condition ⟨ψ_2 | ψ_1 ⟩ = 0 has three solutions: 1) β=0 and α + γ = (2n_1+1)π/3, 2) β=π and α - γ = 2n_1π/3, and 3) = π (2n_1+2n_2+1)/6 and γ = π (2n_1 - 2n_2+1)/6 for some integer numbers n_1 and n_2. The three solutions have the same expression |ψ_2⟩ = (a,b,c,-a) for some complex numbers a,b,c with a ≠ 0. By direct calculation, we get that ⟨ψ_2| J_z |ψ_1⟩≠ 0 and we can conclude that there are no (3/2,2,1) AC subspaces, and thus no mixed OQRs.

In order to study the effect on the QCRB of a deviation from the condition (<ref>) while satisfying the condition (<ref>), let us consider a family of 2-AC mixed spin-3/2 states in the purity range 1/4 ⩽ (ρ^2) ⩽ 1/2, as derived in Appendix <ref>. In particular, we show there that the 2-AC mixed state with the largest purity P=1/2 is the rank-two state ρ_2-AC= (ρ_ψ_1+ρ_ψ_2)/2 with

    |ψ_1⟩ = 1/√(2) (1,0,1,0)   ,
          |ψ_2⟩ = 1/√(2) (0,-1,0,1).

As we have already shown above, none of these 2-AC states have an image (ρ) spanning a 1-AC subspace.

Despite the absence of OQRs, we can compare the respective QCRBs (<ref>) reached by mixed and pure states. In particular, we have that

    I(𝐧,ρ_2-AC)^-1_= 1 ⩾ I(𝐧,ρ_GHZ)^-1_≈ 0.9 ⩾ 3/j(j+1)_= 0.8,

meaning that the pure GHZ state is only slightly better than the 2-AC mixed state ρ_2-AC with purity P=1/2.




 §.§ Spin 2



This is the smallest spin quantum number for which there is a two-dimensional 1-AC subspace. Such a subspace is spanned by the orthogonal pure states

    |ψ_1⟩ = 1/2( 1 , 0, √(2)  i , 0 ,1), |ψ_2⟩ = 1/2( 1 , 0, -√(2)  i , 0 ,1) .

Since both states |ψ_k⟩ are 2-AC <cit.>, so is any mixture ρ = _1 |ψ_1⟩⟨ψ_1|+ (1-_1) |ψ_2⟩⟨ψ_2|, which therefore defines a mixed OQR. By varying the eigenvalues of ρ through _1, we can obtain mixed OQRs with purity (ρ^2) ∈ [1/2 , 1]. 
We prove in Appendix <ref> that the (2,2,1) AC subspace spanned by the states (<ref>) is unique up to a rotation. As a result, mixed OQRs are all of the above form. For mixed states of higher rank and smaller purity, the QCRB increases as the purity decreases. We can confirm this trend by considering the uniparametric family of mixed states

    ρ(ξ) = {[                 ξρ_ψ_1 + (1-ξ)ρ_ψ_2 ,                         ξ∈[ 1/2 , 1 ]; (5ξ-1/3)(ρ_ψ_1 + ρ_ψ_2) + (1-2ξ/3)1 ,                        ξ∈[ 1/5 , 1/2] ]. ,

where the states |ψ_k⟩ are given in Eq. (<ref>). For ξ∈[ 1/2 , 1 ], ρ(ξ) is an OQR, but not for 
ξ∈[ 1/5 , 1/2]. The purity of ρ(ξ) increases monotonically with ξ as

    P(ρ(ξ)) = {[          1+  2 ξ(ξ-1)         ξ∈[ 1/2 , 1 ]; 1/3 [ 2 ξ (5 ξ-2)+1 ]        ξ∈[ 1/5 , 1/2] ].  .

We also find that the QFI for ρ(ξ) is independent of 𝐧 and its inverse is equal to

    I(𝐧,ρ(ξ))^-1 = {[                 1/2       ξ∈[ 1/2 , 1 ]; 3 (ξ+1)/4 (1-5 ξ)^2      ξ∈[ 1/5 , 1/2] ].   .

