Correlation matrices of two-mode bosonic systems

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Pirandola, Stefano , Alessio Serafini, and Seth Lloyd. “Correlation

matrices of two-mode bosonic systems.” Physical Review A 79.5

(2009): 052327. (C) 2010 The American Physical Society.

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http://dx.doi.org/10.1103/PhysRevA.79.052327

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American Physical Society

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Correlation matrices of two-mode bosonic systems

Stefano Pirandola,1Alessio Serafini,2 and Seth Lloyd1,3 1

Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA 2

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom 3

Department of Mechanical Engineering, MIT, Cambridge, Massachusetts 02139, USA

共Received 17 February 2009; published 20 May 2009兲

We present a detailed analysis of all the algebraic conditions an arbitrary 4⫻4 symmetric matrix must satisfy in order to represent the correlation matrix of a two-mode bosonic system. Then, we completely clarify when this arbitrary matrix can represent the correlation matrix of a separable or entangled Gaussian state. In this analysis, we introduce alternative sets of conditions, which are expressed in terms of local symplectic invariants.

DOI:10.1103/PhysRevA.79.052327 PACS number共s兲: 03.67.⫺a, 03.65.Ud, 42.50.⫺p, 02.10.Ud

I. INTRODUCTION

Statistical moments of second order represent a key ele-ment of the quantum mechanical paradigm. Besides provid-ing the “language” in which the uncertainty principles are expressed, they serve as indicators in a number of applica-tions of the theory, with both applied and fundamental inter-est. In particular, second moments are central to the descrip-tion of bosonic fields in second quantizadescrip-tion共as is the case, for instance, in quantum optics, where second order coher-ence is characterized in terms of second moments兲 and of nonrelativistic particles in first quantization. Such systems do, in fact, share the same formal description 关1兴, which

hence extends its domain to a variety of fields ranging from atomic physics to quantum optics, from superconductors’ physics to nanomechanical systems.

During the last decade, the rise of quantum information science has renewed the focus on these areas because of their potential for coherent quantum manipulations, and has con-comitantly brought new problems and questions to the atten-tion of theorists, which resulted in the birth of the field of “continuous variable” 共CV兲 quantum information 关2兴 共see

Refs.关3–5兴 for some literature on CV quantum computation,

CV quantum teleportation, and CV quantum key distribution, respectively兲. A systematic analysis of the properties of con-tinuous variable quantum states inferred from the structure of their second moments has been thereby carried out, which lead to a well-established, extensive theoretical picture 共see, for instance关6–9兴兲. Such an analysis proved to be most

rel-evant also in view of the experimental prominence of the class of Gaussian states关10,11兴, which are completely

deter-mined by their first and second moments. In first, seminal endeavors关6,7兴, the qualitative characterization of the

quan-tum correlations共“entanglement”兲 of Gaussian states of two degrees of freedom has been successfully achieved. Now, while very well established and relatively simple, this result is still often expressed in a nonrigourous or incomplete man-ner, which is prone to confound the unacquainted reader. Because it constitutes one of the basic building blocks on which the theoretical characterization of Gaussian states has been constructed, it seems to us extremely important for it to be re-derived and re-expressed in a rigourous manner and full detail: this is one of the motifs and central aims of the present paper.

In general, the main question we will address and rigour-ously answer is the following: 共i兲 What are the algebraic conditions that a 4⫻4 real symmetric matrix V must satisfy in order to represent the correlation matrix of a two-mode bosonic system 关12兴? Then, we shall move on to answer a

closely connected question: 共ii兲 What are the algebraic con-ditions to be satisfied by V in order to represent the correla-tion matrix of a separable 共or entangled兲 Gaussian state of two bosonic modes?

Both these questions are thoroughly answered providing complete sets of conditions, which are expressed in terms of global or local symplectic invariants. In particular, the set of local conditions, i.e., given in terms of local invariants, is completely new in literature. Then, by specifying some of these algebraic results in the case of positive-definite matri-ces, we can make a direct comparison with the previous work of Ref.关6兴 and provide a rigorous and correct

interpre-tation of its seminal results.

The paper is organized as follows. In Sec. II we review basic notions about bosonic states, correlation matrices, and symplectic transformations. In Sec.IIIwe present the math-ematical tools to be used in the derivations of Secs.IVandV. In Sec.IVwe provide two sets of algebraic conditions for the physical genuinity of the correlation matrix of two bosonic modes. These conditions are expressed in terms of global or local symplectic invariants. In Sec. V we provide similar conditions for separability. Next, in Sec.VI, we specify some of our results for making a direct comparison with the pre-vious achievements of Ref.关6兴. Finally, Sec.VIIis for con-clusions. Notice that in the paper we will denote byM共n,R兲 the set of n⫻n real matrices. Then, we use the compact notation

S共n,R兲 = 兵M 苸 M共n,R兲:M = MT_{其, 共1兲} for the set of the n⫻n symmetric real matrices, and

