Smallest state spaces for which bipartite entangled quantum states are separable

According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators. If we relax the requirement that the local density operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators. This form of separability can be used to construct classical models and simulation methods when only restricted set of measurements are available. With such motivations in mind, we ask what are the smallest such sets of local operators such that a pure bipartite entangled quantum state becomes separable? We find that in the case of maximally entangled states there are many inequivalent solutions, including for example the sets of phase point operators that arise in the study of discrete Wigner functions. We therefore provide a new way of interpreting these operators, and more generally, provide an alternative method for constructing local hidden variable models for entangled quantum states under subsets of quantum measurements.

According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators. If we relax the requirement that the local density operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators. This form of separability can be used to construct classical models and simulation methods when only restricted set of measurements are available. With such motivations in mind, we ask what are the smallest such sets of local operators such that a pure bipartite entangled quantum state becomes separable? We find that in the case of maximally entangled states there are many inequivalent solutions, including for example the sets of phase point operators that arise in the study of discrete Wigner functions. We therefore provide a new way of interpreting these operators, and more generally, provide an alternative method for constructing local hidden variable models for entangled quantum states under subsets of quantum measurements.

OVERVIEW
Entangled quantum systems are a powerful resource for many quantum information processes because of their non-classical correlations. The joint statistics of separated measurements cannot be described by probability distributions over 'local hidden variables' that may underlie individual quantum systems [1], and in some quantum computational models it is known that entanglement is a prerequisite for non-classical computation [2].
However, if the measurements made on quantum states are restricted, as is the case in many settings, then we can sometimes recover such a local description even when the quantum states are maximally entangled. For instance, it has been known since the time of Bell that a two-qubit EPR pair has a local hidden variable (LHV) model for the Pauli measurements [3]. This fact can be reinterpreted as a statement that the EPR pair is separable (i.e. nonentangled) with respect to a more general single-system state space, consisting of cubes of Bloch vectors that enclose the usual Bloch sphere [4]. While the operators that correspond to these 'cube' Bloch vectors are not always physical, they can be considered as valid state descriptions if the measurements we have available are restricted to the Pauli operators.
The investigation of generalized state spaces resides in the study of generalized probabilistic theories [5,6]. This field of research considers physical theories more general than quantum theory by describing single-and multi-particle systems in terms of tables of probabilities for various measurement outcomes that satisfy natural physical constraints, such as not allowing instant signaling. In principle, such theories do not necessarily have an underlying structure in terms of Hilbert spaces and operators, and can exhibit correlations that are more powerful than quantum theory.
In this paper our goal will be to build upon some of these ideas to construct non-quantum local descriptions for entangled quantum states. The trade-off is that we can only make restricted local measurements. Unlike the more general formalism of generalized probabilistic theories, however, the state spaces that we consider still have some quantum structure, in that they are sets of operators of the same dimensions as the quantum density operators.
Constructing the local description is equivalent to asking what local state spaces of operators are required in order for an entangled bipartite quantum state to be considered separable with respect to this new state space. While such local state spaces necessarily contain operators that do not correspond to physical quantum states, in some contexts they can be considered to possess a generalized form of positivity. For instance, if we restrict ourselves to local measurements such that the measurement probabilities obtained from these non-quantum operators are non-negative, then these local operators, while not being physical density operators, still allow us to sample the outcomes of the measurements that we are interested in.
Our goal is to work backwards through this approach, and to try to find the smallest local state space such that a quantum entangled state can be considered separable, on the basis that they will typically be the most useful for constructing local descriptions of entangled states under restricted measurements. The methods that we use have strong connections to the study of cross norm measures of entanglement [7,8].