In Fig. <ref>, we plot I(𝐧,ρ(ξ))^-1 as a function of P(ρ(ξ)). We can see that, for P <1/2, the inverse of the QFI starts to increase until infinity is reached for the maximally mixed state (ξ=1/5). Therefore, the QCRB increases (and thus the metrological power of the state decreases) as the purity decreases and is below 1/2, as announced above.










§ ILLUSTRATIVE EXAMPLES OF AC SUBSPACES AND MIXED OQRS




We now present several examples of AC subspaces found numerically by using the objective function (<ref>) and, subsequently, explore the possibility to get a mixed OQR. 



 §.§ 1-AC subspaces




  §.§.§ Lg AC subspaces



There are several (5/2,2,1) AC subspaces. A first example is the space V spanned by the states

    |ψ_1⟩=   1/√(8)( √(3),0,0,0,√(5),0 )   , 
        
    |ψ_2⟩=   1/√(8)( 0,-√(5),0,0,0,√(3))   .

The candidates for mixed OQRs are given by a mixture of a generic basis of V, ρ= ∑_k=1^2_k |ψ̃_k⟩⟨ψ̃_k|, with 

    |ψ̃_1⟩ =    cos( θ/2) |ψ_1⟩ + 
    sin( θ/2) e^i ϕ|ψ_2⟩  , 
    
    |ψ̃_2⟩ =    sin( θ/2) |ψ_1⟩ -
    cos( θ/2) e^i ϕ|ψ_2⟩  .

The function 𝒜_2(ρ) for this family reads

    𝒜_2(ρ) = 
    9/128[ 
    (_1 - λ_2)^2 cos (2 θ ) + 4 λ_1 λ_2 +13
    ]
      ,

with maximum 𝒜_2(ρ) = 63/64 ≈ 0.984375 achieved for _1 = _2 = 1/2 and any (θ , ϕ ) values, or for θ=0 and any weights (_1 , _2) and phase ϕ. 
We can see that these mixed states are almost 2-AC, but do not fulfill condition (<ref>). Hence, they are not a mixed OQR.

A second example of a (5/2,2,1) AC subspace is the vector space V spanned by the states

    |ψ_1⟩ =   1/4( 
        0, α,0,- β,0, γ)   ,
    
    |ψ_2⟩ =   1/4( 
     γ,0, β,0, α,0
    ).

with α=√(20-5 √(7)), β=√(10 (√(7)-2)) and γ=√(16-5 √(7)). Again, we search for a 2-AC mixed state ρ given by a mixture of a generic basis of the AC subspace V. We get that the maximum of 𝒜_2 (ρ) is equal to

    𝒜_2(ρ) =     
    3/32(7 √(7)-8) ≈ 0.9863   ,

achieved for the equally weighted mixture _1=_2=1/2 for any basis of V. Once again, the corresponding mixed state is not 2-AC, so is not a mixed OQR. 

Although there are mixed 2-AC states (see Appendix <ref>), we observe numerically (see also Fig. <ref>) that none of them has an image (ρ) forming a 1-AC subspace and, consequently, there are no OQRs of spin 5/2.



  §.§.§ Lg AC subspace



The first 3-dimensional 1-AC subspace V emerges for j=3, and is spanned by the states

    |ψ_1⟩ = √(2/5)| 3,3⟩ +
        √(3/5)| 3,-2 ⟩  , 
       |ψ_2⟩ = -√(3/5)| 3,2 ⟩ +
        √(2/5)| 3,-3 ⟩  , 
       |ψ_3⟩ = | 3,0⟩  .

The measure of 2-anticoherence 𝒜_2 of a general coherent superposition |ψ⟩=a|ψ_1⟩+b|ψ_2⟩+c|ψ_3⟩, with  a,b,c∈ℂ and |a|^2+|b|^2+|c|^2=1, is given by

    𝒜_2(ρ_ψ)=3/25[8-4 | a| ^2 | b| ^2-| c| ^4+4 Re(a b (c^*)^2)]   .

