Correlation minor norms, entanglement detection and discord

In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement detection scheme is believed to be advantageous in comparison to other methods because correlations have a simple, intuitive meaning and in addition they can be directly measured in experiment. Moreover, it is demonstrated to be stronger than the CCNR criterion. We also illustrate the relation between the Correlation Minor Norm and entanglement entropy for pure states. Finally, we discuss the relation between the Correlation Minor Norm and quantum discord. We demonstrate that the CMN may be used to define a new measure for quantum discord.

www.nature.com/scientificreports/ In 40,41 , an approach for detecting entanglement using symmetric polynomials of the state's Schimdt coefficients has been studied. It was shown to be a generalization of the well-known CCNR criterion (computable cross-norm or realignment; first defined in 42,43 ), according to which the sum of all Schmidt coefficients is no greater than 1 for any separable state. The symmetric polynomial approach equips each one of these polynomials with some upper bound, and if the polynomial exceeds its bound then it follows that the state is entangled. Therefore, the sum of all Schmidt coefficients with the upper bound 1 is a special case of this approach.
In this paper, we construct for a given quantum state its quantum correlation matrix, and examine the norms of its compound matrices. Since the compound matrix in our case is constructed from minors of a certain correlation matrix, we call the proposed entanglement detectors "Correlation Minor Norms". Seeing that these norms are invariant under orthogonal transformations of the observables, they can be regarded as a family of physical scalars which can be readily derived from bipartite correlations. Next, for each Correlation Minor Norm (CMN) we find an upper bound, such that if the CMN exceeds this bound it is implied that the state is entangled. Our proposed method is shown to generalize the symmetric polynomial approach. We also provide results and conjectures regarding the states that saturate the bounds. Moreover, we explore how the CMN relates to entanglement entropy. Next, we construct a novel measure for quantum discord based on the CMN. In a particular case, it is identical to geometric quantum discord. We conclude by discussing possible generalizations for multipartite scenarios.

Construction of the correlation matrix
Let two remote parties, Alice and Bob, share a quantum system in H A ⊗ H B , the tensor product of Hilbert spaces. Denote d A := dim H A , d B := dim H B , and let A : be an orthonormal basis of the (real) vector space of d A × d A Hermitian operators, w.r.t. the Hilbert-Schmidt inner product. Similarly, B := B j d 2 B j=1 is an orthonormal basis of the d B × d B Hermitian matrices. Note that such a basis always exists, since the real vector space of n × n Hermitian matrices is simply the real Lie algebra u(n) , which is known to have dimension n 2 . Here we regard u(d A ), u(d B ) simply as inner product spaces, ignoring their Lie algebraic properties. Consequentially, we require the normalization tr A i A j = δ ij (and similarly for Bob)-without the factor of 2, which is normally taken to make the structure constants more convenient. For example, for d = 3 one could take A 9 = 1 √ 3 and where γ i are the ("standard-normalization") Gell-Mann matrices. The (cross-)correlation matrix of A, B , denoted by C , is defined by: where ρ is the density matrix shared by Alice and Bob.
As we shall see in the next section, the information contained in C regarding the strength of nonlocal correlations is encoded entirely in its singular values. An equivalent characterization is provided by a noteworthy relation between the singular value decomposition (SVD) of C and the operator-Schmidt decomposition of the underlying state, which we describe hereinafter. The operator-Schmidt decomposition of any state ρ is defined as its unique decomposition of the form: where each k ≥ 0 is a real scalar, and the sets {G k } and {H k } are orthonormal sets of d A × d A , d B × d B Hermitian matrices respectively. It can be shown that the singular values of C are precisely the Schmidt coefficients k ; moreover, the sets {G k } and {H k } are related to the sets {A i } and B j through the orthogonal matrices U, V of the SVD, respectively. Extended definitions and proof may be found in Section I.A of the supplementary information.

