Gaussian entanglement revisited

Gaussian entanglement revisited

Ludovico Lami    Alessio Serafini    Gerardo Adesso
Abstract

We present a novel approach to the problem of separability versus entanglement in Gaussian quantum states of bosonic continuous variable systems, as well as a collection of closely related results. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of vs modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of multimode Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for Gaussian states of vs modes, for arbitrary . We then proceed to show the novel result that Gaussian states invariant under partial transposition are separable. Next, we provide a previously unknown extension of the PPT-separability equivalence to arbitrary Gaussian states of vs modes that are symmetric under the exchange of any two modes belonging to one of the parties. Further, we include a new proof of the sufficiency of the PPT criterion for separability of isotropic Gaussian states, not relying on their mode-wise decomposition. In passing, we also provide an alternative proof of the recently established equivalence between separability of an arbitrary Gaussian state and its complete extendability with Gaussian extensions. Finally, we prove that Gaussian states which remain PPT under passive optical operations cannot be entangled by them either; this is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject. This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem.

plus 0.3ex

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Gaussian states have played a privileged role in quantum optics and bosonic field theories, essentially since the very early steps of such theories, due to their ease of theoretical description and relevance to experimental practice. Over the last twenty years, such a privilege has carried over to quantum information science, where Gaussian states form the core of the ‘continuous variable’ toolbox introeisert (); biblioparis (); adesso07 (); weedbrook12 (); adesso14 (); bucco (). The analysis of quantum Gaussian states from the information theoretic standpoint brought up new subtle elements and much previously unknown insight into their structure. For, while Gaussian dynamics may essentially be dealt with entirely at the phase space level (typically by normal mode decomposition, so that Gaussian dynamics are often trivialised as ‘quasi-free’ in field theory), the analysis of quantum information properties requires one to confront the Hilbert space description of the quantum states. Hence, while Gaussian dynamics might well be exactly solvable with elementary tools, the properties of Gaussian states related to the Hilbert space and tensor product structures are far from being equally transparent.

The problem of Gaussian separability, that is, determining whether a bipartite Gaussian state is separable or entangled adesso07 (), exemplifies such a situation very well. Necessary and sufficient conditions for Gaussian separability in the general vs -mode case are available Werner01 (); Giedke01 (), yet they are recast in terms of convex optimisation problems whose solution (albeit numerically efficient) does not admit, in general, a closed analytical form. For non-separable states, a closely related question is whether their entanglement is distillable or bound HorodeckiBound (). In the case of arbitrary multimode Gaussian states, while entanglement can never be distilled by Gaussian operations alone nogo1 (); nogo2 (); nogo3 (), it is known that entanglement distillability under general local operations and classical communications is equivalent to violation of the positivity of the partial transposition (PPT) criterion Giedke01 (); GiedkeQIC (). In turn, the PPT criterion, which is as well efficiently computable at the level of covariance matrices for Gaussian states, and is in general only necessary for separability Peres (), has been proven to be also sufficient in some important cases, notably when the bipartite Gaussian state under examination pertains to a vs -mode system Simon00 (); Werner01 (), when it is ‘bi-symmetric’ Serafini05 (), i.e. invariant under local permutations of any two modes within any of the two subsystems, and when it is ‘isotropic’, i.e. with a fully degenerate symplectic spectrum of its covariance matrix holwer (); botero03 (); giedkemode (). Outside of these special families, bound entangled Gaussian states can occur, as first shown in the vs -mode case in Werner01 ().

In this paper we provide significant advances towards the characterisation of separability and entanglement distillability in Gaussian quantum states. On one hand, we revisit the existing results, providing in particular a new compact proof for the equivalence between PPT and separability in vs -mode Gaussian states, which encompasses the seminal vs -mode case originally tackled by Simon Simon00 () and its extension settled by Werner and Wolf Werner01 (). Key to our proof is the intensive use of Schur complements, which have enjoyed applications in various areas of (Gaussian) quantum information theory Giedke01 (); giedkemode (); nogo1 (); nogo2 (); nogo3 (); eisemi (); gian (); Simon16 (); Lami16 (), and — as further reinforced by this work — may be appreciated as a mathematical cornerstone for continuous variable quantum technology.

On the other hand, we derive a number of novel results. In particular, a marginal extension of the techniques applied in the aforementioned proof allow us to prove that Gaussian states invariant under partial transposition are necessarily separable, a result previously known only for the partial transposition of qubit subsystems sep 2xN (). We then show that the vs -mode PPT-separability equivalence can be further extended to a class of arbitrary bipartite multimode Gaussian states that we call ‘mono-symmetric’, i.e., invariant under local exchanges of any two modes on one of the two subsystems (see Fig. LABEL:mononucleosi). This result, which (to the best of our knowledge) is observed and proven here for the first time, generalises the case of bi-symmetric states studied in Serafini05 (), providing as a byproduct a simplified proof for the latter as well.