The maximum of 𝒜_2(ρ_ψ) in Eq. (<ref>) under the normalization constraint is equal to 24/25 [The maximum is achieved e.g. for a=b=1/2 and c=1/√(2).]. Hence, no pure state |ψ⟩∈ V is 2-AC. On the other hand, the statistical mixture of the states (<ref>) with weights _k yields a ρ with 𝒜_2(ρ)= 3(2-_3)(4+3_3)/25. This then shows that for _3=1/3, ρ is 2-AC and therefore a mixed OQR. This OQR, explicitely given by

    ρ = _1 ρ_ψ_1 + (2/3-_1 ) ρ_ψ_2 +
    1/3ρ_ψ_3,

is the first example of a mixed OQR without any pure OQR in its image ( ρ).




 §.§ 2-AC subspaces


The (j,k,2) AC subspaces V are particularly interesting because any convex combination of any orthonormal basis of V yields a mixed OQR. In this subsection, we give some examples of 2-AC subspaces for small j values.




  §.§.§ Lg AC subspace








The first instance of a 2-AC subspace is found for j=7/2. An example is the space spanned by the states

    |ψ_1⟩ =   √(3/10) | 7/2,7/2⟩ +
        √(7/10) | 7/2,-3/2⟩  , 
        
    |ψ_2⟩ =   √(7/10) | 7/2,3/2⟩ -
        √(3/10) | 7/2,-7/2⟩  .






  §.§.§ Lg AC subspace



For spins larger than j=7/2, we did not numerically find other (j,2,2) AC subspaces up to the value j=5. One example is spanned by the states

    |ψ_1⟩=   1/√(7)( 
        √(2)| 5 , 5 ⟩ + √(5)| 5 , -2 ⟩)   , 
        
    |ψ_2⟩ =   1/√(7)( 
        -√(5)| 5 , 2 ⟩ + √(2)| 5 , -5 ⟩).









 §.§ AC subspaces of arbitrary dimension






Previously, we used the 1- and 2-AC subspaces to create families of mixed OQRs. 
Here, we take a different approach and show that one can find AC subspaces of order t=1 and 2 of any dimension k with sufficiently large spin using techniques similar to those outlined in Ref. <cit.>. 
For a spin-j system with j a half-integer number, the subspace spanned by the vectors

    |ψ_a⟩= 1/√(2)( 
            0_2a , 1 , 0_2j-4a-1 , 1 , 0_2a)

 with a=0 , 1 , …, ⌊j-1/2⌋ is a (j,k,1) AC subspace. Here, 0_n is a string of n zeroes in the vector, with the convention that 0_0 means we do not add any extra zeroes. In the case where j is an integer and 2j-3-4 ⌊j-1/2⌋ >0, we can add the following extra state to increase the dimension of the 1-AC subspace

    ( 
    0_j , 1 , 0_j
    )   .

The dimension of these generic 1-AC subspaces, denoted by k_1, is given by

    k_1= {[          ⌊j-1/2⌋ +1 for half-integer  j;          ⌊j-1/2⌋ +2      for integer  j ].    ,

and grows linearly as j/2 (see e.g. Fig. <ref>). As an illustration, we write the vectors spanning the (j,k_1,1) AC subspaces for j=2,7/2,4 in Table <ref>. The zeroes in the states components ensure that ⟨ψ_a| T_1 ± 1|ψ_a'⟩∝⟨ψ_a| J_±|ψ_a'⟩ = 0 for any pair of states, where we used Eq. (<ref>). On the other hand, ⟨ψ_a| T_1 0|ψ_a'⟩∝⟨ψ_a| J_z|ψ_a'⟩ = 0 because |ψ_a⟩ = ( |j,m_a⟩ + |j,-m_a⟩)/√(2) with m_a ≠ m_a' for a ≠ a'.



