Generalised versions of separable decompositions applicable to bipartite entangled quantum states

The investigation of separable states in quantum theory has been driven by the notion that they are highly classical, in that they do not demonstrate nonlocality, and are in some contexts unable to support non-classical computation. The converse question, the extent to which entangled states do or do not support non-classical information processing, is less well understood. Motivated by this question we extend the notion of quantum separability into the entangled quantum states, by constructing separable decompositions that describe them with the ‘smallest’ possible sets of non-physical local operators. We consider a few ways to define the word ‘smallest’ and present techniques for obtaining them. The methods involve calculating certain forms of cross norm. The results generalise significantly the results obtained in our previous work on this topic (2015 New J. Phys. 17 093047), and can be be used to construct classical simulation methods and local hidden variable models for subsets of local measurements on entangled quantum states.


Background and overview
A central goal of quantum theory is to determine when quantum systems can or cannot be described by a 'classical' model. The term 'classical' can be defined in many ways, but in the context of multipartite quantum systems two of the most common definitions are a 'non-locality' definition, in which a system is described as being classical if it has a local hidden variable model [1,2], or a 'computational complexity' definition, in which a system is described as being classical if it can be simulated efficiently using a classical computer [3][4][5].
The notion of quantum entanglement plays an important role in studies of these problems. In the context of non-locality, for example, it is easy to show that if a state A B C , , ,  The intuition behind such classical models and simulation algorithms is that for non-entangled systems the correlations are mediated entirely by the classical probabilistic weights p k , followed by extra 'local' sampling of the measurement of the local density matrices. When the classical weights can be efficiently sampled classically, as has been shown to be the case for gate model computers with non-entangling gates [8], then the extra step of sampling the local outcomes from the , , , ... Motivated by these observations, in this work we will consider extending the notion of a separable decomposition to entangled quantum states in order to find new ways of constructing classical models. The way that we will extend the notion of separability is by relaxing the requirement that the local , , , ... k C k r r r be quantum states (e.g. by relaxing the requirement that they be positive). This will permit us to find 'generalised separable' decompositions for entangled quantum states. Without any further considerations this would be a trivial task, as any multipartite operator can be expanded as a probabilistic sum of non-positive operators. However, we will be interested in looking for decompositions in which the local operators are 'as positive as possible', with the aim of eventually using them for classical models. This naturally leads to the question, what criteria should we use to assess 'as positive as possible'? An approach that we took in our previous investigation of this problem [10] was to look for separable decompositions involving small sets of local operators, primarily for describing maximally entangled states. In this work we will extend those results significantly, in some cases to all bipartite quantum states. The results we present will show that separable decompositions of non-entangled quantum states are part of a more general family of separable decompositions, which at the other extreme of entanglement includes discrete Wigner function local hidden variable models for maximally entangled states. As with [10], our results are strongly connected to the notion of cross-norms that has been applied to entanglement theory [11,12].
Our results may be summarised as follows: (i) for any bipartite entangled states we compute all crossnorm achieving decompositions for certain families of cross norm, (ii) on the basis of these decompositions we construct local state spaces for which a given bipartite entangled state is separable, such that the state spaces cannot be made smaller while still supporting a separable decomposition, (iii) for some families of bipartite state we find separable decompositions involving state spaces that are the duals of the largest possible sets of local measurements. Our hope is that in future work these methods will provide useful alternative descriptions of bipartite quantum states for use in classical models, such as generalisations of discrete Wigner functions local hidden variable models, or as alternative variables in classical simulation algorithms.
Structure-This paper is structured as follows. In the next section we discuss the relationship of our work to other literature. In sections 3 and 4 we describe at a broad level the idea of generalised separability, and present some separable decompositions for Bell diagonal states that illustrate and motivate the problems that we will address in this work. In section 5 we describe in detail the four problems (Problems 1, 2, 2 * , 3) that we consider. In section 6 we describe some norms and notation that we use. In section 7 we present a brief list of our main results, which constitute three Theorems. Theorem 1 characterises all solutions to Problem 1 for all bipartite states for all norms that we consider. Theorem 2 solves Problem 2 for all bipartite states. Theorem 3 presents solutions to problems 2 * and 3 for certain families of bipartite state (including the pure states in a region around the maximally entangled state). The proofs are presented in sections 8-10. They may be skipped by those not interested in the proofs. The appendix contains a brief discussion of some notions of generalised positivity.