P共n,R兲 = 兵M 苸 S共n,R兲:M ⬎ 0其, 共2兲 for the set of the n⫻n positive-definite real matrices. We will also consider the set共group兲 of proper rotations

SO共n兲 = 兵M 苸 M共n,R兲:MT

M = I, det M = 1其. 共3兲

PHYSICAL REVIEW A 79, 052327共2009兲

II. BOSONIC SYSTEMS AND SYMPLECTIC TRANSFORMATIONS

A. Correlation matrix of a bosonic system

Let us consider a bosonic system composed by n modes, labeled by an index k. Such a system can be described by an infinite-dimensional Hilbert space H=丢k=1

n _H

k and a vector of quadrature operators xˆTª共qˆ1, pˆ1,¯ ,qˆn, pˆn兲. In particular, these operators satisfy the commutation relations

关xˆl,xˆm兴 = 2i⍀lm, 共4兲 where l , m = 1 ,¯ ,2n and ⍀lmare the entries of the simplec-tic form ⍀ =

丣

k=1 n ␻, ␻ª

冉

0 1 − 1 0

冊

. 共5兲 Let us denote byD共H兲 the space of density operators acting on H. It is known that an arbitrary density operator ␳ 苸D共H兲 has an equivalent representation in a real symplectic space K=K共R2n_{,⍀兲 called the phase space. This is a real} vector space which is spanned by the singular eigenvalues

xT=共q1, p1,¯ ,qn, pn兲 of xˆT 共representing the “continuous variables” of the system兲 and associated to a symplectic product u · v = uT_{⍀v. In this space,␳ corresponds to a} qua-siprobability distribution known as the Wigner function W = W共x兲. In general, such a function is fully characterized by the entire set of its statistical moments关13兴. However, in the

particular case of Gaussian states, the Wigner function is Gaussian and, therefore, fully characterized by the first and second moments only. These two moments are also known as the displacement vector dª具xˆ典 and the correlation matrix 共CM兲 V, whose generic entry is defined by

Vlmª 1

2具⌬xˆl⌬xˆm+⌬xˆm⌬xˆl典, 共6兲 where⌬xˆlªxˆl−具xˆl典. According to the definition of Eq. 共6兲,

the CM of n bosonic modes is a real and symmetric matrix in 2n dimension, i.e., V苸S共2n,R兲. As a direct consequence of Eq.共4兲, such a matrix must also satisfy the uncertainty

prin-ciple关9,14兴

V + i⍀ ⱖ 0. 共7兲

In other words, an arbitrary V苸S共2n,R兲 is a bona fide quan-tum CM if and only if Eq. 共7兲 holds. Equivalently, we can

introduce the set of n-mode quantum CM’s to be defined as

qCM共n兲 ª 兵V 苸 S共2n,R兲:V + i⍀ ⱖ 0其. 共8兲

Notice that the condition of Eq. 共7兲 implies a first relevant

constraint on the matrix V:

Lemma 1 (Definite positivity of V兲. For every V

苸S共2n,R兲 satisfying V+i⍀ⱖ0, one has

V⬎ 0. 共9兲

Proof. Let u苸R2n_{, then 0ⱕu}T_{Vu + iu}T_⍀u=uT_{Vu because} ⍀ is antisymmetric. Hence Vⱖ0. To prove definite positiv-ity suppose, ad absurdum, that a nontrivial real vector u0 exists such that u₀TVu0= 0. Another vector u1苸R2nsuch that

u₁T⍀u0⫽0 always exists as nul共⍀兲=0. As a consequence,

one can always construct a set of complex vectors z = u₀ + iau1, for a苸R, such that

0ⱕ 共zⴱ兲T共V + i⍀兲z = 2au₁T⍀u0+ a2u₁TVu1. 共10兲 Values of a such that the inequality above is violated can always be found regardless of the values of u₁T⍀u0 and

u₁TVu1. This implies uTVu⫽0 for every u苸R2nand,

there-fore, V⬎0. 䊏

According to lemma 1, we then have

qCM共n兲 債 P共2n,R兲. 共11兲

Furthermore, it is trivial to show positive-definite matrices which violate Eq.共7兲, so that we actually have

qCM共n兲 傺 P共2n,R兲. 共12兲 Notice that definite positivity is the only requirement for a real symmetric matrix to be a classical correlation matrix.

B. Symplectic transformations

The most general real linear transformation of the quadra-tures

S:xˆ→ xˆ

⬘

ª Sxˆ, 共13兲 must preserve Eq. 共4兲 in order to be a physical operation.

