Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions
Simon Morelli Santiago Llorens Jens Siewert









smorelli@bcamath.org
BCAM - Basque Center for Applied Mathematics,
Mazarredo 14, E48009 Bilbao, Basque Country - Spain




santiago.llorens@uab.cat
Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain




jens.siewert@ehu.eus

Department of Physical Chemistry and EHU Quantum Center, University of the Basque Country UPV/EHU,
E-48080 Bilbao, Spain
IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain





The state of a finite-dimensional quantum system is described by a density matrix that can be decomposed into a real diagonal, a real off-diagonal and and an imaginary off-diagonal part. The latter plays a peculiar role. While it is intuitively clear that some of the imaginary coordinates cannot have the same extension as their real counterparts the precise relation is not obvious. We give a complete characterization of the constraints in terms of tight inequalities for real and imaginary Bloch-type coordinates. Our description entails a three-dimensional Bloch ball-type model for the state space. We uncover a surprising qualitative difference for the state-space boundaries in even and odd dimensions.



Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions
    Jens Siewert
    April 29, 2024
=========================================================================================================






§ INTRODUCTION


From the very beginning, quantum mechanics was naturally represented in terms of complex numbers. The state of a quantum system was identified with an element of a complex Hilbert space <cit.>.
Whether this complex formulation is actually necessary or merely more elegant than a hypothetical real version was an open question until very recently. 
Assuming the dimension of the system to be bounded, it is not difficult to design experiments that show that complex numbers are needed to describe the observed data. If the dimension is bounded by two, a sequential Stern-Gerlach experiment will suffice to come to this conclusion <cit.>.
However, if no constraints on the dimension are assumed, it is always possible to explain the measurement data of a single system by a real system of higher dimension. This is even possible for two parties that are restricted to local operations <cit.>.
Only recently it was shown that real quantum theory can be experimentally refuted by designing an experiment in a quantum network involving three parties <cit.>.

Thus, while complex numbers are intrinsically needed to describe quantum mechanics, the role of 'imaginarity' has yet to be fully understood, cf., e.g., Refs. <cit.>.
One might ask 'how real' or 'how imaginary' a quantum state can be. The resource theory of imaginarity <cit.> measures imaginarity as a quantity that cannot be increased under the set of free operations, real quantum channels.
We approach this fundamental question from a different perspective. In analogy with the qubit Bloch ball description based on the three Pauli matrices <cit.> we define a diagonal, a real off-diagonal and an purely imaginary coordinate and ask what weight can be put on these coordinates to be compatible with a positive operator. We then completely characterize the set of attainable values for these coordinates and obtain a Bloch ball-type model <cit.> of the quantum state space.

The Bloch representation of a qubit suggests the quantum space state to be rotation invariant with respect to these three  coordinates. As it turns out, this only holds for the real diagonal and real off-diagonal coordinate in arbitrary dimensions, while the imaginary coordinate has to be treated separately.
In this article we give a precise quantitative bound on the imaginary Bloch length of a quantum state compared to both its real diagonal and real off-diagonal Bloch components. Surprisingly, we thereby find a qualitative difference between systems of even dimensions and systems of odd dimensions. This difference is most pronounced for small system size and vanishes in the asymptotic limit as one might expect. It is noteworthy that the generic form for systems of odd dimension appears only from dimension 5 onwards. 





§ NOTATION


Every quantum state can be written in the following way,


    ρ = 1/d(𝕀_d+D+X+I)  ,


where D is a diagonal matrix, X a real off-diagonal matrix and I a purely imaginary (off-diagonal) and skew-symmetric matrix. The operators D, X and I are all Hermitian, traceless, and orthogonal to each other, that is (D X)=(D I)=(X I)=0.
For dimension d=2 we recover the Bloch representation in terms of the Pauli-matrices, where D=⟨σ_Z⟩σ_Z, X=⟨σ_X⟩σ_X and I=⟨σ_Y⟩σ_Y.
Let us introduce the quantities


    S_D=√( D^2/d),     S_X=√( X^2/d)     S_I=√( I^2/d)


as weights of the diagonal, real off-diagonal and the imaginary part of the state ρ. For a qubit we recover S_D=⟨σ_Z⟩, S_X=⟨σ_X⟩ and S_I=⟨σ_Y⟩.
It holds that
 
 
    ρ^2    = 1/d^2(d+ D^2+ X^2+ I^2)
        = 1/d(1+S_D^2+S_X^2+S_I^2)  .