Relationship to other work
To set the scene it is useful to compare our results to related literature in the foundations of physics, entanglement theory, and quantum information. In an attempt to understand why quantum theory permits highly non-classical correlations, various authors have investigated the kinds of correlations that can occur in hypothetical theories more general than quantum theory. In the study of generalised probabilistic theories [13,14] for example, various reasonable axioms are postulated for physical theories (such as no instant signalling), without necessarily demanding that the theory has an underlying Hilbert space or operator space structure. In such theories correlations can arise that are stronger than quantum [14,15], and the question of why such correlations do not seem to occur in nature has attracted much attention. One way of constructing such non-quantum theories is to start from a quantum setting in which local parties can only perform quantum measurements in certain restricted directions, but then to allow the joint states in the theory to be represented by non-positive and hence non-quantum operators (e.g. allowing them to have negative eigenvalues [16]). As long as the measurements are restricted so that the Born rule gives only positive numbers, then the non-positivity of the states will not show up in any calculations, and the theory will yield valid probabilities. The correlations obtained in this way can be stronger than in quantum theory [16].
In a similar vein, in this work we will consider 'state spaces' that do not consist of positive quantum states. However, in contrast to these works, in this article we will only consider joint states that are physical quantum states (even though our results apply to some non-quantum joint states too), instead it will be the local state spaces that are not quantum (e.g. containing operators with negative eigenvalues). Our motivation is hence not whether the correlations are stronger or weaker than quantum correlations, but is more in the spirit of the use of Wigner functions [17] in the discrete setting [18], in which non-positive operators can be used to provide classical local hidden variable models for mutually unbiased basis measurements [19]. A related motivation comes from the work of [20], which studied the nature of entanglement when the local observables are restricted to certain algebras. Settings with restricted local measurements are physically important because in most applications the measurements available will be restricted due to technological constraints, selection rules, imperfections, or by design in order to enable quantum error correction.
We will however not start with a specific set of measurements or dynamical processes in mind. We will instead start with a given bipartite entangled and usually non-local state, and ask how we may represent the state so that the largest set of measurements or processes may be considered classical. In formulating this problem we will draw upon our previous work on this topic [10], which drew connections with the use of cross norms [11,12,21] to study entanglement. This will supply us with the main technical tools that we need. Our previous work [10] primarily focussed on maximally entangled states and some bipartite pure ones. The results we derive here are much more general. Some of the problems we consider have natural mathematical parallels in the study of decompositions in theory of frames [22,23].
Our original motivation for this work was to understand the extend to which the notion of separability can be applied to entangled states, with the aim of developing new variables that could be useful in the classical description of quantum systems, e.g. in the description of valence bonds in PEPS states [24][25][26]. However the separable decompositions we present can also be used to construct local hidden variable models for restricted sets of measurements. However, in this sense our models are weaker than other models in the literature-it has long been known that there are entangled quantum states for which there are hidden variable models for all measurements [6,7], and more recently systematic algorithms have been developed for constructing local hidden variable models where they exist [27]. These methods typically make no (or few) restrictions on the measurements. However, the separable decompositions that we present do have some appealing properties-they are simple analytic models that are a continuation of separable decompositions into the entangled states, they apply broadly, and they do not depart from the linear structure of quantum theory. The linearity in particular may make it easier to use these models as building blocks to describe more complex systems.