As for isotropic Gaussian states, in the traditional approach the sufficiency of PPT for their separability follows from a well known ‘mode-wise’ decomposition of pure-state covariance matrices holwer (); botero03 (); giedkemode (), and from the fact that the covariance matrix of an isotropic state is just a multiple of the covariance matrix of a pure Gaussian state. Here, we derive the sufficiency of the PPT criterion for isotropic Gaussian states following a completely different and arguably more direct approach. Main ingredients of this novel proof are advanced matrix analysis tools such as the operator geometric mean, already found to be useful in the context of quantum optics Lami16 ().

We also consider the well known class of Gaussian passive operations (i.e., the ones that preserve the average number of excitations of the input state, such as beam splitters and phase shifters), which play a central role in quantum optics introeisert (); adesso14 (); bucco (), and we prove that a Gaussian state that always remains PPT under such a set of operations must also always stay separable. This novel result complements the seminal study of passive (), in that the latter only considered the possibility of turning a PPT state into a non-PPT one through passive operations — essentially, the question of generating distillable entanglement — which is not the same as the question of generating inseparability, because Gaussian PPT bound entangled states do exist Werner01 (). Here we settle the latter, more general and fundamental question.

All the previous results enable us to substantially extend the range of equivalence between Gaussian separability and PPT in contexts of strong practical relevance. Last but not the least — in fact, first and foremost in the paper — we address the separability problem directly, and derive a simplified necessary and sufficient condition for Gaussian separability. For a bipartite state, this requires convex optimisation over marginal covariance matrices on one subsystem only, yielding a significant simplification over the existing criteria, which instead require optimisation on both parties Werner01 (); Giedke01 (); eisemi ().

Figure 1: Mono-symmetric Gaussian states of two parties (with modes) and (with modes) are invariant under exchange of any two modes within party . By means of a suitable symplectic transformation on subsystem , these states can be reduced to a vs -mode Gaussian state and a collection of uncorrelated single-mode states on ’s side. Since PPT is equivalent to separability for vs -mode Gaussian states, it follows that PPT is necessary and sufficient for separability of all vs -mode mono-symmetric Gaussian states. In the schematics, entanglement between pairs of modes from the same party is depicted as a single solid (black) line, while entanglement across a mode from and a mode from is depicted as a double (dark red) line.

This paper is organised as follows: in Sec. LABEL:methods, the definition and basic properties of Schur complements and matrix means that will be used in our derivations are recalled, and the Gaussian notation is set; Sec. LABEL:secp contains our main novel finding: a simplified necessary and sufficient condition for Gaussian separability in the general vs -mode case; Sec. LABEL:ppt contains a new proof of the sufficiency of the PPT criterion for separability of vs -mode Gaussian states achieved in a few, swift Schur complements’ manipulation; Sec. LABEL:inva shows that invariance under partial transposition implies separability; in Sec. LABEL:symm we prove that mono-symmetric states are amenable, under local unitary operations, to the vs -mode case, thus implying that PPT is sufficient for them too (see Fig. LABEL:mononucleosi); in Sec. LABEL:mode-wise we provide the reader with a new proof of the sufficiency of PPT for separability of isotropic Gaussian states, not relying on their mode-wise decomposition; Sec. LABEL:pass contains our novel analysis of entanglement generation under passive operations. Sec. LABEL:outro concludes the paper with a brief summary and some future perspectives related to this work.

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One of the messages of the present paper is to lend further support to the fact that methods based on Schur complements and matrix means can be successfully exmployed to derive fundamental results in continuous variable quantum information, following a streak of applications to various contexts including separability, distillability, steerability, entanglement monogamy, characterisation of Gaussian maps, and related problems Giedke01 (); giedkemode (); nogo1 (); nogo2 (); nogo3 (); eisemi (); gian (); Simon16 (); Lami16 (). As a divertissement to set the stage, let us present a compact, essential compendium of such methods.

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Given a square matrix partitioned into blocks as

(0)

the Schur complement of its (square, invertible) principal submatrix , denoted by , is defined as

(0)

A useful reference on Schur complements is the monograph ZHANG05 (). Here we limit ourselves to stress some of the properties we will make use of in the present paper. As it turns out, Schur complements are the answer 42 () to a number of questions that arise pretty naturally in matrix analysis. Many of these applications stem from the fact that the positivity conditions of hermitian block matrices can be easily written in terms of Schur complements.