This happens when the matrix S苸M共2n,R兲 preserves the symplectic form of Eq.共5兲, i.e.,

S⍀ST=⍀. 共14兲

The set of all the matrices S苸M共2n,R兲 satisfying Eq. 共14兲

forms the so-called real symplectic group

Sp共2n,R兲 ª 兵S 苸 M共2n,R兲:S⍀ST

=⍀其, 共15兲 whose elements are called symplectic or canonical

transfor-mations. As a consequence, the most general real linear

transformation in phase space S : x→x

⬘

ªSx must be sym-plectic. Its action on the Wigner function is simply given by

W共x兲→W共S−1_{x兲, so that the displacement is linearly} modi-fied while the CM is transformed according to the congru-ence

V→ SVST. 共16兲

Symplectic transformations are very important since every S acting in the phase space K corresponds to a Gaussian uni-tary Uˆ 共S兲 acting on the Hilbert space H, i.e., a unitary op-erator preserving the Gaussian statistics of the quantum states. These unitaries are the ones generated by bilinear Hamiltonians and can always be decomposed into single-mode squeezers and multisingle-mode interferometers 关10,15兴. In

particular, local symplectic transformations

S =

丣

k=1 n

Sk苸 Sp共2,R兲丣 ¯ 丣Sp共2,R兲 共17兲 correspond to local Gaussian unitaries

Uˆ 共S兲 =

丢

k=1 n

Uˆk. 共18兲

Local symplectic transformations can always be decomposed as products of local rotations and local squeezings. In fact, thanks to the following characterization

Sp共2,R兲 = 兵S 苸 M共2,R兲:det S = 1其, 共19兲 we have that every S苸Sp共2,R兲 can be expressed as a prod-uct of proper rotations

R共␸兲 ª

冉

sin␸ − cos␸

cos␸ sin␸

冊

, 共20兲 and squeezing matrices

S共␰兲 ª

冉

␰

1/2 ₀

0 ␰−1/2

冊

, ␰⬎ 0. 共21兲 Besides, it is also important to identify which quantities of a CM are preserved under the application of symplectic transformations. In general, for a given CM V, we say that a functional

f:V→ f共V兲 苸 R 共22兲

is a共global兲 symplectic invariant if

f共V兲 = f共SVST兲, 共23兲

for every S苸Sp共2n,R兲. Then, we say that f共V兲 is a local symplectic invariant if Eq. 共23兲 holds for every S

苸Sp共2,R兲丣¯丣Sp共2,R兲 关16兴. Notice that we can extend

the notion of symplectic invariance also to a property of a matrix. For instance, the definite positivity of V is a global symplectic invariant since V⬎0⇒SVST_⬎0.

III. SYMPLECTIC ANALYSIS

Here, we review some basic tools that can be used for the symplectic manipulation of the CM’s. In particular, the cen-tral tool in this trade is Williamson’s theorem 关17兴, which

ensures the possibility of carrying out the symplectic diago-nalization of real matrices in even dimension under the

defi-nite positivity constraint共as in the case of the CM’s兲. Lemma 2 (Williamson’s theorem). For every V

苸P共2n,R兲, there exists a symplectic matrix S苸Sp共2n,R兲 such that SVST=

冢

␯1 ␯1
␯n ␯n

冣

ª W ⬎ 0, 共24兲

where the n positive quantities兵␯1,¯ ,␯n其 are the “symplec-tic eigenvalues” of V, and the diagonal matrix W is its “Wil-liamson form”共or “normal form”兲.

The symplectic spectrum兵␯1,¯ ,␯n其 can be computed as the standard eigenspectrum of the matrix 兩i⍀V兩 where the modulus must be understood in the operatorial sense 关18兴.

The corresponding Williamson form W is unique up to a permutation of the symplectic spectrum 共i.e., of the bosonic modes兲. Fixing this permutation, the diagonalizing symplec-tic matrix S of Eq. 共24兲 is defined up to local rotations

丣k=1 n

Rkwith Rk苸SO共2兲. For the sake of completeness, we report in Appendix A a simple proof of Williamson’s theo-rem, originally presented in Ref. 关19兴. Using this proof, we

show in Appendix B an algorithm which finds the diagonal-izing symplectic matrix S of Eq.共24兲. This algorithm is not

the fastest but it can be helpful in studying problems like the optimal discrimination of Gaussian states关20兴 and the

quan-tum illumination 关21兴.

Let us consider the case of a 4⫻4 positive-definite matrix

V苸P共4,R兲, as in the case of CM’s describing two bosonic

modes. This matrix can be expressed in the block form

V =

冉

A C

CT _B

冊

, 共25兲

where A, B苸S共2,R兲, and C苸M共2,R兲. In this case the symplectic spectrum兵␯₁,␯2其ª兵␯₋,␯+其 can be computed via the simple formula关22兴

␯⫾=

冑

⌬共V兲 ⫾

冑

⌬共V兲

2_{− 4 det V}

2 , 共26兲

where

⌬共V兲 ª det A + det B + 2 det C. 共27兲 Here, the quantities det A, det B, and det C are local sym-plectic invariants, while det V and⌬共V兲 are global symplec-tic invariants, which can also be written as

det V =␯₋2␯₊2, ⌬共V兲 =␯₋2+␯₊2. 共28兲 Another important tool in the symplectic analysis is the re-duction to standard form by local symplectic transformations 关6,7兴. In general, such a reduction holds for symmetric

ma-trices V苸S共4,R兲 with positive diagonal blocks, as we easily show in the following. In particular, it can be applied to positive-definite matrices V苸P共4,R兲 and, therefore, to CMs

V苸qCM共2兲.