Motivating example: discrete Wigner representation of Bell states
It has been known for many years [2] that Bell pairs of two qubits, such as the state have local hidden variable models for local measurements of the Pauli operators. In this section we will review this result in the light of a generalised notion of separability. Given the Pauli measurements in the x y , ,and z directions, one can define a 'dual' set operators as those operators of unit trace which return positive probabilities for these measurements under the Born rule, i.e. the set of operators where the 'Pauli' denotes the set of six projectors onto the Pauli measurement eigenstates, and the * is used to denote the dual set. In some situations it is convenient to remove the restriction that the operators be normalised to unit trace, in which case the set becomes a convex (blunt) cone (i.e. a set that is closed under linear combinations involving positive coefficients-sometimes non-negative coefficients are allowed, in which case the cone is said to be 'pointed' rather than 'blunt'). However for the purposes of the present discussion we will include the requirement of normalisation. This means that for our purposes the set of dual operators is a convex set, but not a cone.
The normalised dual set Pauli * of the Pauli measurements can be visualised as a cube of Bloch vectors enclosing the Bloch sphere, and so for brevity we will refer to the set Pauli * as the 'cube'. Although the cube contains non-positive and hence non-physical operators, if we are considering only Pauli measurements, then these operators give valid probability distributions under the Born rule, and hence can be used to construct local hidden variable models, just as the local quantum states do for separable states of the form of equation (1). Indeed this can be done for the Bell state of equation (2), as its density matrix can be written in a 'cube-separable' form [10,18] where , , x y z s s s are the Pauli x y , ,and z operators. Although the W i and W i T are not quantum states as they are not positive, they are unit trace operators and they correspond to certain corners of the cube. Hence we may say that the decomposition (3) furnishes a cube-separable decomposition for f f ñá + + | | , and consequently it supplies a local hidden variable model for Pauli measurements. In passing we note that any two qubit quantum state that has a local hidden variable model for Pauli measurements must have a cube-separable decomposition [28], and these constructions can be generalised to maximally entangled states of any dimension of particles, under mutually unbiased basis measurements [28,10,18].
However, as we shall now see, there is an alternative way of viewing equation (3) that leads to a local hidden variable model for much more than the Pauli measurements. The reason for this is that the convex hull of the operators W i { } in fact corresponds to a 'tetrahedron' of Bloch vectors with vertices given by four of the corners of the cube (similarly the convex hull of the W i T { } gives a tetrahedron aligned in a different direction for the second qubit). As these tetrahedra are smaller than the cube, they are actually the duals of a larger set of measurement operators containing more than just the Pauli measurements. For instance, in addition to the Pauli measurements on the first qubit we may allow POVMs of the form: and where mñ | is the state with Bloch vector 1, 1, 1 3 T ( ) (sometimes referred to as a type of 'magic' state as it brings the power of quantum computation to some architectures that otherwise can be efficiently simulated classically [29]). Similarly we may allow POVMs that are the transpositions of these for the second qubit. As the 'tetrahedron-separability' of f f ñá + + | |furnishes a local hidden variable model for any set of measurements for which the tetrahedron is the dual, these extra measurements can be included in the local hidden variable model in addition to the Pauli measurements.
These observations motivate the primary goal of this work: loosely speaking we aim to systematically find 'good' separable decompositions for bipartite entangled quantum states. Our hope is that they will lead to not only analytic local hidden variable models, but also classically efficient simulation methods for some types complex quantum systems (see e.g. [24] for an example approach). While we will consider a number of different definitions of the word 'good', one of the main guiding principles will be that the decompositions should involve local state spaces that are as 'small' as possible. In order to illustrate the kinds of results that we will obtain, in the next section we will present a family of separable decompositions that generalise the above tetrahedral-separable representation of pure Bell states to all Bell diagonal mixed states. In doing so we will be able to discuss the senses in which our decompositions are optimal.