Correlation minor norm
The goal of this work is to produce physical scalars from C that would allow for entanglement detection. In the context of this paper, a scalar is considered to be physical if it is invariant under a transformation of the set of measurements. Such a transformation is described by a pair of orthogonal matrices (see discussion in Section I.B of the supplementary information): Introducing into (3) the SVD of C , written as C = V A V B T , yields: Since U A and V A are elements of O d 2 A (and similarly for the matrices with the subscript B), we may observe that a general orthogonal transformation of C = V A V B T reduces to the substitution of V A and V B by any other elements of their respective orthogonal groups. Thus, it is clear that any physical scalar derived from C should depend on its singular values, i.e. the operator-Schmidt coefficients.
The simplest candidates for scalars produced by a matrix are its trace, determinant, and any type of matrix norm. However, tr C is not a physical scalar in the sense described above; and a broad class of matrix norms are given as special cases of the scalars constructed in this section. Thus, for now we wish to consider det C (where the discussion is restricted to d A = d B ). The determinant of a quantum cross-correlation matrix can help detect www.nature.com/scientificreports/ entanglement, and may also serve as a measure of entanglement for two-qubit pure states (see Section I.D of the supplementary information) and two-mode Gaussian states [44][45][46] . However, in more general scenarios, there are states in which the mutual information between Alice and Bob stems from specific subspaces of their respective vector spaces (in pure states, the dimension of these subspaces is given by the Schmidt rank). To accommodate these cases, one should go over all possible subspaces of some given dimension and consider the determinant of the matrix comprised of correlations between their basis elements. Then, one could construct a measure as some function of all those determinants. One way of doing so is treat them as entries of a matrix and take its norm.
In light of the observations above, we define the Correlation Minor Norm with parameters h and p = 2: where denotes the set of b-combinations of [a] (this notation is common in the Cauchy-Binet formula), C R,S is the matrix whose rows are the rows of C at indices from R and whose columns are the columns of C at indices from S, The meaning of the parameter p will become clear shortly, when the above definition is generalized.
where we have numbered the sets' elements: Such a matrix N is known as the h-th compound matrix of C , and is denoted by C h C . Now, recall the Schat- where d = min {d A , d B } , and σ k C denotes the k-th singular value of C . This implies that M h,p is indeed a physical scalar. Note that the Schatten p-norm of C itself is obtained as a special case, for h = 1 . Another thing to note is that for h = d 2 , the CMN M h,p is equal to the product of all singular values, irregardless of p; in this case we denote it by M h=d 2 . If d A = d B , this is simply det C.

Entanglement detection using the correlation minor norm
For general mixed states, there are a few known links between Schmidt coefficients and entanglement detection; the best-known is probably the CCNR criterion: If d 2 k=1 k > 1 , then ρ is entangled 47 . The Correlation Minor Norm allows for an equivalent formulation: if M h=1,p=1 > 1 , then ρ is entangled.
The CCNR criterion has an additional immediate consequence regarding the CMN: since M h,p is a monotonically increasing function of the operator-Schmidt coefficients k , there is an upper bound for the value it may obtain without violating the inequality d 2 k=1 k ≤ 1 . Thus, for all h and p, there exists some positive number This implies the Correlation Minor Norm can be used to detect entanglement by the following procedure: given a state ρ , the corresponding correlation matrix C is obtained -either by computation or by direct measurement; then, the SVD of C is used to find the singular values, and these are substituted in (8)   www.nature.com/scientificreports/ In 40 , Lupo et al. generalize the CCNR criterion in the following way: they construct all elementary symmetric polynomials of the Schmidt coefficients k of ρ , and find bounds on these assuming ρ is separable. The h-th elementary symmetric polynomial of n variables is defined as follows: i.e., the sum of all distinct products of h distinct variables. Clearly, M h,p=1 = S h σ 1 , . . . , σ d 2 .
A more recent work 41 which cites 40 , makes the following important claim: assuming d A = d B , they find a tight bound on the h-th symmetric polynomial (for separable states), and prove that as an entanglement detector it is no stronger than the CCNR criterion. Since the conjectures presented in this section imply this is true for the CMN with p = ∞ as well, it seems likely that for d A = d B and any value of p, the CMN is no stronger than the CCNR criterion as an entanglement detector.
However, in the case where d A = d B , it seems the CMN may detect entanglement in cases where CCNR does not. Let us define following [48][49][50]  Proof may be found in Section IV.B of the supplementary information. Moreover, we conjecture the following theorem still holds with the assumption of the state being in FNF removed. If proven, this conjecture would have explained the upper bounds presented in 40 for d A = d B , which had been found numerically.
Before presenting the next result, let us introduce quantum designs 51 A quantum design is regular with r = 1 if all projections are pure (i.e. one-dimensional); it is coherent if the sum k P k is proportional to the identity operator; and it has degree 1 if there exists µ ∈ R such that ∀k � = l, tr(P k P l ) = µ . If a quantum design has all three qualities, then µ = v−b b(v−1) . A regular, coherent, degree-1 quantum design with r = 1 having v elements, is simply a set of v "equally spaced" pure states in the same space. For example, such a quantum design in dimension d containing d 2 elements is known as a symmetric, informationally complete, positive operator-valued measure (SIC-POVM) 52 .
The following theorem tells us how to construct a separable state saturating (10) using quantum designs. Then, the operator-Schmidt coefficients of ρ are α with multiplicity one and β with multiplicity d 2 − 1.
The proof appears in Section IV.C of the supplementary information. Note the last two theorems have the following special case: for h = d 2 , they imply that the above state maximizes the product of all Schmidt coefficients; i.e., it maximizes M h=d 2 ,p for all p.
Furthermore, we have similar claims for p = ∞.