In the same way, it is possible to construct a generic (j,k_2,2) AC subspace for j≥ 5 of dimension 

    k_2= min( ⌊j-κ-3/3⌋    , κ-2/3) +1    ,

with 

    κ = ⌈ j - √(j(j+1)/3) ⌉  ,

spanned by the states

    |ψ_a⟩ = ( 0_3a , α_a , 0_κ , β_a , 0_2j-3-2κ-6a , β_a , 0_κ , α_a , 0_3a) ,

with a=0, …, k_2-1. Again, the zeroes in the states components guarantee that ⟨ψ_a| T_μ|ψ_a'⟩ = 0 for every =1,2, any μ≠ 0, and any pair of states |ψ_a⟩,|ψ_a'⟩. This because the respective T_μ with μ≠ 0 contain one or two ladder operators (<ref>).
In addition, the zeroes of the state components guarantee that ⟨ψ_a| T_ 0|ψ_a'⟩=0 for different states. On the other hand, ⟨ψ_a| T_1 0|ψ_a⟩=0, with T_10∝ J_z, is fulfilled by the property that the coefficients α_a and β_a are associated with eigenstates |j, ± m⟩ whose magnetic quantum number is opposite. Finally, it is possible to satisfy the equality ⟨ψ_a| T_20|ψ_a⟩=0, where T_20∝ j(j+1)1 - 3 J_z^2 by choosing the parameters α_a and β_a, which we can considered as positive real numbers, such that 2α_a^2 + 2β_a^2 =1 (normalization of the state) and

    j(j+1) - 6 [  (j-3a)^2α_a^2 + (j-3a-1 -κ)^2β_a^2  ] = 0.

Indeed, the above equation always has a solution for the value of κ defined in Eq. (<ref>). The dimension of the (j,k_2,2) AC subspace (<ref>) grows asymptotically as k_2 ≈ (√(3)-1)j/(3√(3)). Some examples of (j,k,2) AC subspaces constructed using this approach are given in Table <ref>. 

In Fig. <ref> we show the dimension of the 1- and 2-AC subspaces given by our construction as a function of the spin quantum number j.  We note that both grow slowly but linearly in the asymptotic limit j≫ 1. We also plot the largest dimension k_max of a (j,k,t) AC subspace found numerically for t=1,2 and j ⩽ 17/2. The value k_max for 1-AC subspaces puts a lower bound on the purity of optimal OQRs, (ρ^2)⩾ 1/k_max. Our numerical results show that the Corollary 2 in Ref. <cit.>, stating that the largest 1-AC subspace of spin j states has dimension ⌊ j+1 ⌋, is not tight.






§ NEGATIVITY AND ANTICOHERENT SUBSPACES



The quantum improvement in metrological power is generally attributed to entanglement. To find out if this is the case for mixed OQRs, we discuss in this section the relationship between the negativity _t(ρ) of mixed states ρ constructed from t-AC subspaces, particularly when they define mixed OQRs. 
Despite the classical character of a mixture of states, we will show that a mixed state such that (ρ) is a (j,k,t) AC subspace maximizes all the negativities _t'(ρ) for t'⩽ t. In particular, the negativity of these mixed states with respect to the bipartitions t'|N-t' with t'⩽ t is equal to that of the most entangled pure states (<ref>). Because of the relevance of this result, we express it as a theorem whose proof is provided in Appendix <ref>:


    Let ρ∈ (^(j)) be a mixed state such that (ρ) is a (j,k,t) AC subspace. Then ρ, seen as a N-qubit symmetric state, maximizes the negativity with respect to the bipartition t|N-t with _t(ρ)= t/2.






An important consequence of (the proof of) Theorem <ref> is an upper bound on the dimension of (j,k,t) AC subspaces, given by

    k_max⩽⌊2j-t+1/t+1⌋  .

In particular,  for 1-AC subspaces, it gives a more strict bound k_max⩽⌊ j ⌋ than the one given in Corollary 2 of Ref. <cit.>. We plot this upper bound in Fig. <ref>. On the other hand, we plot the negativity for some (j,k,t) AC subspaces presented in the previous section in Fig. <ref>. We can observe the constant value of the negativity for the bipartitions t|N-t associated to the anticoherence order of the AC subspaces but not for the others which decrease when the purity decreases. In summary, mixed OQRs, which come from 1-AC subspaces, are maximally entangled states with respect to the negativity for the bipartition 1|N-1, despite their variable purity.