Lemma 3 (Standard form). For every

V =

冉

A C

CT B

冊

苸 S共4,R兲, 共29兲

with A , B⬎0, there exists some S苸Sp共2,R兲丣Sp共2,R兲 such

that SVST=

冢

a c+ a c− c+ b c− b

冣

ª VI , 共30兲

where the real parameters a , b , c+, c−satisfy

det A = a2, det B = b2, det C = c+c−, 共31兲 and

det V = det VI=共ab − c₊2兲共ab − c₋2兲. 共32兲

CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS PHYSICAL REVIEW A 79, 052327共2009兲

Proof. Let us consider a pair of single-mode symplectic

transformations SA, SB苸Sp共2,R兲 and a pair of single-mode proper rotations R共␪A兲, R共␪B兲苸SO共2兲. By applying the local symplectic transformation

S = R共␪A兲SA丣R共␪B兲SB 共33兲

to the matrix V, we get

SVST=

冉

A

⬘

C

⬘

C

⬘

T B

⬘

冊

, 共34兲 where A

⬘

ª R共␪A兲共SAASA T_兲RT_{共␪A兲, 共35兲} B

⬘

ª R共␪B兲共SBBSB T_兲RT_{共␪B兲, 共36兲} C

⬘

ª R共␪A兲共SACSB T_兲RT_{共␪B兲. 共37兲} Since A , B苸P共2,R兲, we can apply Williamson’s theorem. This means that we can choose SAand SBsuch that

A

⬘

= R共␪A兲aI RT_{共␪A兲 = aI, 共38兲}

B

⬘

= R共␪B兲bI RT共␪B兲 = bI, 共39兲

where a共b兲 is the symplectic eigenvalue of A共B兲 while the angle ␪A共␪B兲 is arbitrary. Since the pair 兵␪A,␪B其 can be cho-sen freely, we can always choose a pair兵¯␪A,␪¯B其 in Eq. 共37兲 such that C

⬘

= diag共c+, c−兲 关23兴. As a consequence, Eq. 共34兲 is

globally equal to Eq. 共30兲. Finally, since the transformation

of Eq.共33兲 is local and symplectic, all the determinants

rela-tive to the blocks and the global matrix are preserved, so that Eqs.共31兲 and 共32兲 are trivially implied. 䊏

IV. GENUINENESS OF A TWO-MODE CORRELATION MATRIX

By applying the symplectic tools of Sec. III, we can now derive very simple algebraic conditions for characteriz-ing the genuineness of a two-mode CM. In other words, starting from a generic 4⫻4 real and symmetric matrix 关V苸S共4,R兲兴, we give the algebraic conditions that such a matrix must satisfy in order to represent the CM of two bosonic modes, i.e., a bona fide two-mode quantum CM 关V苸qCM共2兲兴. As a consequence of Williamson’s theorem, we have the following algebraic conditions in terms of

glo-bal symplectic invariants关9兴.

Theorem 4. An arbitrary V苸S共4,R兲 is a quantum CM if

and only if it satisfies

V⬎ 0, ␯₋ⱖ 1, 共40兲

or, equivalently,

V⬎ 0, det V ⱖ 1, ⌬共V兲 ⱕ 1 + det V. 共41兲

Proof. For every V苸P共4,R兲, the application of

William-son’s theorem to Eq.共7兲 gives V+i⍀ⱖ0⇔␯₋ⱖ1 共recalling

that ␯−is the smallest symplectic eigenvalue兲. Since V+i⍀ ⱖ0⇒V⬎0 共see lemma 1兲, we can write V+i⍀ⱖ0 ⇔共V⬎0∧␯−ⱖ1兲 which proves the bona fide condition of

Eq. 共40兲 for a generic V苸S共4,R兲. Under the definite

posi-tivity assumption V⬎0, one can also use Eq. 共26兲 to prove

the equivalences

␯−ⱖ 1 ⇔ ⌬共V兲 − 2 ⱖ

冑

⌬共V兲2− 4 det V⇔ max兵2,2

冑

det V其 ⱕ ⌬共V兲 ⱕ 1 + det V ⇔

再

det Vⱖ 1

2

冑

det Vⱕ ⌬共V兲 ⱕ 1 + det V.

冎

共42兲 According to Eq. 共28兲 the condition 2

冑

det Vⱕ⌬共V兲 in Eq.

共42兲 corresponds to 2␯−␯+ⱕ␯−2+␯+2, which is trivially satis-fied. Then, we have

␯−ⱖ 1 ⇔

再

det Vⱖ 1

⌬共V兲 ⱕ 1 + det V,

冎

共43兲 which states the equivalence between to Eqs.共40兲 and 共41兲,

where the underlying assumption V⬎0 is also shown. 䊏 Notice that, crucially, Williamson’s theorem could be ap-plied to V because of its definite positivity, which thus im-plies the existence of well-defined symplectic eigenvalues 关24兴. The condition ␯−ⱖ1 alone is therefore not, by itself,

fully equivalent to the uncertainty principle V + i⍀ⱖ0, un-less definite positivity is also assumed. The essential role of the prescription V⬎0, often neglected in the literature, is especially clear in the formulation of Eq. 共41兲: in fact, the

other two inequalities in Eq.共41兲 only depend on the squared

symplectic eigenvalues and cannot thus distinguish between positive and negative eigenvalues.