Theorem 3 Let ρ be a separable state in FNF, and h ≥ √
Dd . Then: Proof may be found in Section IV.D of the supplementary information. As in Theorem 1, we conjecture this theorem still holds without the assumption that ρ is in FNF. Evidence for why we believe this conjecture to be true may be found in Section IV.E of the supplementary information. The following theorem yields a way of saturating the bound (12): , P B k h k=1 be sets of pure projections comprising regular, coherent, degree-1 quantum designs with r = 1 , in dimensions d A , d B respectively and having h elements each. Define a state: We hypothesize that upper bounds over M h,p for any value of p may be characterized using quantum designs. If this hypothesis is proven, then separable states built using such quantum designs are, in a way, on the "edges" of the convex separable set. However, one should note that quantum designs in a given dimension with a given number of elements do not always exist; the above theorems hold only in the cases where they do exist.

Further results and examples
Relation to entanglement entropy for pure states. Let |ψ� be a pure state, and let s 1 , . . . , s d denote its "pure-state-Schmidt coefficients" (i.e., the ones arising when writing the Schmidt decomposition for pure states of |ψ� ). Then, its operator-Schmidt coefficients are s k s l , i.e. all the pairwise products of pure-state-Schmidt coefficients (if k = l , s k s l appears as an operator-Schmidt coefficient with multiplicity 2; for proof please refer to in Section I.C of the supplementary information.
For pure states, the Correlation Minor Norm is linked to the state's Schmidt rank by the following observation: for all t ∈ [d] , M h=t 2 ,p � = 0 iff the state's pure-state-Schmidt rank is at least t. Thus, the Correlation Minor Norm may be used to find the Schmidt rank in pure states. Figure 1 illustrates a comparison between M h=t 2 ,p and entanglement entropy for all two-qutrit pure states.
Moreover, for any pure state of dimension 2 × D , the CMN and entanglement entropy are only functions of s 2 1 , and both functions have the same monotonicity w.r.t. this parameter (i.e. they increase / decrease in the same domains). This may be demonstrated by noting that effectively, the qubit is only correlated with a two-dimensional subsystem of Bob's system. Using the same reasoning that appears in Section I.D of the supplementary information, it could be argued that M h=4 may indeed quantify entanglement in this scenario.
However, it is clear that not all Correlation Minor Norms are useful for this purpose; in fact, the relation between operator-Schmidt coefficients and pure-state-Schmidt coefficients implies: Figure 1. Several CMNs M h,p and the von-Neumann entanglement entropy S(ρ A ) , plotted for the two-qutrit states with pure-state-Schmidt coefficients given in spherical coordinates: s 1 = sin θ cos φ , s 2 = sin θ sin φ and s 3 = cos θ (to avoid repetition, only the domain 0 ≤ θ , π ≤ π/2 is plotted). The area where S(ρ A ) vanishes, θ = 0, π/2 , alludes to the domain where the state's Schmidt rank is 1, and the same goes for M h=2 2 ,p . However, M h=3 2 ,p also tells us where the Schmidt rank is 2, i.e., φ = 0, π/2 . (This figure was created using MATLAB R2016A). Relation to quantum discord. Since the CMN seems to capture some value related to quantum correlations, it is intriguing to ask whether it may somehow be be used to measure their strength. The geometric measure for quantum discord (GQD) with respect to Alice's subsystem is defined 38 as: i.e. the shortest squared Euclidean distance between ρ and any classical-quantum state (the expression for discord w.r.t. Bob's subsystem is defined similarly, where the minimization goes over all quantum-classical states).
Motivated by this definition and by the expression for GQD derived in 39 , we suggest the following measure for discord w.r.t. Alice's subsystem, based on the CMN: where the maximization goes over all projective measurements on Alice's subsystem is the state obtained from ρ by performing the measurement A and obtaining the appropriate ensemble of the projections i (i.e., the state is measured but not "collapsed"). The following result suggests that D A h,p may be thought of as a measure for discord: Moreover, for any state ρ , we have D A G (ρ) = D A h=1,p=2 . The proof for this fact, as well as for the theorem above, appears in Section V of the supplementary information. As evident in the proof, D A G (ρ) = 0 ⇒ D A h,p (ρ) = 0 for h > 2 as well. Figure 2 illustrates several of the measures D A h,p (ρ) for a two-parameter family of states given in 53 . The states appear in Section V of the supplementary information. It is also worth noting in this context, that the two-qubit separable state with maximal discord has the same operator-Schmidt coefficients as the Werner state with c = 1/3 37,54 (hence they are unitarily equivalent); and it is precisely the state maximizing the CMNs with p = 1 . In other words-the entanglement, discord and CMNs for the two-qubit Werner states are all monotonically increasing functions of the parameter c ∈ [0, 1] , and for the critical value of c (above which the states are entangled) the Werner state is precisely the one for which the CMN obtains its separable upper bound. This situation occurs for the two-qutrit Werner state as well, where the critical value of c is 1/4 (when constructed as in 55 ).