GENERALIZED SEPARABILITY
In the conventional quantum description of entanglement, a quantum state of two or more particles is said to be entangled if it cannot be written as a probabilistic mixture of products of single particle quantum states [9]. The textbook example of an entangled quantum state for two d-level quantum particles is the maximally entangled state, denoted here by |φ d , which by a suitable local basis choice can be written in the form: It is well known that this state cannot be written in the form of a separable state, where ρ A j and ρ B j are drawn from the sets Q A and Q B of quantum states on each subsystem, and p j forms a probability distribution. However, if we relax the restriction that the local operators ρ A j and ρ B j be drawn from Q A and Q B , then we can indeed find separable decompositions for |φ d . If two convex sets of operators V A and V B are such that a given quantum state Ψ can be written as: where ρ A j ∈ V A and ρ B j ∈ V B , and p j forms a probability distribution, then we will say that Ψ is V-separable, where V denotes the pair of sets (V A , V B ).
This generalized notion of separability [6] can adopt practical significance if the local operators in the separable decomposition exhibit some form of what can be referred to as generalized positivity. In the usual study of quantum separability, the notion of positivity amounts to matrix-positivity of the local operators so that they can correspond to density matrices. However, in other contexts, other notions of positivity may be useful. In this work we will consider a type of positivity defined with respect to a subset of quantum measurements, which has physical significance when the measurements that we have available are restricted. Consider the general case of a positive-operator valued measurement (POVM) performed on a single quantum particle: We define the dual set, denoted by M * , of this measurement as the set of operators that give valid probability distributions for these measurements: We say that the elements of M * are positive with respect to M, or more concisely M-positive. Such definitions appears in the context of generalized probabilistic theories [5]. If instead of a single POVM we consider a collection F of POVMs measurements on the same particle, then we define the dual F * as the intersection of sets of operators that give valid probability distributions for all of the measurements in the collection: If the local operators appearing in a generalized separable decomposition are positive with respect to the measurements on each particle, then this can lead to a classical representation for the state of a bipartite (or, for that matter, multipartite) system where each particle is measured in restricted ways. If, for example, the sets V A and V B are the dual sets F * A and F * B of local collections of quantum measurements F A and F B made on particles A, B respectively, and if the quantum state Ψ is V-separable, then the separable decomposition supplies a local hidden variable model for measurements from F A and F B made on Ψ. Moreover, in some cases such separable decompositions can enable one to efficiently classically simulate quantum entangled systems under restricted measurements, see for example [4,10].
With such applications in mind, our goal in this work will be to try to identify the smallest choices for V A and V B such that the maximally entangled state |φ d is Vseparable. While we make our definitions of 'small' more precise in the rest of the paper, the motivation for this problem is: the smaller the state space, the more positive the operators will typically be. Consider for example positivity with respect to collections of measurements: if we consider two sets V A ⊂ W A , then (V A , V B )-separability implies (W A , V B )-separability, and hence supplies a LHV model for a no smaller class of measurements.
While there can be many definitions of 'size' for the sets V A , V B , we will choose definitions that enable us to make analytical progress, and in the process identify choices of V A , V B that are the 'smallest' possible-in that no strict subsets of them can be chosen while keeping |φ d separable. However, we will see that there are many such inequivalent solutions. Amongst them is the set of phase point operators which are used to describe the discrete Wigner function [12].

VARIANTS OF THE PROBLEM
The usefulness of the sets V A , V B for providing local hidden variable models or classical simulation tools depends upon the measurement operators that we are interested in, and this means that there are many different ways that one could define the 'size' of the sets V A , V B . We will not consider specific measurements in this work, so we will take a broader approach where we will use various norms to define the size of an operator set. Our aim is to identify state spaces that could be useful in a broad range of situations, even though they may not be best choice for specific cases.
Ideally, the method we use to quantify the size of the sets V A , V B should reflect how far from matrix-positive the operators within V A , V B are. This is because if V A , V B consist only of matrix-positive operators (which is of course not possible for an entangled state such as |φ d ), and if they are normalized to have unit trace, then they will be in the dual of all quantum measurements and hence have a LHV model for all quantum measurements. If we additionally restrict ourselves to sets V A , V B of Hermitian unit trace operators only (note that an operator has to be unit trace to be in the dual of a POVM), then the trace norm, denoted by · 1 , captures matrixpositivity in a satisfying way. Indeed, for unit trace Hermitian operators the quantity ( X 1 − 1)/2 ≥ 0 is equal to the sum of the negative eigenvalues. Moreover, because the trace norm is multiplicative for tensor products of operators (i.e. A ⊗ B 1 = A 1 B 1 ), it behaves well when considering composite systems. This suggests that if we restrict our attention to sets V A , V B of unit trace operators, then we could define the size of a single set S by and then define the size of both sets V A , V B together by the product of their individual sizes V A 1 V B 1 , we refer to this joint size as the product size. However, finding smallest sets using the trace norm and incorporating the requirement of Hermiticity and unit trace appears to be difficult. So initially we will begin by analyzing a different problem where we abandon the condition of unit trace, allow ourselves to consider more general norms · in place of the trace norm (so that now we define the size of a single set by S · · = sup{ X | X ∈ S}), although we consider the problem with or without the restriction of Hermiticity.
Problem 1: For a quantum state Ψ and for a suitable norm · what is the infimum product size V A V B of all pairs of sets (V A , V B ) (with or without the Hermitian constraint) such that Ψ is V-separable?
We will find that (at least for multiplicative norms that satisfy the cross property A ⊗ B = A B ) this problem is equivalent to computing the so-called projective tensor norm (also known as greatest cross norm) that has been used already in the study of entanglement [7,8]. This will enable us to draw on explicit formulas that have already been derived in previous works, as well as make strong connections to entanglement measures.
A drawback of Problem 1, whichever norm we use, is that there could be two choices V A , V B and V A , V B with the same size as measured by a norm, while still having (say) V A ⊂ V A , V B ⊂ V B . In such cases, the sets V A , V B will be the preferential choice, as they have a strictly larger dual set of measurements. So ideally we would like to know if there are smallest choices for (V A , V B ), in the following sense: We will see that there are many such 'smallest' solutions in the case of the maximally entangled state |φ d φ d |, and we will present methods of constructing a number of them. We will do this by initially tackling Problem 1 where the norm is chosen to be the 2-norm, and then showing how the solutions to Problem 1 contain also solutions to Problem 2. In fact, we will find that for the 2-norm we are able to incorporate the requirement of unit trace and Hermiticity quite straightforwardly.