Lemma 1.

Consider a hermitian matrix

(0)

Then is strictly positive definite () if and only if and . Then, by taking suitable limits, is semidefinite positive () if and only if and for all .

A consequence of this result that will be relevant to us is the following.

Corollary 2.

Let be a hermitian matrix partitioned as in (LABEL:H_part). Then, if ,

(0)

Here we mean that the matrix set on the right-hand side has a supremum (i.e. a minimum upper bound) with respect to the Löwner partial order ( if and only if is positive semidefinite), and that this supremum is given by the Schur complement on the left-hand side.

We note in passing that from the above variational representation it follows immediately that is monotone and concave in .

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Somehow related to Schur complements are the so-called matrix means. As one might expect from their name, these are functions taking two positive matrices as inputs and yielding another positive matrix as output. For an excellent introduction to this topic, we refer the reader to (BHATIA, , Chapter 4). Here, we review only some basic facts that we will find useful throughout the paper. Given two strictly positive matrices , the simplest mean one can define is the arithmetic mean , whose generalisation from scalars to matrices does not present difficulties. Another easily defined object is the harmonic mean parallel sum (); ando79 (), denoted by and given by

(0)

Incidentally, the harmonic mean can be also defined as a Schur complement, with the help of the identity , which immediately implies that is monotone and jointly concave in and , i.e. concave in the pair .

The least trivially defined among the elementary means is undoubtedly the geometric mean between strictly positive matrices geometric mean (); ando79 (), which can be constructed as

(0)

where the above maximisation is with respect to the Löwner partial order, and the fact that the particular set of matrices we chose admits an absolute maximum is already nontrivial. With a bit of work one can show that is explicitly given by

(0)

Having multiple expressions for a single matrix mean is always useful, as some properties that are not easy to prove within one formulation may become apparent when a different approach is taken. For instance, the fact that is covariant under congruences, i.e.  for all invertible , is far from transparent if one looks at (LABEL:geom_expl), while it becomes almost obvious when (LABEL:geometric) is used. On the contrary, the fact that when is not easily seen from (LABEL:geometric), but it is readily verified employing (LABEL:geom_expl).

As it happens with scalars, the inequality

(0)

holds true for all . In view of the above inequality, it could be natural to wonder, how the geometric mean between the leftmost and rightmost sides of (LABEL:hga) compares to . That this could be a fruitful thought is readily seen by asking the same question for real numbers. In fact, when it is elementary to verify that . Our first result is a little lemma extending this to the non-commutative case. We were not able to find a proof in the literature, so we provide one.

Lemma 3.

For strictly positive matrices, the identity

(0)

holds true.

Proof.

We start by defining , . It is easy to see that , for instance because . Therefore, the identity holds. Now, on the one hand the congruence covariance of the geometric mean implies that

On the other hand,

Putting all together, we see that

Conjugating by , we obtain

(0)

where the last step is an application of (LABEL:geom_expl). ∎

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In the remainder of this Section, we provide a brief introduction to the main concepts of the Gaussian formalism. Quantum continuous variables just describe quantum mechanics applied to an infinite-dimensional Hilbert space equipped with position and momentum operators () satisfying the so-called canonical commutation relations (in natural units, ). Such a Hilbert space describes, for instance, a collection of quantum harmonic oscillators, or modes of the electromagnetic radiation field. The operators are often grouped together to form a single vector of operators . The canonical commutation relations then take the form

(0)

An important object one can form is the displacement operator. For any , we define

(0)

It turns out that Nature has a special preference for quadratic Hamiltonians. A prominent example is the free-field Hamiltonian . Not surprisingly, thermal states of quadratic Hamiltonians are extremely easily produced in the lab, in fact so easily that they deserve a special name: Gaussian states introeisert (); biblioparis (); adesso07 (); weedbrook12 (); adesso14 (); bucco (). As the name suggests, they can be fully described by a real displacement vector and a real, quantum covariance matrix (QCM) , defined respectively as and . By quantum covariance matrix we mean a real, symmetric, strictly positive matrix that moreover satisfies the Heisenberg uncertainty relation simon94 ()

(0)

Note that (LABEL:Heisenberg) can equivalently be written as upon applying transposition (as , ).

The Gaussian state with QCM and displacement vector admits the representation

(0)

which justifies the alternative definition of Gaussian states as the continuous variable states associated with a Gaussian characteristic function.