In dimensions d>2 the quantities S_D, S_X and S_I can be seen as a coarse-graining of the Bloch vector of ρ.
Every quantum state ρ can be expanded in an orthogonal matrix basis as


    ρ = 1/d(𝕀_d+∑_k=1^d^2-1 v_k μ_k)  ,


where we call a basis {μ_k}_i=0^d^2-1 satisfying μ_0=𝕀_d and 
(μ_kμ_l^†)=dδ_kl a Bloch basis <cit.>.

The vector [𝐯]_k=v_k=⟨μ_k⟩ is called the Bloch vector of the state ρ, where v_0=1 is not included.
Choosing a matrix basis that includes only diagonal elements μ^d_k, strictly real off-diagonal elements μ^x_k and strictly imaginary elements μ^i_k, e.g., the generalized Gell-Mann matrices, the state can be expanded as


    ρ = 1/d(𝕀_d+∑_k=1^d-1 v^d_k μ^d_k+∑_k=1^d(d-1)/2 v^x_k μ^x_k+∑_k=1^d(d-1)/2 v^i_k μ^i_k),


where


    D=∑_k=1^d-1 v^d_k μ^d_k ,  
        X=∑_k=1^d(d-1)/2 v^x_k μ^x_k ,  
        I=∑_k=1^d(d-1)/2 v^i_k μ^i_k .


It follows immediately that 


    S_D^2=∑_k=1^d-1 |v^d_k|^2 ,  
        S_X^2=∑_k=1^d(d-1)/2 |v^x_k|^2 ,  
        S_I^2=∑_k=1^d(d-1)/2 |v^i_k|^2 .








§ BOUNDS


We now investigate which values of S_D, S_X and S_I are compatible with a quantum state.
The purity ρ^2 of a quantum state is bounded by one and therefore from Eq. (<ref>) it follows that,


    S_D^2+S_X^2+S_I^2 ≤ d-1  .


For qubits this is the only restriction and the quantum set, that is, the set {(S_D(ρ), S_X(ρ),S_I(ρ):ρ≥0, ρ=1}, is rotation invariant. We will see that the same applies in higher dimension only for the values S_D and S_X,  while the coordinate S_I plays a special role.
We therefore start our investigation by bunching the two real coordinates together to S_R=√(S_D^2+S_X^2). It turns out that the constraints we get for the S_R and S_I values are the only ones that exist, and that the real coordinates
are subject to rotational symmetry.

Every quantum state ρ can be split into its real and imaginary parts,

    ρ = 1/d(R+I)  ,

with R=d(ρ+ρ^T)/2 and I=d(ρ-ρ^T)/2 <cit.>.
We introduce the operator R=𝕀+D+X, a real  positive, symmetric operator with R=d which is orthogonal to the imaginary part, I. 
Note that, while R includes the identity 𝕀, the length S_R does not contain its contribution. That is,  
.


For every finite-dimensional quantum state it holds that the quantities S_R and S_I satisfy


    S_I^2 ≤ 1 + S_R^2  .






It follows from the positivity of ρ and ρ^T that

    0   ≤ d (ρρ^T)=1/d[(R+I)(R-I)]
       =1/d( R^2- I^2)=1+S_R^2-S_I^2  .







The bound is saturated by states of the form


    ρ = [    α_2  -iα_2; iα_2 i    α_2;           α_4  -iα_4;          iα_4    α_4;             ⋱;           α_d  -iα_d;          iα_d    α_d ]


for even dimension and states of the form


    ρ = [     α_2   -iα_2;    iα_2     α_2;             α_4   -iα_4;            iα_4     α_4;               ⋱;           α_d-1 -iα_d-1;          iα_d-1   α_d-1;               0 ]


for odd dimensions if S_R≥ 1/√(d-1). This is easy to see by noting that in both cases R^2= I^2=d^2∑α_k^2. This bound is shown in blue in Figs. <ref>–<ref>.