Example decomposition: Bell diagonal qubit states
Bell diagonal mixed states are the two qubit states that are diagonal in the Bell basis: In [30] it is shown that quantum states that are diagonal in this basis may be given the following representation: ) are three real parameters. The representation gives a valid quantum state when the vector t t t t , , Within this convex hull the vectors with t t t 1 correspond to quantum separable states, while the pure maximally entangled Bell states have t t t 1 (please note that although the convex hull of the 1, 1, 1 , 1, 1, 1 , 1, 1, 1 , ) is a tetrahedron, it is conceptually distinct from the tetrahedra described elsewhere in this work, which are defined with reference to only one state particle, not two). For convenience we rewrite the representation (6) as follows where we have absorbed any negative signs into the i s of the second qubit, calling the resulting operator i s . Consider a given Bell diagonal state AB r . The techniques we describe later in this paper can be used to write down the following 'separable' decomposition for the state: and the B k are defined through the same expressions, but replacing σ with s . We define the 'local state space' for particle A as the convex hull of the A k and for particle B as the convex hull of the B k , and we denote these state spaces as V A and V B respectively. We describe the decomposition (8) as a 'separable decomposition for AB r w.r.t. V A and V B '. The state spaces are all 'tetrahedral' (i.e. simplices) in shape, and the operators A k and B k interpolate between the discrete Wigner representation for pure Bell states when t t t 1 at one extreme, and quantum-separable decomposition when t t t 1 at the other, corresponding to the threshold at which the Bell diagonal states become quantum-separable. In this sense we see that the ordinary notion of quantum separability seems to have quite natural analogous extensions for all Bell diagonal quantum states, matching the discrete Wigner representation (3) in the extreme case of pure Bell states.
As we will prove later, provided that an odd number of the t i are non-zero (which is generically the case), the decompositions described by equation (6) have a number of notable properties which we think of as being 'good': 1. The operators appearing in the decompositions (8) have the smallest norm possible, in the sense that it is not possible to provide a separable representation of any given AB r involving operators of lower average 2-norm.  1), then no smaller convex state spaces can be found that contains the quantum states while admitting a separable decomposition for AB r . The proof of this and the previous property will be presented in footnote 6 , as it requires a slight modification of the arguments presented in the main text.

No convex strict subsets of V
These three properties illustrate the three main notions of 'goodness' that we will use to identify separable decompositions for bipartite entangled quantum states, and the goal of this work will be to find decompositions for entangled quantum states that share these properties. All three notions were considered in our previous work [10], where we derived separable decompositions with these properties for certain families of bipartite pure state.
In this work we extend these results to much larger classes of state: we obtain all separable decompositions Figure 1. The image illustrates a convex hull of a tetrahedron and the sphere of quantum Bloch vectors. The tetrahedron is the convex hull of the phase point operators in the discrete Wigner representation. This local state space is a solution to Problem 2 * for Bell states. 6 A slight rewriting of (6) gives us the operator Schmidt coefficients for the state. As the t i s can be negative, we absorb any negative signs into the definition of the Paulis, and we include factors of 2 to normalise the Pauli basis: The decompositions also solve Problems 2 * and 3, as we now show. The cross norm AB I I , r   is given by the sum of the operator Schmidt coefficients: which is greater than 1 when the state is quantum entangled, as for those states t t t 1 . The decomposition (8) achieves this cross norm value, because each operator A k or B k has the same 2-norm: Suppose now that we consider the convex hulls of the A k { } and B k { } with the local quantum states and attempt to construct a separable decomposition. All the operators in these convex sets have unit trace. However, as quantum states have 2-norm less than 1, by strictness of the triangle inequality the only operators in these state spaces with norm equal to t t t 1 2 As long as either one or all three of the t i are nonzero, or equivalently an odd number of the t i are non-zero, the number of distinct operators A k { } and B k { } matches the operator-Schmidt rank. Hence in these cases the state spaces cannot be made smaller (while remaining convex and containing the local quantum states) and still enable a separable decomposition of AB r , and so we have solutions to Problem 2 * . The conic hulls of these spaces give solutions to Problem 3. Hence we see that the local state spaces constructed from the convex hulls of the A k and B k in equation (8) and the local quantum states give separable decompositions for all Bell diagonal states, and when the states are quantum entangled they give separable decompositions that solve Problems 2 and 3.
achieving the first property for all bipartite states, we obtain separable decompositions achieving the second property for all bipartite states, and we obtain separable decompositions achieving the third property for some bipartite states.