Clearly, linear transformations that preserve the commutation relations (LABEL:CCR) play a special role within this framework. Any such transformation is described by a symplectic matrix, i.e. a matrix with the property that . Symplectic matrices form a non-compact, connected Lie group that is additionally closed under transposition, and is typically denoted by pramana (). The importance of these operations arises from the fact that for any symplectic there is a unitary evolution on the Hilbert space such that . Most importantly, such a unitary is the product of a finite number of factors , where is a quadratic Hamiltonian, and as such it can be easily implemented in laboratory. Under conjugation by , Gaussian states transform as

(0)

It turns out that all Gaussian states can be brought into a remarkably simple normal form via unitary transformations induced by quadratic Hamiltonians. In fact, a theorem by Williamson willy (); willysim () implies that for all strictly positive matrices there is a symplectic transformation and a diagonal matrix such that

(0)

The diagonal elements , each taken with multiplicity one, are called symplectic eigenvalues of , and are uniquely determined by (up to their order, which can be assumed decreasing by convention with no loss of generality). Accordingly, we will refer to as the symplectic spectrum of . Notably, Heisenberg uncertainty relation (LABEL:Heisenberg) can be conveniently restated as , or equivalently for all . A Gaussian state can be shown to be pure if and only if all of its symplectic eigenvalues are equal to , which corresponds to the matrix equality . Correspondingly, a QCM satisfying for all (or equivalently ) will be called a pure QCM. Note that pure QCMs are themselves symplectic matrices, , and are the extremal elements in the convex set of QCMs.

Finally, note that displacement vector is often irrelevant since it can be made to vanish by local unitaries, resulting from the action of the displacement operator of (LABEL:displacement) on each individual mode. Since all the physically relevant informational properties such as purity and entanglement are invariant under local unitaries, all the results we are going to present will not depend on the first moments. Therefore, in what follows, we will completely specify any Gaussian state under our investigation as in terms of its QCM alone.

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The QCM of a Gaussian state pertaining to a -mode bipartite system can be naturally written in block form according to the splitting between the subsystems and :

(0)

According to the same splitting, the matrix appearing in (LABEL:CCR) takes the form

(0)

with and .

The entanglement properties of a bipartite Gaussian state can thus be conveniently translated at the level of QCMs. Recall that, in general, a bipartite quantum state is separable if and only if it can be written as a convex mixture of product states, , with being probabilities Werner89 (). For a Gaussian state of a bipartite continuous variable system, we have then the following.

Lemma 4 (Proposition 1 in Werner01 ()).

A Gaussian state with -mode QCM is separable if and only if there exist an -mode QCM and an mode QCM such that

(0)

In view of the above result, a QCM satisfying (LABEL:sep_eq) for some marginal QCMs , will itself be called separable from now on. The criterion in (LABEL:sep_eq) is necessary and sufficient for separability of QCMs, and can be evaluated numerically via convex optimisation Giedke01 (); eisemi (), however such optimisation runs over both marginal QCMs, hence scaling (polynomially) with both and .

The first main result of this paper is to show that the necessary and sufficient separability condition (LABEL:sep_eq), for any and , can be further simplified. This result is quite neat and of importance in its own right. In particular, it allows us to recast the Gaussian separability problem as a convex optimisation over the marginal QCM of one subsystem only (say without loss of generality), resulting in an appreciable reduction of computational resources, especially in case party comprises a much smaller number of modes than party .

Theorem 5 (Simplified separability condition for an arbitrary QCM).

A QCM of modes is separable if and only if there exists an -mode QCM such that

(0)

In terms of the block form (LABEL:V_explicit) of , when the above condition is equivalent to the existence of a real matrix satisfying

(0)

If is not invertible, we require instead for all .

Proof.

Since both sets of QCMs defined by (LABEL:sep_eq) and (LABEL:simp_sep_1) are clearly topologically closed, we can just show without loss of generality that their interiors coincide. This latter condition can be rephrased as an equivalence between the two following statements: (i) for some QCMs ; and (ii) for some QCM .

Now, once is fixed, the supremum of all the matrices satisfying is given by the Schur complement , as the variational characterisation (LABEL:variational) reveals. Therefore, statement (i) is equivalent to the existence of such that . This is the same as to require , as the positivity conditions of Lemma LABEL:pos_cond immediately show.

Until now, we have proven that the separability of can be restated as for some appropriate QCM . Employing Lemma LABEL:pos_cond, we see that this is turn equivalent to (LABEL:simp_sep_2), or to its -modified version when is not invertible. ∎

Remark 1.