For a quantum state in odd dimension it holds that the quantities S_R and S_I satisfy

    √(d) S_I ≤ √(d-1) +  S_R  ,

whenever S_R≤1/√(d-1).



Before proving this bound in full generality, we first show the case S_R=0.
Every state satisfying this can be written as


    ρ = 1/d(𝕀+I)  .


The operator 𝕀+I needs to be positive, that is, its eigenvalues need to be positive. Let U be a unitary diagonalizing I. Since unitary transformations do not change the eigenvalues we can write


    U(𝕀+I)U^† = 𝕀+UIU^† = 𝕀+I_diag  .


Therefore it must hold for the eigenvalues of I that |λ_k(I)|≤ 1. Since I is traceless it further holds that ∑λ_k(I)=0. The quantity I^2=∑λ_k(I)^2 is convex, hence maximizing it gives a solution on the boundary.
Under the constraints ∑λ_k(I)=0 |λ_k(I)|≤ 1 we find the optimal solution to be λ_2k-1(I)=-λ_2k(I)=1 with λ_d(I)=0 for odd dimension. Therefore,


    S_I^2 = 1/d I^2 ≤ 2/d⌊d/2⌋  ,


so that S_I^2≤ 1 for even dimension and S^2_I≤ (d-1)/d otherwise.






Subsequently we prove Proposition 2 in the general case.

We know that I is a skew-symmetric matrix and that it has the same eigenvalues as its transpose. Therefore the eigenvalues of I come in pairs with opposite sign, with an additional 0 for odd dimensions. Let λ_k(I) be ordered such that λ_2k(I)=-λ_2k-1(I).
We also know that both R+I and R-I are positive matrices.
Let U be a unitary diagonalizing I=UI_diagU^†. It then holds that U^† RU± I_diag≥0.

Since R and I are orthogonal to each other, also R=U^† RU and I_diag are orthogonal,


    0   =(R I)=(R UI_diagU^†)=(U^† R UI_diag)=
       =(R I_diag)=∑_k=1^dR_kk(I_diag)_kk=∑_k=1^dR_kkλ_k(I) .


It holds that R_kk≥|λ_k(I)| and it is known that the diagonal R_kk is majorized by the eigenvalues λ_k(R)=λ_k(R). It then holds that ∑_k=1^dR_kk^2≤∑_k=1^dλ_k(R)^2
and we conclude


    I^2=∑λ_k(I)^2≤∑_k=1^d-1R_kk^2≤∑_k=1^d-1λ_k(R)^2≤ R^2 .


With this we recover the previous result. The first and the second inequality are saturated if all eigenvalues satisfy λ_2k-1(I)=-λ_2k(I)=λ_2k-1(R)=λ_2k(R). For the third inequality to be saturated we have to choose λ_d(R)=0 in odd dimensions.
But this implies S_R≥ 1/√(d-1) so that the bound cannot be tight for S_R<1/√(d-1).
Set now the smallest eigenvalue of R λ_d(R)=t≤1, it immediately follows that R^2≥ d(d-2t+t^2)/(d-1) where the minimum is attained for λ_k(R)=(d-t)/(d-1).
For a given R^2 it must therefore hold that 


    t ≥ 1-√((d-1) R^2 -d^2+d)/√(d) = 1-√(d-1)S_R .


It now holds in odd dimensions for S_R≤1/√(d-1),


    dS_I^2   = I^2=∑λ_k(I)^2≤∑_k=1^d-1λ_k(R)^2= R^2-t^2≤
       ≤ d+dS_R^2-(1-√(d-1)S_R)^2=(√(d-1)+S_R)^2,


which proves the result.





The inequality (<ref>) for odd dimensions and S_R≤ 1/√(d-1) is saturated by states of the form


    ρ = [        α      -iα;       iα        α;                 α      -iα;                iα        α;                 ⋱;                 α      -iα;                iα        α;          1-(d-1)α ]


with 1/(d-1)≥α≥1/d. To see this it suffices to note that  R^2=d^2(d(d-1)α^2-2(d-1)α+1)
and .
Figures <ref>–<ref> show this bound in green. 