Besides the algebraic requirements of the previous theo-rem, we can derive an alternative set of conditions by apply-ing the reduction to standard form. These conditions are ex-pressed in terms of local symplectic invariants.

Theorem 5. An arbitrary

V =

冉

A C

CT B

冊

苸 S共4,R兲 共44兲

is a quantum CM if and only if it satisfies

A,B⬎ 0, 共45兲

⌬共V兲 ⱕ 1 + det V, 共46兲 2

冑

det A det B +共det C兲2ⱕ det V + det A det B. 共47兲

Proof. Under the assumption A , B⬎0, the matrix V can be

reduced to the standard form VI of Eq.共30兲 via a local

sym-plectic transformation S. Since S⍀ST_{=⍀, the Heisenberg} principle can be written in the equivalent form关25兴

V + i⍀ ⱖ 0 ⇔ VI_{+ i⍀ ⱖ 0. 共48兲} Since the matrix VI_{+ i⍀ is Hermitian, its four eigenvalues} ␭+

+

,␭₋+,␭₊−,␭₋−are real. It is then easy to show that 2␭_⫾+

= a + b +

冑

␮⫾ 2

冑

␯, 共49兲 2␭_⫾− = a + b −

冑

␮⫾ 2

冑

␯, 共50兲 where

␮ª 4 + 共a − b兲2_{+ 2共c} + 2_{+ c} − 2_{兲 ⱖ 4, 共51兲} and ␯ª 4共a − b兲2_{+共c++ c−兲}2_{关4 + 共c+− c−兲}2_{兴 ⱖ 0. 共52兲} Since␭₊−is the minimum eigenvalue, we have that

VI+ i⍀ ⱖ 0 ⇔ ␭₊−ⱖ 0 ⇔ a + b −

冑

␮+ 2

冑

␯ⱖ 0 ⇔

再

共a + b兲2ⱖ␮+ 2

冑

␯ a + bⱖ 0

冎

⇔

冦

关共a + b兲2_−␮兴2_{ⱖ 4␯} 共a + b兲2_{−␮ⱖ 0} a + bⱖ 0.

冧

共53兲 Last condition a + bⱖ0 in Eq. 共53兲 is trivially included in

A⬎0 and B⬎0 which gives a⬎0 and b⬎0 共for congruence

with the diagonal matrices aI and bI兲. The other two condi-tions

关共a + b兲2_−␮兴2_{ⱖ 4␯, 共54兲} and

共a + b兲2_{−␮ⱖ 0, 共55兲} can be recast in terms of the local symplectic invariants. In fact, by inserting Eqs.共51兲 and 共52兲 in Eq. 共54兲, we get

a2+ b2+ 2c+c−ⱕ 共ab − c+2兲共ab − c−2兲 + 1, 共56兲 which is equivalent to Eq. 共46兲 by using Eq. 共32兲 and

⌬共V兲=⌬共VI_兲=a2_{+ b}2_{+ 2c}

+c−. Finally, by inserting Eq. 共51兲 in Eq.共55兲, we get 2ab−c₊2− c₋2ⱖ2, which is equivalent to

2a2b2− ab共c₊2+ c₋2兲 ⱖ 2ab, 共57兲 since ab⬎0. In terms of local symplectic invariants, last inequality is equal to

2 det A det B − I4ⱖ 2

冑

det A det B, 共58兲 where

I4ª Tr共A␻C␻B␻CT␻兲 = ab共c+ 2

+ c₋2兲 共59兲 is another local symplectic invariant. In fact, the quantity I4 is connected to the other local symplectic invariants by

det V = det A det B +共det C兲2− I4, 共60兲 which holds for every V苸S共4,R兲. Using Eq. 共60兲 in Eq.

共58兲, we then get Eq. 共47兲. 䊏

V. SEPARABILITY OF A TWO-MODE CORRELATION MATRIX

Few years ago, Ref.关6兴 showed how to extend the partial

transposition and the Peres entanglement criterion 关26兴 to

bipartite bosonic systems. In fact, partial transposition PT:␳AB→˜␳AB corresponds in phase space to a “local time reversal” which inverts the momentum of only one of two subsystems. This means that we have the following transfor-mation for the Wigner function:

PT:W共x兲 → W˜ 共x兲 ª W共⌳x兲, 共61兲 where ⌳ ª

冉

1 1

冊

丣

冉

1 − 1

冊

. 共62兲 For the corresponding CM V苸qCM共2兲, the PT transforma-tion is given by

PT:V→ V˜ ª ⌳V⌳, 共63兲 where the partially transposed matrix V˜ belongs to P共4,R兲 关25兴 but not necessarily to qCM共2兲. By writing V in the

block form of Eq.共25兲, one easily checks that the action of

the PT transformation⌳ reduces to the following sign flip det C→ − det C, 共64兲 at the level of the local symplectic invariants. As a conse-quence, the positive-definite matrix V˜ has