It has been recently observed Bhat16 () that condition (LABEL:simp_sep_2) is equivalent to the corresponding Gaussian state with QCM being completely extendable with Gaussian extensions. We remind the reader that a bipartite state is said to be completely extendable if for all there exists a state that is: (i) symmetric under exchange of any two systems; and (ii) an extension of in the sense that . When the original state is Gaussian, it is natural to consider extensions of Gaussian form as well. Interestingly enough, the above Theorem LABEL:simp_sep_lemma provides a simple alternative proof of the remarkable fact (also proven in Bhat16 ()) that Gaussian states are separable if and only if completely extendable with Gaussian extensions.

Remark 2.

It is worth noticing that both Lemma LABEL:sep and Theorem LABEL:simp_sep_lemma extend straightforwardly to encompass the case of full separability of multipartite Gaussian states. In the case of Lemma LABEL:sep, this extension was already formulated in Werner01 (); 3-mode sep (). As for Theorem LABEL:simp_sep_lemma, the corresponding necessary and sufficient condition for the full separability of a -partite QCM would read for appropriate QCMs .

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We now focus on investigating known and new conditions under which separability becomes equivalent to PPT for Gaussian states, so that the problem of deciding whether a given QCM is separable or not admits a handy formulation.

For any bipartite state , recall that the PPT criterion provides a useful necessary condition for separability Peres ():

(0)

where the suffix denotes transposition with respect to the degrees of freedom of subsystem only. In finite-dimensional systems, PPT is also a sufficient condition for separability when H3 ().

In continuous variable systems, the PPT criterion turns out to be also sufficient for separability of QCMs when either or is composed of one mode only Simon00 (); Werner01 ().

Theorem 6 (PPT is sufficient for Gaussian states of vs modes).

Let be a bipartite QCM such that either or are composed of one mode only. Then is separable if and only if

(0)

which amounts to the corresponding Gaussian state being PPT, .

For completeness, we recall that the partial transpose of an -mode QCM (i.e., the covariance matrix of the partially transposed density operator ) is given by , where with respect to a mode-wise decomposition on the subsystem the matrix can be written as , with Simon00 (). Accordingly, we can say that the QCM is PPT if and only if is a valid QCM obeying (LABEL:Heisenberg), which is equivalent to (LABEL:PPT).

The original proof of Theorem LABEL:PPT_thm came in two steps. Firstly, Simon Simon00 () proved it in the particular case when both and are made of one mode only by performing an explicit analysis of the symplectic invariants of ; this seminal analysis is quite straightforward to follow and particularly instructive, but eventually a bit cumbersome, since it requires to distinguish between three cases, according to the sign of , where is the off-diagonal block of the QCM partitioned as in (LABEL:V_explicit). Later on, Werner and Wolf Werner01 () reduced the problem for the vs -mode case with arbitrary to the vs -mode case; the proof of this reduction is geometric in nature and rather elegant, but also relatively difficult.

Our purpose in this Section is to use Schur complements to provide the reader with a simple, direct proof of Theorem LABEL:PPT_thm. Before coming to that, there is a preliminary lemma we want to discuss.

Lemma 7.

Let be hermitian matrices. There is a real symmetric matrix satisfying if and only if , where denotes complex conjugation.

Proof.

The only complex entry in a hermitian matrix is in the off-diagonal element. Suppose without loss of generality that and (both conditions in the statement are in fact symmetric under complex conjugation of or ). It is easy to verify that a such that and always exists, and we see that is a real symmetric matrix. Moreover, since belongs to the segment joining and we conclude that . ∎

Remark 3.

Lemma LABEL:2x2_interval admits an appealing physical interpretation which also leads to an intuitive proof. This interpretation is based on the fact that hermitian matrices can be seen as events in -dimensional Minkowski space-time through the correspondence . Furthermore, translates in Minkowski space-time to ‘ is in the absolute future of ’, since the remarkable determinantal identity holds true. Now, the complex conjugation at the matrix level becomes nothing but a spatial reflection with respect to a fixed spatial plane in Minkowski space-time. Thus, our original question is: is it true that whenever both an event and its spatial reflection are in the absolute future of a reference event then there is another event which is: (i) in the absolute future of ; (ii) in the absolute past of both and ; and (iii) lies right on the reflection plane? The answer is clearly yes, and there is a simple way to obtain it. Start from and shoot a photon to the location of that event between and that will happen on the other side of the reflection plane. After some time the photon hits the plane, and this event clearly satisfies all requirements.

Now we are ready to give our direct proof of the equivalence between PPT and separability for vs -mode Gaussian states, leveraging the simplified separability condition of Theorem LABEL:simp_sep_lemma.

Proof of Theorem LABEL:PPT_thm.