§ CONCLUSIONS




In this work, we established the necessary conditions, as outlined in Table <ref>, for a mixed spin-j state to achieve optimal sensing capabilities for infinitesimal rotations about arbitrary axes, as dictated by the QCRB averaged over all rotation axes (<ref>). While pure states are OQRs when they are 2-AC, the mixed scenario adds the additional condition that (ρ) must be a 1-AC subspace (see Eq. (<ref>)), introducing the need for anticoherent subspaces to achieve the optimal sensitivity in the measurement of rotations.
As a result, we studied the existence of t-AC subspaces and, in particular, the maximum dimension that 1-AC subspaces can have for a given value of j (see Fig. <ref>). This value is important because we have shown that it sets a lower bound on the purity of mixed OQRs. Our detailed study of mixed OQRs for small spin quantum numbers shows that the first mixed OQR is found for spin j=2 (see Eq. (<ref>)). Furthermore, we presented various examples of AC subspaces and mixed OQRs, including 2-AC subspaces that generate a continuous family of mixed OQRs, and cases where the mixed OQRs arise from a mixture of non-OQR pure states (see Eq. (<ref>)). We showed how to construct 1- and 2-AC subspaces of any dimension for sufficiently high spin by providing explicit subspaces bases (see Tables <ref>–<ref> for selected examples). For these subspaces, we showed that the dimension increases linearly with the spin quantum number j, and we have compared it to the maximum dimension k_max obtained numerically for the same spin (see Fig. <ref>). Additionally, we provided an upper bound on k_max (Eq. (<ref>)) in terms of the spin j and the anticoherence order t. One of the main conclusions of our work is that mixed states can achieve the same sensitivity as optimal pure states. This property can be related to entanglement by interpreting spin states as symmetric multiqubit states. More precisely, our Theorem <ref> establishes that any state obtained by mixing pure states from a t-AC subspace, in particular mixed OQRs, maximizes all negativities for the bipartitions t'|N-t' with t'⩽ t. Therefore, mixed spin-j OQRs not only have genuine multipartite entanglement, as was already known to be necessary for pure states to be OQRs <cit.>, but they are also maximally entangled states with respect to the negativity for the bipartition 1|N-1 among all symmetric states.







§ ACKNOWLEDGMENTS

ESE acknowledges support from the postdoctoral fellowship of the IPD-STEMA program of the University of Liège (Belgium). JM acknowledges the FWO and the F.R.S.-FNRS for their funding as part of the Excellence of Science programme (EOS project 40007526). CC would like to acknowledge partial financial support from the
DGAPA-UNAM project IN112224.







§ EXISTENCE OF ANTICOHERENT MIXED STATES TO ANY ORDER




In this Appendix, we prove the existence of mixed AC states of any order t other than the maximally mixed state. More generally, we prove the existence of mixed states with (ρ  T_μ)=0 for any subset of indices ∈ℐ⊂{ 1 , … , 2j } and μ = - , … ,. We start with the maximally mixed state ρ_0 =1/(2j+1) and a mixture  

    ρ = ρ_0 + ϵ A   ,

with an operator A such that

    A= ∑_L=1
     L≠ℐ^2j∑_M=-L^L A_LM T_LM  ,

with arbitrary coefficients A_LM making A Hermitian. The state ρ is Hermitian, with trace (ρ)=1, and positive for a sufficiently small ϵ >0. In fact, if we denote _min the lowest (and therefore potentially negative) eigenvalue of A, then ρ is a valid state for ϵ≤ [(2j+1)|_min|]^-1. In particular, if we consider ℐ= { 1, 2, … ,t}, the state defined above is t-AC.




§ 2-AC SPIN-3/2 MIXED STATES




Here, we characterize the set of 2-AC spin-3/2 mixed states. By definition, they have an expansion in the multipole operator basis of the form

    ρ = 1/41 + ∑_μ= -3^3 ρ_3 μ T_3 μ  .