Definitions and Problems considered
Consider a convex set of local operators A  acting on system A, and convex set of local operators B  acting on system B. We say that a bipartite state AB r is , where the p k form a probability distribution, and the A k and B k are operators drawn from A  and B  respectively. We refer to A  and B  as 'local state spaces'. In the literature a bipartite quantum state is usually said to be separable if it has a separable decomposition w.r.t. local quantum state spaces (i.e. the local unit trace positive operators). However, to distinguish this notion from more general ,

A B
  ( )-separability we will call such states quantum-separable rather than just separable.
If we are to attempt to use the decomposition to provide a local hidden variable model for an entangled quantum state, or to provide a route to a classical simulation algorithm, then we would like to find decompositions involving local state spaces that are 'positive enough' for whatever our intended purpose is. There are many inequivalent ways of defining the phrase 'positive enough', depending upon the measurements or dynamics that one is considering. In the earlier examples of local hidden variable models, we saw that 'positive enough' meant being contained within the dual set of the measurements being considered. However, there are many inequivalent sets of measurements that one may consider, and moreover, if instead of local hidden variable models one seeks variables to use in classical simulation algorithms, then other notions of positivity can apply (see e.g. [24], and appendix). In spite of the wide variety of possible generalised definitions of positivity, many of them have the property that if a set of operators is positive, then so is any subset of it. This means that if we have two candidate separable decompositions for a given quantum states, such that the local state spaces for one decomposition are contained strictly within the local state spaces of the other, then the former decomposition will be preferred as the local state spaces can only be 'more' positive. The comparison of cube and tetrahedral separability for the pure Bell state was an example of this: the smaller tetrahedral state space was preferred as it provided a more powerful local hidden variable model.
Our main aim in this work will hence be to find 'small' separable decompositions for entangled quantum states. There are four variants of this problem that we will consider, depending upon different definitions of the word 'small', although two of them (Problem 2 * and Problem 3 below) are equivalent and are defined separately for later convenience. All four problems were considered in our previous work [10], where solutions were presented primarily for maximally entangled states. In this work we will generalise those results significantly. The problems are defined as follows.
• Problem 1: for a given bipartite quantum state AB r consider all pairs of local convex state spaces , r is separable, and work out the minimum (in all cases considered in this paper the minimum exists) which can be of a different form for each subsystem A and B.
In [10] this was shown to be equivalent to computing a form of cross-norm [11,12] built from •  (readers that are not familiar with cross norms may see equation (17) for a definition of the versions that we consider here). Depending upon the choice of norm, this problem may not have strong physical meaning by itself. However, we use it as a mathematical tool to solve the other variants of the problem that we consider. As discussed above, the generalised positivity of a set of operators usually implies generalised positivity of any of its subsets, so it is in our advantage to look for separable decompositions involving local state spaces that cannot be made smaller. This motivates the second of our problems.
Problems 1 and 2 contain a couple of drawbacks that we remedy by defining the following two problems, Problem 2 * and 3. While Problem 2 * and Problem 3 were both considered in [10], only Problem 3 was explicitly defined there. However they are mathematically equivalent.
• Problem 2 * : the same as Problem 2, but with the additional constraint that we identify smallest state spaces under the restriction that they must contain the local quantum states and they only contain operators with unit trace.
• Problem 3: the same as Problem 2, but with the additional constraint that the state spaces ,