For even dimension d>2, Eqs. (<ref>) and (<ref>) describe the boundary, they intersect at S_R=√((d-2)/2). For odd dimension the boundary 
(for S_R≤ 1/√(d-1))  is described by Eq. (<ref>), it tangentially touches the bound in Eq. (<ref>) at S_R= 1/√(d-1).

Going back to our original coordinates S_D, S_X and S_I, we immediately find the following result.


    The three quantities S_D, S_X and S_I satisfy the bounds 
    
    
    S_I^2    ≤ 1 +S_D^2+S_X^2
    
           √(d) S_I    ≤ √(d-1) +√(S_D^2+S_X^2)  ,

    
    where the last inequality holds only for S_D^2+S_X^2≤1/(d-1).



    The bounds follows trivially from Eq. (<ref>) and (<ref>) by replacing S_R^2=S_D^2+S_X^2.



    The bounds in Corollary <ref> define a tight upper bound. Together with Eq. (<ref>) they are optimal, in the sense that there exist no further constraints on the quantities S_D, S_X and S_I that do not trivially follow from them.



The states in Eq. (<ref>),(<ref>) and (<ref>) have no real off-diagonal contribution and thus saturate the bounds in Corollary <ref> for S_X=0, together with the pure states β|0⟩+i√(1-β^2)|1⟩, where 0≤β≤ 1.

We will now define a continuous orthogonal transformation that shifts weight from the diagonal component to the real off-diagonal component as long as S_D>0. Note that any  orthogonal (i.e., real) transformation leaves the coordinates S_R and  S_I unchanged. 
Consider an orthogonal transformation O_kl(θ)= cosθ(|k⟩⟨k|+|l⟩⟨l|)+sinθ(|k⟩⟨l|-|l⟩⟨k|) acting on the subspace spanned by the states {|k⟩,|l⟩}. 
Start acting with this transformation on the first and the last entry until either one of the new diagonal entries becomes 1/d.
If it is the last entry we perform a second orthogonal transformation from |0⟩ to |d-2⟩ until one of them reaches 1/d. If it is the first entry we apply the transformation from |1⟩ to |d-1⟩. We repeat this process until all diagonal entries are equal to 1/d.
This procedure continuously shifts weight from the diagonal coordinate S_D onto the real off-diagonal coordinate S_X until the diagonal coordinate vanishes. The orbits of the states in Eqs. (<ref>), (<ref>) and (<ref>) form the boundary.
This concludes the proof.










§ BLOCH BALL FOR A QUDIT


We now discuss  the geometric implications
of our investigation. In recent years considerable effort was devoted to the question of constructing models for the quantum state space of dimension d, e.g., Refs. <cit.>. That is, one asks for the shape of this (d^2-1)-dimensional object if it is represented in a lower dimension, thereby maintaining as many geometric properties of the original state space as possible. Because of the reduction in the number of parameters for dimensions d≥ 3 it is clear that any model, while correctly displaying some of the properties of the state space, will suffer from a loss of information and inadequately represent other features. Consequently, any model is the result of a judicious choice of the coordinates, depending on the set of properties that are to be maintained. The corresponding objects may have rather different shapes.

Our work provides a striking illustration for these considerations. As is evident from Fig. 2, our parametrization of the qudit state Eqs. (<ref>), (<ref>) in imaginary, real diagonal and off-diagonal coordinates leads to a three-dimensional model for the qudit state space for all d≥ 2. While, e.g., the model in Ref. <cit.> emphasizes the simplex of diagonal states (in d=3) the present model focuses on the bounds for the imaginary coordinates and, for example, is not convex for d≥ 3.
Note that convexity is restored for d=3 if one uses just one  real coordinate S_R (instead of S_X and S_D), as can be observed in Fig. <ref> (a). 

Another salient feature of the present model is its rotation invariance about the imaginary axis, cf. Fig. <ref>. It is a direct consequence of the unitary invariance of the purity, Eq. (<ref>): Any transformation that leaves S_I unchanged, necessarily preserves also the radius S_R in the real plane.