Conclusions
The task of entanglement detection is important for basic quantum science, as well as various quantum technologies. The current work was motivated by the following question: since bipartite entanglement can be characterized by correlations between all of the parties' observables, can it also be detected via some norm of these correlations? As demonstrated by our results, the answer is likely to be affirmative.
We have defined the Correlation Minor Norm and explored its characteristics. This has allowed us to propose an approach for detecting entanglement both in pure and mixed states. Furthermore, it was shown that for pure states, the Correlation Minor Norm allows one to determine the Schmidt rank, and in some cases also quantify the strength of quantum correlations. Given the dimensions of the two parties' respective systems, one may choose a single set of operators which can be used for detecting entanglement in any state, be it pure or mixed.
Additionally, we have shown that the CMN with h = 1, p = 2 admits a natural relation to geometric quantum discord. This affinity motivated a definition of a more general measure for quantum discord which is based on the CMN. Some of these measures might mitigate the known issues with existing discord measures [56][57][58][59] .
One optional direction for future research may include development of dynamical equations for the Correlation Minor Norm. This may be interesting, as the correlation matrix contains exactly the same information as the density matrix.
Another possible generalization is considering multipartite systems. In 60 , the authors consider detection of genuine multipartite entanglement and non-full-separability using correlation tensors. Specifically, they consider tensors comprising all multipartite correlations between orthonormal bases to the traceless observables; and they   www.nature.com/scientificreports/ find upper bounds on norms of matricizations of these tensors, such that exceeding these bounds implies the state is genuine multipartite entangled, or non-fully-separable. This paper may hint as to how our work may be generalized to the multipartite case: one could consider the full correlation tensor (i.e. correlations between bases to the entire space of observables, not just the traceless ones); then, the CMN with parameters h, p may be defined as the Schatten p-norm of the hth compound matrix of a certain matricization of this tensor. The bounds shown in 60 could then be utilized to find two upper bounds on each of the CMNs-one for non-genuinely-entangled states, and another for fully-separable states. The question of which matricization should be used remains to be determined. Moreover, further work is required to find the states saturating these bounds.
Received: 23 July 2020; Accepted: 19 January 2021 Figure 2. The CMN-motivated measures for geometric quantum discord D A h,p , for the family of states given in 53 , with parameters q, r ∈ [0, 1] . The discord is computed using the formula given on 53 . The similarity is not a coincidence, as the geometric discord of these states is given by a product of two operator-Schmidt coefficients (times a factor of 2). (This figure was created using MATLAB R2016A).