Suppose without loss of generality that is composed of one mode only. As in the proof of Theorem LABEL:simp_sep_lemma, since both sets of QCMs defined by (LABEL:sep_eq) and (LABEL:PPT) are topologically closed, we can assume that is in the interior of the PPT set, i.e. that . Our goal will be to show that in this case belongs to the separable set, as characterized by Theorem LABEL:simp_sep_lemma. Since is taken to be invertible, the PPT condition reads

Now, define and , and observe that . Thanks to Lemma LABEL:2x2_interval, we can find a real matrix such that

Choosing the negative sign in the above inequality, we see that the second condition (LABEL:simp_sep_2) in Theorem LABEL:simp_sep_lemma is met, and therefore is separable. ∎

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As a further example of application of Theorem LABEL:simp_sep_lemma, we study here the separability of a special class of PPT Gaussian states, i.e. those that are invariant under partial transposition of one of the subsystems. This problem has an analogue in finite-dimensional quantum information, already studied in sep 2xN (), where it was shown that bipartite states on that are invariant under partial transpose on the first system are necessarily separable.\hyper@linkstartlink\@currentHref-\@mpfn.\@textsuperscript0\hyper@linkend\hb@xt@1.8em\@textsuperscript0The proof reported in sep 2xN () is rather long, so here we provide a shorter one, again based on Schur complements. A state on that is invariant under partial transposition on the first subsystem can be represented in block form as . By a continuity argument, we can suppose without loss of generality that . Rewrite . Both terms are positive by Lemma LABEL:pos_cond. Since the second one is separable, let us deal only with the first one, call it . We have , where is hermitian. Denoting by its spectral decomposition, we obtain the following manifestly separable representation of :

\@finalstrut

Here we show that for Gaussian states an even stronger statement holds, in that invariance under partial transposition implies separability for any number of local modes.

Corollary 8.

A bipartite Gaussian state that is invariant under partial transposition of one of the two subsystems is necessarily separable.

Proof.

Without loss of generality, we can assume that the partial transpose on the system leaves the state invariant. We now show that under the this assumption the separability condition (LABEL:simp_sep_2) is immediately satisfied, since the rightmost side is already a real, symmetric matrix. In fact, equating the original QCM (LABEL:V_explicit) with the one obtained after partial transpose on the system, we get the identities and , where, as previously set, according to a mode-wise decomposition of the system, and . As a consequence,

where we used also . This shows that is equal to its complex conjugate, and is therefore (despite appearances) a real symmetric matrix. Hence the separability condition (LABEL:simp_sep_2) is satisfied with . ∎

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Throughout this Section, we show how the PPT criterion is also necessary and sufficient for deciding the separability of bipartite Gaussian states of vs modes that are symmetric under the exchange of any two among the first modes. These states will be referred to as mono-symmetric (with respect to the first party ). As can be easily seen, this novel result (see Fig. LABEL:mononucleosi for a graphical visualisation) is a generalisation of both Theorem LABEL:PPT_thm and of one of the main results in Serafini05 (), where the subclass of bi-symmetric states was considered instead, bi-symmetric meaning that they are invariant under swapping any two modes either within the first or within the last (that is, they are mono-symmetric in both and ).

Theorem 9 (Symplectic localisation of mono-symmetric states).

Let be a mono-symmetric Gaussian state of modes, i.e. specified by a QCM that is symmetric under the exchange of any two of the modes of subsystem . Then there exists a local unitary operation on corresponding to a symplectic transformation that transforms into the tensor product of uncorrelated single-mode Gaussian states () and a bipartite Gaussian state of vs modes. At the QCM level, this reads

(0)

The separability properties of and are equivalent, in particular is separable if and only if it is PPT.

Proof.

We will prove (LABEL:monolocale) directly at the QCM level, by constructing a suitable local symplectic . By virtue of the symmetry under the exchange of any two modes of subsystem , if we decompose as in (LABEL:V_explicit), the submatrices and have the following structure:

(0)

where each one of the blocks in (LABEL:strucaz) is a real matrix, with and symmetric adescaling ().

We can now decompose the real space of the first modes as . According to this decomposition, we may rewrite and as follows:

(0)

where , with denoting the standard basis for . Observe that the symplectic form on subsystem decomposes accordingly as . If is an orthogonal matrix such that , we easily see that on the one hand , i.e.  is symplectic, while on the other hand

(0)
(0)

Therefore, the initial QCM has been decomposed as a direct sum of one-mode QCMs , and of one -mode QCM , via a local symplectic operation on subsystem , given precisely by . This proves (LABEL:monolocale) constructively. Applying Theorem LABEL:PPT_thm, one then gets immediately that the PPT condition is necessary and sufficient for separability in this case. ∎

Remark 4.