CONNECTIONS TO CROSS NORMS
In this section, we show that there is a direct connection between the above problems and the notions of cross norms. Consider any norm · such that on two states spaces it satisfies the cross property. For instance, the well-studied family of Schatten p-norms X p = tr(|X| p ) 1/p , where 1 ≤ p < ∞, obey the cross property. For any fixed norm · on a state space, we consider the projective tensor norm, denoted by · γ , on the tensor product of two spaces: where the infimum is over finite sums of arbitrary operators A i , B i (not necessarily in V A or V B , respectively). If the norm · inside the sum is equal to the p-norm · p , then we denote the corresponding projective norm by · γ,p .
It has been shown by one of us [7] that a bipartite quantum state ρ is quantum separable if and only if ρ γ,1 = 1. Moreover, the cases of · γ,1 and · γ,2 have been used in the definition of entanglement measures [7] while · γ,2 was also used in [8] to formulate a computable separability criterion. Now let Ψ be a bipartite quantum state (in this section we do not need to assume that it is |φ d φ d |) and · be a (fixed) norm on the local state spaces. We now show that Ψ γ is exactly the minimum possible value of V A V B in Problem 1.
If Ψ is V-separable, consider any separable decomposition of Ψ: From the definition of the cross norm in Eq. (5), it follows that Hence, we see that the infimum product size of the local sets is lower bounded by the corresponding cross norm. We now show that the opposite inequality holds. For simplicity we assume that infimum of equation (5) is an achievable minimum (the following argument can be modified to hold even when the infimum is not achievable). Under this assumption let be the finite decomposition of Ψ that achieves Ψ γ in Eq. (5). We may define a probability distribution by p i · · = A i B i / Ψ γ . Then we have that: Now let V A be the convex hull of the set of all operators Ψ γ A i / A i and similarly V B be the convex hull of the set of all operators Ψ γ B i / B i . In the separable decomposition (8), all the local operators in the product terms from the sets (V A , V B ) have the same norm, equal to Ψ γ , hence showing that V A V B ≤ Ψ γ as this is an explicit separable decomposition. Hence, there are choices of state spaces This connection allows us to use existing results on the calculation of the cross norm. In particular, the value of · γ,1 and · γ,2 has been calculated for (among others) all pure bipartite states [7], with or without the requirement of Hermiticity.
We will now build upon these results to provide solutions to Problem 2, including the requirement of unit trace. In the next section, we will begin this analysis by rederiving some of the results of [7] for · γ,2 in the case of interest to us (the maximally entangled state). We will use these observations to provide a variety of optimal solutions V A , V B , and then also provide solutions to Problem 2.