A B
  ( ) must be convex cones of positive trace operators that contain the quantum states (in this case the weights appearing in the separable decomposition need only be positive-whether they form a normalised probability distribution is irrelevant as we are now dealing with cones). The motivation for this problem is explained in [10], but for convenience we recap the idea in footnote 7 . Problem 2 * and Problem 3 are mathematically equivalent because the conic hulls 8 of solutions to Problem 2 * automatically give solutions to Problem 3, and a unit trace slice through a solution to Problem 3 gives a solution to Problem 2 * . Problem 3 is only included here as it is expressed in terms of the language of conic sets, which is sometimes a more natural framework in which to consider generalisations of quantum theory [20]. However the problem we will consider directly is Problem 2 * . The reasons for defining these modified problems is to address two drawbacks with Problems 1 and 2. The first drawback is that they do not consider the fact that local quantum states may be freely used in the separable decompositions because they are positive under any notion that we will consider. This means that if one set of separating local state spaces differs from another set simply by containing (convex combinations) with additional local quantum states, it should be considered equally 'good', even though it involves larger state spaces. This is why we have added the requirement that the state spaces contain the local quantum states. Moreover as discussed in [10], including the requirement of the local quantum states means that if we have a solution to Problem 2 * it is not possible to find smaller separating local state spaces for which the dual is a larger set of local measurements. The second drawback with Problems 1 and 2 is that some solutions lead to local operators of zero trace, for instance the decomposition x y z s s s cannot be generalised positive for a complete POVM other than the trivial identity measurement, as they are traceless. Including the requirement of unit trace in Problem 2 * means that, provided that the local operators are Hermitian and bounded (which can be easily enforced for most of the constructions that we present), there will be some non-trivial POVMs (e.g. ones with measurement operators almost proportional to the identity) for which the state spaces is dual.
In order to solve these problems we will need to exploit connections to certain families of cross-norm, which we define in the next section, alongside the notation that we use.

Cross-norms and notation
In this work we will characterise the local operators and the entangled state using norms that are given by the 2-norm of the output of fixed invertible linear transformations. We denote them in the following way: where Λ is an invertible linear transformation (which will always be displayed in upper case, although the font may vary), and X 2 L   ( ) refers to the 2-norm of X L( )(the 2-norm of an operator Y is defined as Y Y tr( ) † ). As described in the description of Problem 1, we can use norms such as these to define the size of sets  of operators via the definition 7 Problem 3 is primarily motivated by the construction of LHV models. The reason for considering cones as opposed to convex sets only is that if a set of operators, say A i { }, is generalised-positive for a given process, then so is the cone of operators r A i i { } generated by multiplying the operators by arbitrary positive numbers r i . This means that if our motivation is simply to construct LHV models for a particular class of measurements without considering any transformation L (i.e. the process is M ,  ( )) then a separable decomposition w.r.t. a given pair of convex sets does not imply a LHV model for any more measurements than separability w.r.t. the cones generated by those sets. The other conditions of Problem 3 are included to ensure that the state spaces are positive for a non-trivial class of quantum measurements:the condition that the cones contain the quantum states is added to ensure that we only consider state spaces that are generalised positive for quantum effects, and the condition of positive trace is added in order to ensure that all such effects can be turned into complete measurements (in the sense that for a positive trace operator A, positivity w.r.t. quantum POVM element M automatically implies positivity w.r.t. a complete measurement of the form cM I cM , There are two reasons why norms such as (14) are useful to us: firstly the triangle inequality is strict for them, i.e.
X Y X Y , 1 5 unless X=cY for some positive number c 9 , and secondly we can often calculate the so-called cross norms [11,12,21] resulting from them explicitly, as well as all cross norm achieving decompositions. Given two invertible linear transformations , L G we define the , L G cross-norm as: The arguments presented in [10] show that minimising A )-separable is equivalent to calculating a cross norm of AB r . In particular the following identity holds inf , 18 where the infimum is taken over all pairs of sets , We will utilise the fact (in analogy to the Schmidt decomposition or Singular value decomposition for pure quantum states) that a bipartite density matrix may be given an operator Schmidt decomposition, i.e. it can be written as: where the X i and Y i are orthonormal operators bases (i.e. X X Y Y tr t r , the operator-Schmidt coefficients s i satisfy s 0 i > , and the operator-Schmidt rank is given by D, which satisfies H dim B are the dimensions of particle A and particle B respectively. In the rest of this work will usually assume that D H H dim dim ) , but the arguments can easily be extended to all permissible values of D and all finite dimensions.
We will also use the basis occurring in the operator Schmidt decomposition to represent linear operators on each subsystem. Consider an operator A on the first particle and an operator B on the second particle. We may expand where the a b , i i form a vector of expansion coefficients, which we will denote by bold font (column) vectors a a a , , . The complex conjugate is included on the b i for later convenience. In this representation the norms are given by: where v  denotes the usual complex Euclidean norm of vector v.