We can reduce the number of ρ_3μ variables by a global rotation rendering ρ_33=0 and ρ_32∈R. Moreover, by the hermiticity of the state, we also get ρ_3-3=0 and ρ_32∈R. We now perform the change of variables 

    ρ_30=w  c_0   ,   ρ_31= -ρ^*_3-1 = w  c_1 e^i ϕ  ,
    
       ρ_32= ρ_3-2= w  c_2   ,

so that

    ρ = 1/4 + w [     c_0 T_30 + c_1  ( e^i ϕ T_31 - e^-i ϕ T_3-1)
        
        + c_2 ( T_32+ T_3-2) ]   ,

with c_0^2 + 2c_1^2 + 2c_2^2 =1 and where all the new variables (w,c_0,c_1,c_2,ϕ) are real. Finally, ρ must be a valid state, , its eigenvalues
 
    _±± = 1/4±w/2√(1 ±2/5√(X))

 must be positive, where
 
    X=     2 c_1^2 (-2 √(30) c_2 c_0 cos (2 ϕ )+8 c_0^2+15 c_2^2)
         
        
         + 21 c_1^4+4 c_0^2 (c_0^2+10 c_2^2).

The positivity of ρ is guaranteed if the lowest eigenvalue _-+ is positive, which is fulfilled for 
w ⩽√(5)/ ( 2 √(5+ 2 √(X))).
In particular, if we are interested in the 2-AC state with the highest purity (ρ^2)= 1/4 + w^2, we must
maximize the variable w for valid states, , for states with _-+⩾ 0. To fulfill this, we minimize the factor of the variable w in _-+, √(1+ 2/5√(X)), achieved for X=0. The minimum is given by (c_0 , c_1 , c_2 , ϕ) = (0,0,1/√(2),0) and the corresponding state has doubly-degenerated eigenvalues _±±= 1/4±w/2. Then, we can take w=1/2, leading to the state

    ρ = 1/2( |ψ_1⟩⟨ψ_1| + |ψ_2⟩⟨ψ_2|)   ,

with eigenvectors (<ref>) and purity (ρ^2) = 1/2.






§ UNIQUENESS OF THE LG AC SUBSPACE



Here, we prove that, up to a rotation, the (2,2,1) AC subspace V spanned by the states (<ref>) is unique. We begin by considering a generic (2,2,1) AC subspace V with basis {|ψ_1⟩ , |ψ_2⟩}. By definition, the |ψ_k⟩ states are 1-AC. We then rotate the subspace to Ṽ, now with basis |ψ̃_k⟩ = |ψ_k⟩, such that |ψ̃_1⟩ has the form of a generic spin-2 1-AC state (as given in Ref. <cit.>)

    |ψ̃_1⟩ = 1/√(2+|μ|^2)(1,0,μ , 0,1)   ,

where μ∈C. The unnormalized vectors J_a|ψ̃_1⟩ for a=x,y,z read

    J_x |ψ̃_1⟩∝   ( 1 + √(32)μ) (0,1,0,1,0)   ,
    
        
      J_y |ψ̃_1⟩∝   ( 1 - √(32)μ) (0,1,0,-1,0)   ,
    
    
      J_z |ψ̃_1⟩∝    (1,0,0,0,-1) .

We can see that for μ≠±√(2/3), the set

    ℬ = {|ψ̃_1⟩ , J_x|ψ̃_1⟩ , J_y|ψ̃_1⟩ , J_z|ψ̃_1⟩}

consists of 4 orthogonal vectors. Its orthogonal complement ℬ^⊥ has dimension 1 and, by direct calculation, consists of the state

    1/√(4+2 |μ|^2)( μ^*,0,-2 ,0 , μ^*)
      .

The latter state must be equal to |ψ̃_2⟩= |ψ_2⟩ because Ṽ is 1-AC subspace and must obey the condition (<ref>). The explicit expressions of the states |ψ̃_k⟩ show that Ṽ is the same subspace spanned by the states (<ref>) whatever the value of μ. In the special case μ = ±√(2/3), it is shown in Ref. <cit.> that these states are, up to a rotation, equal to the state |2,0⟩. We can then study this case by starting with |ψ̃_1⟩= (0,0,1,0,0). Generating the set ℬ in the same way, which now contains 3 vectors, we obtain that |ψ̃_2⟩∈ℬ^⊥ must have the form
 
    |ψ̃_2⟩∝ (a,0,0,0,b)   ,

with a,b ∈C. However, since |ψ̃_2⟩ must be 1-AC, we must have that b=a e^i ϕ. Consequently, Ṽ is spanned by the states

    (0,0,1,0,0)   ,   1/√(2)(e^-i ϕ/2,0,0,0,e^i ϕ/2)   ,

which, after a rotation by -ϕ/2 about the z axis, is equivalent to the subspace spanned by the states

    (0,0,1,0,0)   ,   1/√(2)(1,0,0,0,1 )   .