SMALLEST STATE SPACES FOR MAXIMALLY ENTANGLED STATES UNDER THE 2-NORM
We begin by expanding a d×d matrix of a single system operator X ∈ S in an orthogonal basis of d 2 Hermitian operators C i : where the expansion coefficients x i ∈ C form a d 2dimensional vector x · · = (x 1 , x 2 , . . . , x d 2 ), and the operator basis is chosen to satisfy the condition tr(C i C j ) = d δ ij . An example of such a basis for qubit systems is the set of Pauli operators plus the identity. In such an expansion, the square of the 2-norm of the operator X is given by tr( where x is the standard Euclidean norm of the vector x. Theorem 1 Consider any convex sets of operators Proof: Parts (a) and (c) are existing results, either following from the calculation of · γ,2 in [7], or from the Schmidt decomposition applied to the operator space. However, we will also rederive them as it will help us to prove the remaining parts. First, note that if there is a convex operator decomposition such as: where p k is a probability distribution and A k ∈ V A and B k ∈ V B then the Hilbert-Schmidt inner product of both sides with the basis of Hermitian operators C i ⊗C T j must match. Expressing A k = α k i C i and B k = (β k j ) * C T j , where α k and (β k ) * (the conjugate is incorporated into the definition for later convenience) are complex expansion vectors representing A k and B k , respectively, and using tr(C i C j ) = tr(C T i C T j ) = dδ ij and the identity φ d | X ⊗ Y |φ d = tr(XY T )/d we must have: All the statements of the theorem are short consequences of the above identity. In particular, summing over the d 2 terms involving i = j gives: where β k , α k represents the inner product. This means that the average of the inner products of α k and β k is equal to 1. Hence, by convexity and the Cauchy-Schwarz inequality, it must be the case that max k α k β k ≥ 1, and hence, using the fact that S is no less than · for one of its elements, gives V A 2 V B 2 ≥ d, proving (a). If we now restrict our attention to only sets satisfying V A 2 V B 2 = d, this means that we must have α k β k ≤ 1. Hence. by convexity and the Cauchy-Schwarz inequality, the only way that Eq. (13) can be true is if β k 2 α k = β k , and hence β k , α k = 1 for all k. This implies (b), and shows that there is a trade-off-the smaller the 2-norm of one state space, the larger must be the 2-norm of the other.
To see that at least d 2 operators are required, let us put the fact that β k 2 α k = β k back into Eq. (12) to get where we have defined new unnormalized vectorsα k · · = √ p k β k α k . We may now reinterpret this equation in the following way. For fixed i we consider the coefficients α k i for varying k to be coefficients of a vector of length N , where N (which in principle could be very large) is the number of different values of k in the sum (14). Then Eq. (14) tells us that we have d 2 such vectors of norm 1/d, and they form an orthogonal set (in the dimension N vector space). For it to be possible to pick d 2 orthogonal vectors, k must range over at least d 2 values, hence proving (c).
To show (d) note that setting p k = 1/d 2 and β k j · · = δ jk (and β k = β k 2 α k ) trivially satisfies equation (12), showing that (the convex hull of) any orthogonal basis {C i } of operators satisfying tr(C i C j ) = d δ ij provides suitable choices for V A and V B (by setting V B = V T A ). Finally to show (e) note that for λ ∈ (0, 1) the strict inequality λX are not proportional to each other (this follows from Cauchy-Schwarz, and it does not hold for trace norm), hence, no other operators within the convex hull of the d 2 operators C i can attain a 2-norm of √ d, and hence a strict subset cannot satisfy the necessary condition (b).
These observations provide us with a method of constructing solutions to Problems 1 and 2. However, they have been derived by relaxing the requirement that the operators in V A , V B be Hermitian and of unit trace. It is straightforward (we describe this in the next section) to include these requirements in the context of the 2-norm, but we have not been able to include these constraints in the context of the trace norm.
In App. A, we prove that parts of Theorem 1 also generalize to arbitrary bipartite pure states, providing one  .σ), where x is a three dimensional real vector, and σ is a vector of the three qubit Pauli operators. The eight vertices of the cube correspond to non-quantum operators with x = (±1, ±1, ±1). (b) and (c) are the two sets VA and VB of local operators on systems A and B, respectively, with each set containing four vertices of the cube satisfying the condition tr(CiCj) = 2δij. Specifically, the qubit maximally mixed state is separable with respects to these two sets, solution of Problem 2 for all bipartite pure states.