Summary of results
In this section we list the main results of the paper. Proofs are deferred to later sections. ). Then 9 For convenience we reproduce the following standard argument for showing that the triangle inequality is strict for the norms that we consider: The third line follows from the Cauchy-Schwarz inequality, which is strict unless X c Y L = L ( ) ( ) for some positive number c. As Λ is taken to be invertible this is equivalent to saying that the inequalities are strict unless X cY = .  with the A k linearly independent and the B k linearly independent is a solution to Problem 2, in the sense that the sets:  = å we find that the operator-Schmidt decomposition itself is a solution to Problem 2.
We note that there are close parallels between theorems 1 and 2 and theorems on ensemble decompositions of density matrices (see [32] and theorem 2.6 in [5]). The reasons for these connections will be discussed in the proofs.
Theorem 3. (Solutions to Problem 2 * and Problem 3 for some states.) For all bipartite states it is possible to choose the G appearing in equation (25) so that the operators A B , k k are of unit trace. For bipartite pure states in a region around maximally entangled states, the convex hulls of the operators A k and B k in equation (25) provide solutions to Problem 2 * provided that the p k are picked to be uniform (i.e. all equal to each other), and G is an orthogonal matrix taking the real vector with coefficients x s X tr (which is a unit vector for bipartite pure states) to the unit vector with components p k .

Proof of theorem 1: characterisation of solutions to Problem 1 for some cross norms
Before we present the proof of theorem 1, we present a derivation of a result proven in [11,12]. This will help us to establish notation and adapt the result later in the paper.
. Then the ,   cross-norm (which is usually called the projective cross 2-norm) of AB r is the sum of the operator Schmidt coefficients s , Proof. We present the proof under the assumption that D dim dim It is straightforward to modify the argument for the general case. Consider any finite decomposition of AB r into product operators E F.
where e e e , , n n n 1 2 are vectors of the expansion coefficients. Now, AB r also has an operator Schmidt decomposition In order for the two decompositions (28) and (32) to be equal, the following condition must hold where the right inequality follows from the Cauchy-Schwarz inequality. The sum of the operator Schmidt coefficients s i D i 1 å = therefore gives us a lower bound for the usual cross 2-norm AB ,   r   , but since this lower bound is in fact achieved with the decomposition (32), the inequality (34) is tight. , This result was proven in [11,12]. However we will now extend this result by finding all decompositions achieving the cross norm, by drawing upon parallels to the task of finding decompositions of positive operators [32]. We will then apply the result to obtain all cross norm achieving decompositions for the other cross-norms that we consider. These results will later be used to provide solutions to Problems 2 and Problems 2 * /3.
Our task is to work out all finite decompositions (28) such that Any solution to Problem 1 for which the norm used is the usual 2-norm can be obtained in this way. We may use these solutions to construct all solutions to Problem 1 for the other cross norms that we consider. Suppose that we would like to compute the decompositions that achieve the cross norm: If we define an operator (not necessarily a density matrix): then it is clear that The reason for this is as follows.
However the reverse inequality also holds, because any optimal decomposition achieving AB , and so: Hence all AB , r L G   achieving decompositions can be constructed by first constructing all decompositions achieving AB ,   s   using the methods described above, and then applying the inverse transformations 1 Land 1 G -. Moreover, using the fact that cross norm achieving decompositions are equivalent to separable decompositions solving Problem 1, all solutions to Problem 1 for these norms may be achieved in this way.

