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The Quantum Separability Problem for Gaussian States

Determining whether a quantum state is separable or entangled is a problem of fundamental importance in quantum information science. This is a brief review in which we consider the problem for states in infinite dimensional Hilbert spaces. We show how the problem becomes tractable for a class of Gaussian states.


I. INTRODUCTION
The concept of entanglement arose with the question of completeness of quantum theory [1]. Nowadays entanglement is regarded as a fundamental property of certain quantum states and it appears to be an important physical resource. In some sense, entanglement is synonymous of inseparability because entangled states possess some global properties that cannot be explained in terms of only the parties (subsystems) of the system. Roughly speaking, entangled states possess "strong" correlations among parties that cannot be explained within any classical local theory (because these would imply an instantaneous action at distance). Separable states may also exhibit correlations among parties, but these are purely classical and local, hence "weaker" than those underlying entanglement.
Recently, the role of entanglement became important and often necessary in many different contexts like quantum algorithms, quantum communication protocols, quantum cryptography, etc. (see e.g. [2]). So, the problem of deciding whether a given quantum state is separable or entangled has become of uppermost importance. This can be called the Quantum Separability Problem (QSP). Essentially, it represents an instance of a combinatorial optimization problems called the Weak Membership Problem [3].
Although there exists a number of characterizations of separability, there is still no feasible procedure to solve QSP in its generality (see e.g. [4] and references therein). Concerning its computational complexity, QSP is a "difficult" problem. In fact, QSP has been proved to be NP-hard [5]. However, if we restrict ourselves to specific classes of quantum states, there are examples in which QSP can be efficiently solved. For instance, this is the case of states in Hilbert space of dimension 2 or 3 [6] and certain finite sets of states [7].
In infinite dimensional Hilbert spaces, Gaussian states give rise to an important class of states for which QSP is "easy" (see e.g. [8,9] and the reference therein). In this paper, we review the formulation of QSP for infinite dimensional Hilbert spaces and we show how to tackle the problem for the class of Gaussian states.
The paper is organized as follow. In Sec.II we review some basic notions of Quantum Theory. In Sec. III we formalize the QSP. In Sec.IV we introduce the Gaussian states. In Sec.V we develop a criterion for separability of Gaussian states. Finally, some conclusions are drawn in Sec.VI.

II. BASIC NOTIONS OF QUANTUM THEORY
In this section, we introduce some terms and notions of Quantum Mechanics needed to approach the paper. Of course, the expert reader may skip this section.
In its standard formulation, Quantum Theory takes place in Hilbert spaces [10]. A Hilbert space H is a vector space over the field of complex numbers C endowed with an inner product (which induces a norm), that can have finite or infinite dimension. We use the so-called Dirac notation for a vector |ψ . Its dual is ψ|. Then, the inner product between two states |ψ and |φ reads ψ|φ ∈ C. The norm of a vector |ψ results |ψ = ψ|ψ . The following two postulate fix the mathematical representation of quantum states: Postulate II.1. The space of states of a physical system is a Hilbert space. The states are described by unit norm vectors in such Hilbert space.
Postulate II.2. The space of states of a composite system is the tensor product of Hilbert spaces of subsystems.
The structure of Hilbert space naturally leads, when considering composite systems, to the concept of entanglement. In fact, there exist states of the whole system that cannot be factorized into states of the subsystems.
It is fashinating that this seemingly abstract mathematical notion has a large impact in the description of the quantum mechanical world.
The above postulates can be generalized in terms of mixture of states, {p j , |ψ j }, where p j denotes for the probability for the system to be in the state |ψ j . This can be done by introducing the notion of density operator : Definition II.1. A density operatorρ is a non-negative, self-adjoint, trace-one class operator which is also positive semi-definite (that is ψ|ρ|ψ ≥ 0 ∀|ψ ∈ H.
Thus we can represent the mixture {p j , |ψ j } by the density operatorρ = j p j |ψ j ψ j |.
Definition II.2. A stateρ of a composite bipartite system is said to be separable iff it can be written in the form with non-negative p j 's such that j p j = 1, and whereρ (1) j ,ρ (2) j are density operators of the subsystems; the state is said to be entangled otherwise.
The physical quantities of a system that can (in principle) be measured are called observables. The next postulate fixes the mathematical representation of observables: Postulate II.3. To physical observables correspond self-adjoint operators. The possible measurement results on the observable O are the eigenvalues of the associated self-adjoint operatorÔ. The expectation value is Ô ≡ Tr(Ôρ).
Restrictions on expectation values are imposed by the following famous principle: Principle II.1 (The Uncertainty Principle). Any two observables A and B in H must satisfy, for all quantum states, the following inequality:

III. THE QUANTUM SEPARABILITY PROBLEM
In this section we introduce the Quantum Separability Problem. Let us consider a quantum system with two parties associated to a Hilbert space H 1 ⊗ H 2 ∼ = C M ⊗ C N . Notice that such a Hilbert space is isomorphic to R M 2 N 2 and it is endowed with the Euclidean inner product (X,Ŷ ) ≡ Tr(XŶ ) which induces the corresponding norm X ≡ tr(X 2 ) and distance measure X −Ŷ . Let D ⊂ H 1 ⊗ H 2 denote the set of all density operators. The set of bipartite separable quantum states, S ⊂ D, is defined as the convex hull of the separable pure states {|ψ 1 ψ| ⊗ |η 2 η|} where |ψ 1 (resp. |η 2 ) is a normalized vector in C M (resp. C N ). An arbitrary density matrix in D is parametrized by M 2 N 2 − 1 real variables. Since we deal with continuous quantities, in defining the separability problem we cannot allow infinite precision, so we need to introduce a precision parameter δ ∈ R + .
In this formulation, this problem is equivalent to an instance of a combinatorial optimization problem called Weak Membership Problem [3]. In its complete generality, QSP has been shown to be NP-hard [5]. Thus, any devised test for separability is likely to require a number of computational steps that increases very quickly with M and N . For M N ≤ 6 the positivity under Partial Transpose (see the next section) represents a necessary and sufficient test [6]. Otherwise, there only exist sufficient 'one-sided' tests for separability. In these tests, the output of some polynomial-time computable function ofρ can indicate that this is certainly entangled or certainly separable, but not both (see e.g. [4] and reference therein).

IV. GAUSSIAN STATES
In this section we introduce Gaussian states. Let us now move to M, N → ∞, thus considering two infinite dimensional Hilbert spaces H 1 and H 2 . In such spaces we can introduce continuous spectrum self-adjoint operators corresponding to canonical position and momentum variables [10]. Let us arrange them into four-dimensional column vectorsv The operators inv obey commutation relations [10] that take the compact form with There is a one-to-one correspondence between density operators and c-number Wigner distribution functions in phase space [11], the space of variables z, i.e. R 4 , in this case.
Definition IV.1. For a given density operatorρ in H 1 and H 2 the corresponding Wigner function is defined as follows [18] W (z) := Tr ρT (z) , whereT In turn, it resultsρ A density operatorρ has finite second order moments if Tr(ρq 2 j ) < ∞ and Tr(ρp 2 j ) < ∞ for all j. In this case we can define the vector mean m as m :=Tr(ρv) and the real symmetric correlation matrix V as where {∆v α , ∆v β } ≡ ∆v α ∆v β + ∆v β ∆v α is the anticommutator. It results A given V is the correlation matrix a physical state iff it satisfies as consequence of the Uncertainty Principle 2 and commutation relation (3). The correlation matrix forms a 4 × 4 matrix that transforms as an irreducible second rank tensor under the linear canonical (symplectic) transformations and has 4 invariants. If we write the correlation matrix in the block form the invariants are det A, det B, det C and Tr(AJCJBJC T J). The condition (11) implies A ≥ 1/4 and B ≥ 1/4. Moreover, Eq.(11) can be read as It is also worth remarking that any correlation matrix can be brought into the standard form with a, b, c, d ∈ R, by effecting suitable local canonical transformations corresponding to some element of Sp(2, R)× Sp(2, R) ⊂ Sp(4, R). Now we are ready to give the definition of Gaussian state: Definition IV.2. A stateρ is called Gaussian if its Wigner function takes the form with m a real 4-vector and V a real symmetric 4 × 4 -matrix.
One can show that m is indeed the mean and V is the correlation matrix. These define the Gaussian state uniquely. In what follows, we simply consider the case m = 0, because m can be easily removed by some local displacement and thus has no influence on the separability or inseparability of the state.