This original result yields a substantial enlargement to the domain of validity of PPT as a necessary and sufficient criterion for separability of multimode Gaussian states, reaching beyond any existing literature. In practice, Theorem LABEL:PPt_sym tells us that, in any mono-symmetric Gaussian state, all the correlations (including and beyond entanglement) shared among the whole modes of and the whole modes of can be localised onto correlations between a single mode of vs the whole , by means of a local unitary (symplectic at the QCM level) operation at ’s side only. Being unitary, this operation is fully reversible, meaning that the correlations with can be redistributed back and forth between and the whole set of modes with no information loss. This also means that quantitative results on any measure of such correlations between and encoded in can be conveniently evaluated in the much simpler vs -mode normal form constructed in the proof Theorem LABEL:PPt_sym, ignoring the uncorrelated modes.

In the special case of being the QCM of a bi-symmetric state, i.e. with full permutation symmetry within both and , it is immediate to observe that applying a similar construction by means of a local unitary at ’s side as well fully reduces to a two-mode QCM , with equivalent entanglement properties as the original , plus a collection of uncorrelated single modes. This reproduces the findings of Serafini05 ().

Similarly to what discussed in Remark LABEL:remulti, the results of Theorem LABEL:PPt_sym can also be straightforwardly extended to characterise full separability and, conversely, multipartite entanglement of arbitrary multimode Gaussian states which are partitioned into subsystems, with the requirement of local permutation invariance within some of these subsystems. It is clear that, by suitable local symplectic transformations, each of those locally symmetric parties can be localised onto a single mode correlated with the remaining parties, thus removing the redundancy in the QCM. Gaussian states of this sort generalise the so-called multi-symmetric states studied in moleculo (), where local permutation invariance was enforced within all of the subsystems, resulting in a direct multipartite analogue of bi-symmetric states.

\@xsect

It is well known that the PPT criterion is in general sufficient, as well as obviously necessary, for pure bipartite states to be separable Peres (). This may be seen by a direct inspection of the Schmidt decomposition of a pure state. Let us note, incidentally, that a stronger statement holds, namely any bound entangled state (in any dimension) must have at least rank chen08 ().

The Schmidt decomposition theorem is in fact so important that a Gaussian version of it, that is, the determination of a normal form of pure QCMs under local symplectic operations, is of central importance in continuous variable quantum information. As can be shown at the covariance matrix level holwer (); giedkemode () or at the density operator level botero03 (), every pure bipartite Gaussian state can be brought into a tensor product of two-mode squeezed vacuum states and single-mode vacuum states by means of local unitaries with respect to the vs partition. In particular, by acting correspondingly with local symplectic transformations, any pure QCM (where pure means ) can be transformed into a direct sum of (pure) two-mode squeezed vacuum QCMs and (pure) single-mode vacuum QCMs. More precisely, at the level of QCMs, one can formulate this fundamental result as follows.

Theorem 10 (Mode-wise decomposition of pure Gaussian states holwer (); botero03 (); giedkemode ()).

Let be a bipartite QCM of modes , assuming (with no loss of generality). If is a pure QCM, i.e. all its symplectic eigenvalues are equal to (which amounts to ), then there exist local symplectic transformations , mapping into the following normal form:

(0)

where with and , for a real squeezing parameter , is the pure QCM of a two-mode squeezed vacuum state of modes and , and is the pure QCM of the single-mode vacuum state of mode . In particular, with respect to the block form (LABEL:V_explicit), for any pure QCM the marginal QCMs and have matching symplectic spectra, given by and .

Leaving apart its far-reaching applications, in the context of the present paper this result is mainly instrumental for assessing the separability of so-called isotropic multimode Gaussian states. The QCM of any such state of modes is characterised by the property of having a completely degenerate symplectic spectrum, i.e. formed of only one distinct symplectic eigenvalue (repeated times). This means that the QCM of any isotropic state is proportional by a factor to a pure QCM. Hence, Theorem LABEL:mode-wise_thm tells us that can be brought into a direct sum of two-mode QCMs via a local symplectic congruence (local with respect to any partition into groups of modes and ), as first observed in holwer (). Thanks to Theorem LABEL:PPT_thm, this guarantees the following.

Theorem 11.

The PPT criterion is necessary and sufficient for separability of all isotropic Gaussian states of an arbitrary number of modes.

However, notwithstanding the importance of Theorem LABEL:mode-wise per se, one could strive to seek a more direct way to obtain Theorem LABEL:iso_thm. Our purpose in this Section is in fact to provide an alternative proof of this result, which does not appeal to the mode-wise decomposition theorem at all, and uses directly Lemma LABEL:sep instead, leveraging matrix analysis tools such as the notions of matrix means introduced in Section LABEL:secMM.