Again, this AC subspace is equivalent to the subspace spanned by the states (<ref>).




§ MAXIMAL NEGATIVITY OF MIXED STATES




Before proving Theorem <ref>, let us examine the properties of an orthonormal basis ℬ= {|ψ_i⟩}_i=1^k associated with a (j,k,t) AC subspace V with t⩽ j. Since each state |ψ_i⟩ is t-AC, its Schmidt decomposition for the bipartition A|B≡ t|N-t 
is given by (see Subsection <ref>)
    
    |ψ_m⟩ = 1/√(t+1)∑_α=1^t+1|ϕ^A_α⟩|ϕ^B_mα⟩  ,

    where we are free to choose the common basis {|ϕ^A_α⟩} of ^A for each state |ψ_m⟩, using the non-uniqueness of the singular value decomposition on the degenerated Schmidt eigenvalues <cit.>.
  Here, all the Schmidt numbers are degenerated and, consequently, we can choose the basis of ^A for each state |ψ_m⟩. We now establish a consequence of the anticoherence property of a subspace V over the states |ϕ^B_mα⟩ in the decomposition (<ref>).


    Let V be a (j,k,t) AC subspace, with an orthonormal basis ℬ= {|ψ_m⟩}_m=1^k where each state has a Schmidt decomposition in the bipartition t|N-t, ^(j)⊂^A⊗^B = ^(t/2)⊗^(j-t/2), given by
    
    |ψ_m⟩ = 1/√(t+1)∑_α=1^t+1|ϕ^A_α⟩|ϕ^B_mα⟩  .

Then, the set {|ϕ_mα^B⟩}_(m,α) is orthonormal

    ⟨ψ_m α^B |
        ψ_n β^B⟩ = δ_mnδ_αβ  .


Proof. The Schmidt decomposition of each state |ψ_m⟩ implies that each subset {|ϕ_mα^B⟩}_α =1^t+1 is orthonormal ⟨ϕ_mα^B | ϕ_mβ^B ⟩ = δ_αβ. Now, due to the fact that V is a (j,k,t) AC subspace, each state defined by a linear combination of two states of ℬ,

    c_1 |ψ_m⟩ + c_2 |ψ_n⟩
    = 1/√(t+1)∑_α = 1^t+1|ϕ_α^A⟩( c_1 |ϕ_mα^B⟩ + c_2 |ϕ_nα^B⟩)   ,

is t-AC, and then, it has a fully degenerated Schmidt decomposition. This implies that the states c_1 |ϕ_mα^B⟩ + c_2 |ϕ_nα^B⟩ must be orthogonal

    ( c_1^* ⟨ϕ_mβ^B| + c_2^* ⟨ϕ_nβ^B|)
        ( c_1 |ϕ_mα^B⟩ + c_2 |ϕ_nα^B⟩) = δ_αβ  .

for any c_1 and c_2 values. In particular, we can choose the two cases c_1 = 1/√(2), and c_2 = 1/√(2) and c_2 = i/√(2). After a little  algebra, they read as

    ⟨ϕ_nβ^B | ϕ_mα^B ⟩ = ±⟨ϕ_m β^B | ϕ_n α^B ⟩  .