Intriguingly, the bounds in Eqs. (<ref>), (<ref>) reveal that the geometry of the state space boundary is different in even vs. odd dimensions: While for odd dimensions there is always a flat surface part (of mixed states) close to the most distant point on the imaginary axis, the surface is curved in the corresponding region for even dimensions, cf. Fig. <ref> (b). 
We note that the concave part of the model in d=3 is completely described by the linear bound Eq. (<ref>). Hence the first dimension for which the generic odd-d behavior is observed is d=5: In the vicinity of S_I=√((d-1)/d) the linear bound describes the surface before the quadratic bound, Eq. (<ref>), takes effect for S_R=1/√(d-1).  The linear surface part remains present for all odd dimensions but becomes negligibly small for large d, see Fig. <ref>.
At the same time the restriction imposed by Eq. (<ref>) becomes more stringent. The respective part in (S_R,S_I) coordinates that remains compatible with a quantum state decreases, until asymptotically only half of the original circular area defined by the purity bound in Eq. (<ref>) remains, see Fig. <ref>.




§ RELATION TO RESOURCE THEORY


Recently, the resource theory of imaginarity has received considerable attention <cit.>.
A measure of imaginarity is a quantity that is non-increasing under real quantum channels, i.e.,
completely positive and trace-preserving (CPTP) 
maps that allow a description with purely real Kraus operators.
The robustness of imaginarity is defined as ℐ(ρ)=I/d_1 and the geometric measure of imaginarity is defined as ℐ_G(ψ)=1-sup_ϕ∈ℛ|⟨ϕ|ψ⟩|^2, with the convex roof extension for mixed states.
While these quantities are imaginarity monotones, our imaginary coordinate ρ_I_2 is not an imaginary measure <cit.>. This stems from the fact that the Hilbert-Schmidt norm is generally not contractive under CPTP maps for dimension larger than two <cit.>, in contrast to the trace norm <cit.>.
Our goal is not the study of imaginarity as a resource, rather we characterize the constraints on coordinates describing the quantum state space.
Nonetheless, our analysis establishes a relation between the real weight and the robustness of imaginarity ℐ for odd dimensions. For S_R≥1/√(d-1) there exist states that achieve ℐ=1 (cf. the states saturating the bound in Proposition <ref>), while for S_R=0 no state can reach full imaginarity. The question remains open for the region 0<S_R<1/√(d-1), as none of our presented states reaches maximum imaginarity. On the other hand, for even dimensions, no restriction exists concerning real weight and imaginarity, showing again remarkable differences between even and odd dimensions.




§ CONCLUSION


We generalize the Bloch coordinates of a qubit describing the diagonal, real off-diagonal and imaginary part of a quantum state to arbitrary finite dimensions. We completely characterize the set of coordinates compatible with a quantum state. While for the simplest quantum system of a qubit these coordinates are interchangeable, this property is lost in higher dimensions. The imaginary coordinate plays a distinctive role and is bounded with respect to the other coordinates. Surprisingly, this bound exhibits a qualitative difference between even and odd dimensions, where the general bound for odd dimensions is first observed in dimension 5. With these findings we obtain a three-dimensional model of the generally higher-dimensional quantum state space that joins the already established models. It further illustrates the peculiar properties of finite-dimensional quantum state spaces and represents an important step towards their understanding.




§ ACKNOWLEDGEMENTS


The authors thank Asier Izquierdo for discussions.
S.M. was supported by the Basque Government through IKUR strategy and through the BERC 2022-2025 program and by the Ministry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142-S / MICIN / AEI / 10.13039/501100011033, PID2020-112948GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe".
S.L. acknowledges funding from the Spanish MICINN with funding from European Union NextGenerationEU (PRTRC17.I1) and the Generalitat de Catalunya, and from the Spanish Agencia Estatal de Investigación and the MICCINN (grant PID2022-141283NB-I00). 
J.S. was supported by
Grant PID2021-126273NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe" as well as by
the Basque Government through Grant No. IT1470-22.



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