⌬共V˜ 兲 = det A + det B − 2 det C ª ⌬˜共V兲, 共65兲 and symplectic eigenvalues

␯

˜_⫾=

冑

⌬˜共V兲 ⫾

冑

⌬˜共V兲

2_{− 4 det V}

2 . 共66兲

Once the PT transformation has been extended, also the Peres criterion can be consequently extended via the logical implication

␳AB separable⇒˜␳AB苸 D共H兲 ⇒ V˜ 苸 qCM共2兲, 共67兲 which becomes an equivalence for Gaussian states under 1 ⫻n mode bipartitions 关6,8兴.

Theorem 6 (Separability). Let us consider a Gaussian state ␳AB with CM V苸qCM共2兲. Then, ␳AB is separable if and only if V˜ 苸 qCM共2兲, 共68兲 or, equivalently, ␯ ˜₋ⱖ 1, 共69兲 or, equivalently ⌬˜共V兲 ⱕ 1 + det V. 共70兲

Proof. The proof of Eq. 共68兲 follows exactly the same

steps of the one in Ref. 关6兴, where the P representation is

exploited 共see Ref. 关27兴 for recent connections between P

representation and separability.兲 In order to prove Eqs. 共69兲

and共70兲, let us apply theorem 4 to the positive-definite

ma-trix V˜ 苸P共4,R兲. Then, we get

V˜ 苸 qCM共2兲 ⇔␯˜₋ⱖ 1 ⇔

再

det V˜ ⱖ 1

⌬˜共V兲 ⱕ 1 + det V˜

冎

共71兲 where Eq. 共69兲 is trivially proven. Now, since det V˜

= det共⌳V⌳兲=det Vⱖ1, the first condition in Eq. 共71兲 is

al-CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS PHYSICAL REVIEW A 79, 052327共2009兲

ways satisfied and, therefore, the separability condition is

reduced to Eq. 共70兲. 䊏

Let us now derive the algebraic conditions that a generic symmetric matrix must satisfy to represent the CM of a sepa-rable or entangled Gaussian state. The following corollary gives an easy recipe to check if a symmetric matrix is a good or bad candidate for this aim.

Corollary 7. An arbitrary

V =

冉

A C

CT B

冊

苸 S共4,R兲 共72兲

represents the CM of a separable Gaussian state if and only if it satisfies V⬎ 0, ␯₋ⱖ 1, ˜␯₋ⱖ 1, 共73兲 or, equivalently, V⬎ 0, det V ⱖ 1, ⌫共V兲 ⱕ 1 + det V, 共74兲 or, equivalently, A,B⬎ 0, ⌫共V兲 ⱕ 1 + det V, 共75兲

2

冑

det A det B +共det C兲2ⱕ det V + det A det B, 共76兲 where ⌫共V兲ªdet A+det B+2兩det C兩. Instead, it represents the CM of an entangled Gaussian state if and only if it sat-isfies V⬎ 0, ␯−ⱖ 1, ˜␯−⬍ 1, 共77兲 or, equivalently, V⬎ 0, det V ⱖ 1, ⌬共V兲 ⱕ 1 + det V ⬍ ⌬˜共V兲, 共78兲 or, equivalently, A,B⬎ 0, ⌬共V兲 ⱕ 1 + det V ⬍ ⌬˜共V兲, 共79兲

2

冑

det A det B +共det C兲2ⱕ det V + det A det B. 共80兲

Proof. In order to represent the CM of a separable

Gauss-ian state, the symmetric matrix V苸S共4,R兲 must simulta-neously satisfy

V苸 qCM共2兲, V˜ 苸 qCM共2兲. 共81兲

The bona fide condition V苸qCM共2兲 is equivalently ex-pressed by the conditions of Eqs.共40兲 and 共41兲 in theorem 4.

Then, for every V苸qCM共2兲, the separability condition V˜ 苸qCM共2兲 is equivalent to Eqs. 共69兲 and 共70兲 in theorem 6.

By combining Eq.共40兲 with Eq. 共69兲 and Eq. 共41兲 with Eq.

共70兲, one easily gets Eqs. 共73兲 and 共74兲, where

max兵⌬共V兲,⌬˜共V兲其ⱕ1+det V⇔⌫共V兲ⱕ1+det V. According to theorem 6, for every V苸qCM共2兲 the entanglement con-dition is expressed by ˜␯−⬍1 or, equivalently, by ⌬˜共V兲⬎1

+ det V. Then, it is trivial to derive the corresponding Eqs. 共77兲 and 共78兲. The proof of Eqs. 共75兲 and 共76兲 and Eqs. 共79兲

and共80兲 is the same as before except that now we have to

combine the Eqs. 共45兲–共47兲 of theorem 5 with Eq. 共70兲 for

the separability and with ⌬˜共V兲⬎1+det V for the

entangle-ment. 䊏

VI. RELATION WITH THE PREVIOUS WORK BY SIMON

In order to make a direct comparison with the previous work by Simon关6兴, we have to specify some of our results,

given for arbitrary symmetric matrices V苸S共4,R兲, to the case of positive-definite matrices, i.e., V苸P共4,R兲. As an immediate consequence of theorem 4, we have the following result.