Consequently, they are orthogonal for any m , n which ends the proof of our Lemma. 
□

As an example, we can take the (2,2,1) AC subspace defined by the states

    |ψ_1⟩   = 1/√(3)( |2,2⟩ + √(2)|2,-1⟩)   ,
            
    |ψ_2⟩   = 1/√(3)(√(2)|2,1⟩ -|2,-2⟩)   ,

and equivalent to the subspace defined in Eq. (<ref>) up to a global rotation. Without loss of generality, we can take the basis for the Schmidt decomposition in the partition 1|3 of the subsystem ^A as | ϕ_m^A ⟩ = | 1/2 , ±1/2⟩. The Schmidt decomposition of the states are |ψ_m⟩ =  ( |ϕ_+^A⟩|ϕ_m+^B⟩ + |ϕ_-^A⟩|ϕ_m-^B⟩) / √(2) with

    |ϕ_1+^B⟩ =  √(2)| 3/23/2⟩ 
        + |3/2,-3/2⟩/√(3) 
        ,   |ϕ_1-^B⟩ = | 32 , - 12⟩  ,


    |ϕ_2+^B⟩ =  
         | 32 ,  12⟩
         ,   |ϕ_2-^B⟩ =  
     |3/2,3/2⟩ - √(2)| 3/2 -3/2⟩/√(3)  ,

where we observe that the set of four spin-3/2 states {|ϕ_mα^B⟩}_m,α are orthonormal.

By Lemma <ref>, the existence of a (j,k,t) AC subspace implies the existence of an orthonormal set of k(t+1) states in ^B =^(j-t/2). Then, the dimension of ^(j-t/2) implies the inequality k(t+1) ⩽ (^B) = 2j-t+1. We can use this inequality to obtain the upper bound on the maximal dimension of a (j,k,t) AC subspace mentioned in Eq. (<ref>).

Now let us prove Theorem <ref> following an argument similar to that used for pure states in <cit.>.

Proof of Theorem <ref>. Let us consider a mixed state ρ= ∑_m=1^k λ_m |ψ_k⟩⟨ψ_k| where (ρ) is a (j,k,t) AC subspace. ρ seen as a symmetric N-qubit state of a bipartite system t|N-t reads

    ρ = 1/t+1∑_m=1^k ∑_α ,β=1^t+1λ_m |ϕ_α^A⟩|ϕ_mα^B⟩⟨ϕ_β^A|⟨ϕ_mβ^B|  ,

where {|ϕ_α^A⟩}_α is an orthonormal basis of ^A by the definition of the Schmidt decomposition, and ℬ = {|ϕ_m α^B⟩}_m,α is an orthonormal set (Lemma <ref>) with k(t+1) elements. The partial transpose state ρ^T_B reads

    ρ^T_B = 1/t+1∑_m=1^k ∑_α ,β=1^t+1λ_m |ϕ_α^A⟩|ϕ_mβ^B*⟩⟨ϕ_β^A|⟨ϕ_mα^B*|  ,

where the asterisk means complex conjugation. The (t+1)(2j-t+1) eigenvectors of ρ^T_B are divided in four types

    
  * There are (t+1)k eigenvectors |x_mα⟩≡|ϕ^A_α⟩|ϕ^B*_mα⟩ for α= 1 , … , t+1, m=1,…, k and eigenvalues λ_k / (t+1).
    
  * There are two sets of t(t+1)k/2 eigenvectors given by
    
    |y^±_α , β ,m⟩≡1/√(2)( 
            |ϕ^A_α⟩|ϕ^B*_mβ⟩±|ϕ^A_β⟩|ϕ^B*_mα⟩)   ,

    for α < β with α,β =1 , … , t+1 and m=1,… , k. Each eigenvector has a respective eigenvalue ±λ_m/ (t+1).
    
  * In the case that ℬ^⊥ is not empty, it has dimension h= 2j-k(t+1)-t+1 with a basis labeled by |φ^B_m⟩ with m=1,… , h. Then, there are h(t+1) eigenvectors of ρ^T_B, |z_α m⟩= |ϕ_α^A⟩|φ^B *_m⟩ with eigenvalues equal to zero.

We can note that the negative eigenvalues come from the eigenvectors |y^-_α,β,m⟩. The negativity of ρ is thus equal to

    _t (ρ) = ∑_m=1^k ∑_α,β=1 
    α < β^t+1λ_m/t+1 = t/2 ,
 
, equal to the negativity (<ref>) of a pure t-AC state, which is maximal for symmetric states. □





apsrev4-2