INCORPORATING UNIT TRACE VIA THE 2-NORM SOLUTION AND APPLICATIONS TO LHV MODEL CONSTRUCTION
The previous sections show that any orthogonal basis of d 2 Hermitian operators normalized to tr(C i C j ) = d δ ij provides a solution to both Problem 1 (in the case of the 2-norm) and Problem 2. It is not difficult to find bases of Hermitian operators with unit trace, thereby incorporating the requirement of Hermiticity and unit trace into solutions of Problems 1 and 2. Indeed, if we start from any Hermitian basis C i for which the first element is the identity C 1 = 1 and the remaining C i are traceless, then imposing the requirement that V A , V B contain Hermitian operators with unit trace becomes equivalent to demanding that the expansion vectors α k , β k are real, and that their first components are 1/d. If we consider only solutions that are constructed as the convex hull of d 2 orthogonal operators of 2-norm √ d, then this is equivalent to picking a d 2 × d 2 real orthogonal matrix such that the top row consists of (1/d, 1/d, 1/d, . . . ), and can be solved using the Gram-Schmidt procedure.
Amongst these solutions there exists one that is already widely used in the construction of classical models: the d 2 subsets of the phase point operators [12] that describe discrete Wigner functions. Each such subset provides a Hermitian unit trace orthogonal basis of the correct norm.
However, there are many other ways of constructing Hermitian unit trace orthogonal basis sets that are not unitarily equivalent to subsets of phase point operators. Our analysis shows that the measurements that define such sets as their dual will have local hidden variable models for the maximally entangled state, going beyond constructions available via a discrete Wigner function approach.
In the context of constructing LHV models it is important to note that two sets of operators satisfying V 1 ⊂ V 2 could nevertheless have duals that contain the same sets of quantum measurements (the differences in their duals could be made from other operators), and hence give LHV models for the same scenarios. If we are considering sets of unit trace operators, then the dual of conv(V ∪Q), where Q is the quantum state space, determines the set of quantum measurements in the dual of V. So if our motivation is to construct LHV models, then we should really consider a modified version of Problem 2, where we look for smallest local state spaces under the constraint that they must contain the local quantum states. However, it is not difficult to show that sets given by conv(W ∪ Q) for any of our solutions W to Problem 2 are solutions to this modified problem, because the quantum states have too small a norm to make conv(W ∪ Q) contain W for different W,W , and vice versa. Hence, if we consider any two inequivalent solutions W, W that we have constructed to Problem 2, as is possible for d > 2, then this means that they will supply LHV models for inequivalent sets measurements.
In the case of d = 2 it can be shown that the only unit trace Hermitian operator basis satisfying tr(C i C j ) = dδ ij are tetrahedra that unitary rotations or transpositions of the example presented in Fig. 1. However, for higher dimensions d > 2, there are inequivalent solutions that do not share the same spectrum.

CONCLUSIONS
We have determined small local state spaces that admit a separable decomposition of an entangled pure state |ψ and cannot be made strictly smaller while maintaining separability. In the context of maximally entangled states in particular, where the local state spaces can be chosen to have unit trace, this has applications in constructing local hidden variable models.
Our measure of 'smallest' state space is given by the operator 2-norm not only because it renders the optimization of Problem 1 analytically tractable, but also because it enables solutions of Problem 2. We do note, however, that using the trace norm would be more natural when searching for states spaces of operators that are not very negative; further work is required to explore this option.
We have made a connection between cross norms and generalized separability, and it is likely that these connections can be generalized when considering other notions of positivity for the local state spaces.
It will be interesting to know whether it is possible to extend the method from the bipartite to the multipartite case, where very little is known about classical models for quantum states.
where d is the Schmidt rank of |ψ . Note that tr(C ij C T kl ) = d δ ik δ jl . Suppose that we have a separable decomposition of |ψ as follows: Let us suppose that A k = ij α k ij C ij and B k = ij (β k ij ) * C T ij . Then we may compute: Summing the equation over g, h, i, j gives: where α k and β k are d 2 × 1 vectors of coefficients α k ij and β k ij respectively, and β k , α k symbolizes their inner product. If we make the restriction that the local operators in the separable decomposition are of 2-norm ≤ ( g λ g ) 2 , then the only way that this equation can be true is if each α k and β k are proportional, and have product 2-norm equal to ( g λ g ) 2 /d. We may place this finding back into equation (15) to get: ψ| C gh ⊗ C ij |ψ = d λ g λ h δ gi δ hj , This equation may be reinterpreted as an orthogonality relation between d 2 vectors labeled by i, j, with N components in each vector (the size of the range of k). Hence if we need at least N ≥ d 2 orthogonal vectors of dimension d 2 . If we set N = d 2 then we may arrive at a solution by setting p k = 1/d 2 , and then picking an orthogonal set of vectors normalized such that the vector labeled by i, j has Euclidean norm dλ i λ j . As with the maximally entangled case, any such solution is a smallest one, as no other set of d 2 operators in the convex hull can have a high enough 2-norm.
One disadvantage of this solution is that some of the operators in the local state spaces are traceless, and hence are not positive for any subset of measurements. However, in future work we will explore other notions of positivity (particularly in scenarios involving post-selection) that these local state spaces can nevertheless have.