Corollary 8. An arbitrary V苸P共4,R兲 is a two-mode

quantum CM V苸qCM共2兲, i.e., V+i⍀ⱖ0, if and only if

␯−ⱖ 1, 共82兲

or, equivalently,

det Vⱖ 1, ⌬共V兲 ⱕ 1 + det V, 共83兲 or, equivalently,

det Vⱖ 1, 共84兲

det A det B +共1 − det C兲2− I4ⱖ det A + det B, 共85兲 where I₄ªTr共A␻C␻B␻CT_␻兲.

Proof. By applying theorem 4 under the assumption V

⬎0, one trivially derives the equivalent conditions in Eqs. 共82兲 and 共83兲. In order to prove Eqs. 共84兲 and 共85兲, let us

reduce the positive-definite matrix V to its standard form of Eq. 共30兲. Under local symplectic transformations, we then

have the equivalence

⌬共V兲 ⱕ 1 + det V ⇔ a2_{+ b}2_{+ 2c} +c−ⱕ 1

+共ab − c+2兲共ab − c−2兲. 共86兲 Note that Eq.共86兲 can be equivalently written as

a2b2+共1 − c+c−兲2− ab共c₊2+ c₋2兲 ⱖ a2+ b2. 共87兲 In terms of local symplectic invariants, last relation can be written as in Eq.共85兲, where Eq. 共59兲 has been also used. 䊏

Notice that Eq. 共85兲 corresponds to the Eq. 共17兲 of Ref.

关6兴, up to notation factors 关28兴. In Ref. 关6兴, this condition is

incorrectly claimed to be equivalent to the Heisenberg prin-ciple V + i⍀ⱖ0 共this equivalence is claimed under the posi-tivity contraint V⬎0, which is a sufficient condition for the reduction to standard form used in the corresponding proof of Ref. 关6兴兲. In order to have a full equivalence with the

Heisenberg principle V + i⍀ⱖ0, the supplementary condi-tion of Eq. 共84兲 is mandatory. It is indeed rather simple to

construct a positive-definite matrix V苸P共4,R兲 which satis-fies Eq. 共85兲 but violates V+i⍀ⱖ0. As an example, let us

V共x兲 =1 2

冢

1 + 4x 0 − 1 + 4x 0 0 1 + 4x 0 − 4x − 1 + 4x 0 1 + 4x 0 0 − 4x 0 1 + 4x

冣

, 共88兲

which is positive-definite for every x⬎0. It is easy to verify that the Hermitian matrix V + i⍀ has the following real ei-genvalues: ␭⫾= 1 4共1 + 8x ⫾

冑

17 − 16x + 64x 2_{兲, 共89兲} ␪⫾=1 4共3 + 8x ⫾

冑

17 − 16x + 64x 2_{兲. 共90兲}

Since␭−is the minimal eigenvalue, the Heisenberg principle

V + i⍀ⱖ0 is equivalent to ␭−ⱖ0, which gives

xⱖ 1/2. 共91兲

Now, let us explicitly compute Eqs.共84兲 and 共85兲. It is easy

to show that Eq.共84兲 is equivalent to

x共8x + 1兲 ⱖ 1 ⇔ x ⱖ 共

冑

33 − 1兲/16 ⯝ 0.3, 共92兲

while Eq.共85兲 共Simon’s genuineness condition兲 is equivalent

to 1 2+ x共8x − 5兲 ⱖ 0 ⇔ 0 ⬍ x ⱕ 1 8 ORxⱖ 1 2. 共93兲 From Eq. 共93兲, one can see that Simon’s condition alone

does not exclude the matrices V共x兲 for 0⬍xⱕ1/8, which are clearly unphysical since they violate the Heisenberg con-dition of Eq.共91兲. A complete equivalence with Eq. 共91兲 is

retrieved by coupling Eq.共93兲 with Eq. 共92兲, where the latter

equation excludes the nonphysical region 0⬍xⱕ1/8. This imprecision in Simon’s work leads to a common misunderstanding of the subsequent separability condition 关Eq. 共19兲 of Ref. 关6兴兴, which in our notation corresponds to

det A det B +共1 − 兩det C兩兲2− I4ⱖ det A + det B. 共94兲 In this condition, the Heisenberg principle is erroneously claimed to be included共in fact, it is only partially included兲. Hence, Simon’s separability condition of Eq.共94兲 is actually

valid only if V苸qCM共2兲, i.e., the positive-definite matrix

V苸P共4,R兲 is already known to be a bona fide quantum

CM. In other words, the separability criterion of Eq. 共94兲

must be tested on positive-definite matrices which are al-ready known to describe the second statistical moments of a physical quantum state. However, under this assumption of physicality, Simon’s separability criterion of Eq. 共94兲

dis-plays a redundant modulus and must be simplified to det A det B +共1 + det C兲2− I4ⱖ det A + det B. 共95兲