Note that, in almost all the remainder of this Section, for a single system of modes, we will find it more convenient to reorder the vector of canonical operators as , corresponding to a position-momentum block structure. The symplectic form appearing in (LABEL:CCR) is accordingly rewritten as

(0)

We will then write any QCM , as well as any symplectic operation acting on it, with respect to this alternative block structure, unless explicitly stated otherwise.

Let us start with a preliminary result, equivalent to Proposition 12 of manuceau () or Lemma 13 of Lami16 (). We include a proof for the sake of completeness.

Lemma 12.

Let be a positive matrix. Then is a pure QCM if and only if , and it obeys (LABEL:Heisenberg) if and only if

(0)
Proof.

Let the Williamson form of be given by (LABEL:Williamson), where (in the convention of this Section) , with . Then we can write

where we used in order: (i) the identities , , all consequences of the defining symplectic identity ; (ii) the fact that commutes with ; and (iii) the orthogonality relation . Now the first claim becomes obvious, since if and only if is a pure QCM. In general, as it can be seen from the above expression, and are brought in Williamson form by simultaneous congruences with the same symplectic matrix . Hence, the covariance of the geometric mean under congruence ensures that

where the last passage is an easy consequence of the fact that for all . By comparison with (LABEL:Williamson), we see that the Heisenberg uncertainty relation can be rephrased as , which reproduces (LABEL:WalterWhite), proving the second claim. ∎

Remark 5.

From the above proof it is also apparent how, for any positive , the matrix is a pure QCM (independently of the nature of ).

Now we are ready to explain our direct argument to show separability of PPT isotropic Gaussian states, alternative to the use of the mode-wise decomposition.

Proof of Theorem LABEL:iso_thm.

We start by rewriting the PPT condition (LABEL:PPT) for a QCM as

where in the convention of this Section . Thanks to Lemma LABEL:QCM_geom_lemma, this becomes in turn

and finally

after conjugating by , applying once more the covariance of the geometric mean under congruences, introducing a real parameter (to be fixed later), and defining . Now, we apply Lemma LABEL:lemma_ha=g to the above expression, obtaining

Although it is not yet transparent, we are done, as the right-hand side of the above inequality is exactly of the form when is the QCM of an isotropic Gaussian state. In fact, let be such that is a pure QCM, satisfying , where we have now reverted to a block decomposition with respect to the vs splitting. Then on the one hand since we find

while on the other hand

where we used the definition (LABEL:harmonic) of harmonic mean and the fact that . Putting all together, we find

Since we already observed that is a QCM for any (and analogously for ), a direct invocation of Lemma LABEL:sep allows us to conclude the proof. ∎

\@xsect

Throughout this Section, we finally complete the solution of a problem posed in passive () and there addressed under some additional constraints. Let us start by recalling that symplectic operations can be divided into two main categories, namely those such as squeezers that require an exchange of energy between the system and the apparatus, called active, and those that can be implemented using only beam splitters and phase plates, called passive. A symplectic matrix represents a passive transformation if and only if it is also orthogonal, meaning that (it may be worth adding that symplectic orthogonal transformations form the maximal compact subgroup of the symplectic group). As it turns out, symplectic orthogonal matrices can be represented in an especially simple form if we resort to a position-momentum block decomposition. Namely, one has the parametrisation bucco ()

(0)

where

and is a generic, unitary matrix, with denoting its complex conjugate.

Since the implementation of passive operations is so inexpensive in quantum optics and entangled states so useful for quantum technologies, the question first posed in passive () was a natural one: “What bipartite Gaussian states are such that they can be entangled via a global, passive operation?” However, in this full generality the problem was left unanswered in passive (). Instead, another related question was investigated and answered there, namely whether distillable Gaussian entanglement can be produced in the same fashion. For Gaussian states, as mentioned in the Introduction, distillability is well known to be equivalent to non-positivity of the partial transpose Giedke01 (); GiedkeQIC (), so the authors of passive () proceeded to identify the class of Gaussian states that can be made PPT with a passive transformation. However, it is important to realise that since PPT and separability are not the same for general multimode Gaussian states, the two questions are a priori different. Here we show that the answer to the original question above turns out to be yet another situation where the PPT condition is necessary and sufficient to ensure separability of Gaussian states. In other words, we will prove that a bipartite Gaussian state that cannot be made distillable (i.e. non-PPT) via passive operations is necessarily separable, and thus it stays separable under the application of said passive operations. Let us start with a technical lemma that we deduce from recent results obtained in bhatia15 ().

Lemma 13.