Proof of theorem 2
We will now show that some of the decompositions of theorem 1 also yield solutions to Problem 2. To do this we first demonstrate two lemmas that will be needed to prove the theorem.  Hence by lemma 6 any separable decomposition: for which the A k and B k are linearly independent supplies a solution to Problem 2. Let us now characterise such decompositions. Using the completeness of the operators X i and Y i (assuming that D dim dim , the arguments can be modified straightforwardly for more general cases) we may pick invertible matrices G J , defined such that: Hence from the operator Schmidt decomposition of AB r we require that: Hence we have a decomposition of AB r in terms of linearly independent operators iff

Explicitly this gives:
which is a equivalent to equation (25).
It is worth noting that these decompositions can also achieve AB r dependent cross norms, in a similar manner to the solutions to Problem 1: the cross norm corresponding to the inverse transformations 1

Proof of theorem 3
To obtain solutions to Problem 2 * (and hence also Problem 3) is more difficult than solving Problems 1 and 2. The reason is that solving the problem for all bipartite quantum states would also entail solving the quantumseparability problem, which is known to be NP hard [33]. So we expect to obtain analytical solutions only for a subset of the bipartite states. Nevertheless, our approach to obtaining these solutions will consist of two steps. In the first step we shall attempt to identify solutions to Problem 2 that contain only unit trace operators. We will be able to do this for all bipartite quantum states (in fact the argument works for all bipartite unit trace Hermitian operators). In the second step we will consider the convex hulls of the local quantum states with these state spaces, and show that for sufficiently correlated AB r the local quantum states cannot participate in a separable decomposition as they have a norm that is too small. Hence a separable decomposition can only involve the original operators, and no strict convex subset of the state space admits a separable decomposition while containing the local quantum states. Let us begin by identifying solutions to Problem 2 involving unit trace operators only. From equation (57) we see that we must find choices for G and p k such that the operators of those equations have unit trace. This is equivalent to requiring G and p k to satisfy: tr . ( ) by y, and the unit vector with coefficients p k by g. These vectors cannot be arbitrary, for instance the fact that AB r has unit trace constrains x y 1 T = . However, all three vectors can be chosen to be real (by restricting attention to operator Schmidt decompositions over the real space of Hermitian operators).
In terms of this notation we see that we are looking for an invertible matrix G and a real unit vector g (not necessarily with positive components) such that: The left inequality implies that GG x y. T = Furthermore, as long as G x T is real, then g will automatically be a unit vector because g g x GG x x y 1 T T T T = = = . Hence to find unit trace solutions to Problem 2 we only need to identify an invertible matrix G such that: There are infinitely many suitable solutions for G. If x=y, then G can be any real orthogonal matrix. If x y ¹ then the vectors x y , span a real two dimensional subspace of D  . Let us define a real orthonormal basis of two vectors q and r for this subspace such that Hence, if it were the case that for any local quantum state Ψ (either of particle A or particle B) that 1 1 then adding the quantum states to the local unit trace spaces could not provide any new separable decompositions, and the resulting state spaces would be solutions to Problem 2 * . Let us try to now understand when this is the case. Write X where the coefficients i a and i b form vectors that we denote by α and β, which must be of less than unit norm in order to describe a quantum state, so that:

Conclusions
We have considered the construction of separable decompositions for entangled quantum states that are obtained by relaxing the requirement that the local operators in the decomposition be positive unit trace quantum states. The motivation for this problem is the construction of LHV models and classically efficient simulations for bipartite entangled quantum states, or the multipartite quantum states that are built from them.
In this context it is of interest to find 'smallest' decompositions, where the sets of local operators cannot be made smaller while continuing to allow a separable description. We consider four variants of this problem. The first (Problem 1) uses norms to quantify how 'small' a set of operators is. The second (Problem 2) uses set inclusion (so that one set is smaller than another if it is contained within it). The third (Problem 2 * ) and fourth (Problem 3) variants we add restrictions (requiring the sets to be unit trace, contain the quantum states, or be cones) that are important when constructing classical models. For the first problem we present all solutions for bipartite states for some norms For the second problem we present some solutions for all bipartite quantum states. For the remaining two problems we obtain solutions for some bipartite states, including pure states in a region around the maximally entangled state.
Our results generalise those of [10], and have strong relationships to the study of generalised probabilistic theories with operator spaces [16], the study of discrete phase space distributions [18], and cross norm entanglement measures [11,12]. In the manner of [24] we believe that they may find applications in the study and simulation of entangled many-body quantum states.