V. A SEPARABILITY CRITERION FOR GAUSSIAN STATES
In this section, we describe how to solve QSP for Gaussian states. Let us consider a separable stateρ sep of the form (1) in the Hilbert space H 1 ⊗ H 2 . Let us choose a generic couple of observables for each subsystem, sayr j ,ŝ j on H j (j = 1, 2), withĈ Then, we introduce the following observables on H 1 ⊗ H 2 : with a j , b j ∈ R. From the the Uncertainty Principle 2, it follows that every stateρ on H 1 ⊗ H 2 must satisfy However, for separable states, a stronger bound exists. We have in fact the following theorem [12]: Theorem V.1. For any separable state the following implication holds: where with being Ĉ j k ≡ Tr[Ĉ jρ . The theorem can be proved with the help of a family of linear inequalities which must be always satisfied by separable states. The convolution of such relations gives the condition (19), representable by a region in the (∆û) 2 , (∆v) 2 plane delimited by an hyperbola. Notice that, since the following inequalities hold In particular, Eq. (24) tells us that the bound (19) for separable states is much stronger than Eq. (18) for generic states. Moreover, Eq. (24) gives us a simple separability criterion. In fact, while W is not easy to evaluate directly, as it depends on the type of convex decomposition (1) that one is considering, the right hand side of Eq. (24) is easily measurable, as it depends on the expectation value of the observablesĈ j . Then, we can claim that Eq. (19) is a necessary criterion for separability, i.e.
Example V.1. An important simplification applies when the observableĈ j is proportional to the identity operator, e.g.r j ≡q j andŝ j ≡p j . In such a case, Eq.(18) reduces to while Eq.(19) reduces to Let us now consider the case in whichr j ,ŝ j are linear combinations of canonical observablesq j andp j , i.e.
where a 3 , a 4 , b 3 , b 4 ∈ R are generic real parameters. Then Eq. (19), taking into account Eq.(3), becomes that should be compared with It is easy to verify that, given a j , b j (j = 1, . . . , 4), the "product condition" (29) implies the "sum condition" (30). However, if we require Eqs. (29) and (30) to be verified for all possible values of the coefficients a j , b j , the two are equivalent since it is possible to re-obtain one from another using a convolution trick, like the used with Eqs. (19) and (22) (the one-to-one correspondence between quadratic and linear tests under all circumstances has been also pointed out in Ref. [13]). It turns out that the restriction is necessary and sufficient for separability of Gaussian states [14,15]. However, solving QSP by testing the condition (31) would be hard from a complexity point of view, due to the presence of the universal quantifier at right hand side.
Nevertheless, the condition (31) can be rephrased in a simpler way. First notice that the uncertainty relation satisfied by all (separable and inseparable) states and corresponding toû andv formed from Eq. (19), is equivalent to as much as like Eqs. (29) and (30). Then, what is the relation between conditions (30) and (33)? They are simply related by the partial transpose transform This operation invertsp 2 , leavingq 1 ,p 1 , andq 2 unchanged [19]. In fact, separable states satisfies the usual uncertainty relation (33) and the analogous one obtained under partial transpose; thus these satisfy the condition (30).
On the other hand, the transformation (34) changes the correlation matrix as V →Ṽ = ΛV Λ. Hence, the compact uncertainty relation (11) becomesṼ Expressed in terms of invariants, the condition (35) forṼ takes a form identical to (13). The signature in front of det C in the second term on the left hand side is changed. Thus, if we write the requirement that the correlation matrix of a separable state has to obey (35), in addition to the fundamental uncertainty principle (11), can be stated as follow The necessity follows from theorem V.1. The sufficiency follows from the fact that Gaussian states with correlation matrix having det C ≥ 0 are separable [14].
The statement V.2 is equivalent to the condition (31), but much more effective to be used. Given the standard form (14) of the correlation matrix V we can consider the space of all possible Gaussian states as isomorphic to R 4 , while the set of physical states is a subspace G ⊂ R 4 defined through (11). Furthermore, the equation f (V ) = 0 reads The equation defines the surface S of the subset S ⊂ G of separable states. Then, by simply evaluating f we can say whether a given state (point in G) is within S (hence separable) or not (hence entangled). This is an easy computational task that can be efficiently accomplished. In reality, taking into account a finite accuracy δ, we can only say that the state is almost separable (resp. almost entangled) within δ. Nevertheless, if we want to assert that the state is strictly separable (resp. strictly entangled), we have to be sure that the distance of the stateρ from the surface S is greater than 1/δ. That is According to Sec.III, the distance between two states is considered as ρ −ρ ′ ≡ Tr [(ρ −ρ ′ ) 2 ] and for Gaussian states this can be expressed through Wigner functions (hence correlation matrices) as Such a task can be efficiently accomplished with the aid of geometrical arguments and simple algorithms. For instance, a software package that efficiently find all hyperplanes tangent to the surface S, from which evaluate the l.h.s. of Eq(38), is already available [16].

VI. CONCLUSIONS
Summarizing, we have given a brief review of QSP for Gaussian states of two parties. The problem has been approached by developing tests that involve variances to arrive at an efficient solution based on the invariance (positivity) of only separable states under partial transpose. Notice that this argument can be further generalized to partial scaling transforms to which partial transpose belongs. In fact, while K and V are always invariant under linear canonical transformations, they are not invariant under scale changes on thev that are not contained in Sp(4, R).
The necessary condition for the separability of the state is Notice, however, that for multipartite systems besides separability (resp. inseparability) there can be the possibility of partial separability (resp. partial inseparability), e.g. separability of a subsystem with respect to the others which in turns are entangled [9]. Hence QSP becomes much more subtle and even for Gaussian states it is not completely understood.