Criterion 9 (Separability). Let us consider a two-mode

quan-tum state␳ABhaving quantum CM

V =

冉

A C

CT B

冊

苸 qCM共2兲. 共96兲

The separability of␳ABimplies

⌬˜共V兲 ⱕ 1 + det V, 共97兲 or, equivalently,

det A det B +共1 + det C兲2− I4ⱖ det A + det B. 共98兲 In particular, if␳ABis a Gaussian state, then it is separable if and only if Eq.共97兲 关or Eq. 共98兲兴 holds.

Proof. The proof is straightforward. Since V is a quantum

CM, it is positive-definite and satisfies the condition det V ⱖ1. Now, suppose that the corresponding two-mode state

␳ABis separable. Then we have

␳AB separable⇒␳˜AB苸 D共H兲 ⇒ V˜ 苸 qCM共2兲 ⇔ V˜ + i⍀ ⱖ 0. 共99兲 By applying corollary 8 to the positive-definite matrix V˜ =⌳V⌳, we have

V˜ + i⍀ ⱖ 0 ⇔ det V˜ ⱖ 1, ⌬共V˜兲 ⱕ 1 + det V˜. 共100兲

Since det V˜ =det V, we have that det V˜ ⱖ1 is automatically satisfied in the previous Eq.共100兲. Then, we get

␳AB separable⇒ ⌬˜共V兲 ⱕ 1 + det V, 共101兲 where ⌬˜共V兲ª⌬共V˜兲 is defined in Eq. 共65兲. Using Eq. 共60兲,

one easily proves the equivalence between Eqs. 共97兲 and

共98兲. Finally, the full equivalence which holds for Gaussian

␳ABis a direct application of theorem 6. 䊏 This criterion represents a simplification of Simon’s rability criterion. Now, it is important to notice that the sepa-rability criterion becomes a bit more involved when arbitrary positive-definite matrices V苸P共4,R兲 are considered, with-out any other a priori assumption. For a generic V 苸P共4,R兲, both the Heisenberg principle 共V+i⍀ⱖ0兲 and the separability property共V˜ +i⍀ⱖ0兲 must be explicitly con-sidered and combined together, in order to get a complete set of algebraic conditions. Thanks to these conditions, one eas-ily checks when a positive-definite matrix V苸P共4,R兲 can represent the quantum CM of a Gaussian state␳ABwhich is separable or entangled. By applying corollary 7, we get the following criterion for positive-definite matrices.

Criterion 10. An arbitrary V苸P共4,R兲 represents the CM

of a separable Gaussian state if and only if

det Vⱖ 1, 共102兲

⌫共V兲 ⱕ 1 + det V, 共103兲 or, equivalently,

det Vⱖ 1, 共104兲

det A det B +共1 − 兩det C兩兲2− I4ⱖ det A + det B. 共105兲 Instead, it represents the CM of an entangled Gaussian state if and only if

det Vⱖ 1, 共106兲

CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS PHYSICAL REVIEW A 79, 052327共2009兲

⌬共V兲 ⱕ 1 + det V ⬍ ⌬˜共V兲, 共107兲 or, equivalently,

det Vⱖ 1, 共108兲

共1 + det C兲2_{⬍ det A + det B − det A det B + I4} ⱕ 共1 − det C兲2_{. 共109兲} The proof is a trivial application of corollary 7, together with Eq. 共60兲, used to state the equivalences between Eqs.

共103兲 and 共105兲, and between Eqs. 共107兲 and 共109兲.

Accord-ing to Eq. 共109兲, positive-definite matrices with det Cⱖ0

can only be associated to separable Gaussian states关6兴.

No-tice that the original Simon’s separability criterion, i.e., Eq. 共105兲, must be coupled with the mandatory condition of Eq.

共104兲 in order to investigate correctly the separability

prop-erties of a generic positive-definite matrix.

VII. SUMMARY

In theorem 4, we have rederived and explicitly stated all the precise algebraic conditions a symmetric matrix must satisfy to represent the CM of a two-mode bosonic 共or ca-nonical兲 quantum system, including the 共critical and often neglected兲 definite positivity condition. Such conditions are expressed in terms of global symplectic invariants. In theo-rem 5, we have derived a new and alternative set of condi-tions, which are expressed in terms of local symplectic in-variants. In these local conditions the positivity check is restricted to the submatrices A and B. Finally, in theorem 6, the necessary and sufficient condition for the separability of two-mode Gaussian states has been reviewed and cast in a compact form. We should stress that such a condition is valid only under the assumption that physicality is also met 关V 苸qCM共2兲兴. In corollary 7, both the physicality and separa-bility have been explicitly taken into account. Then, we have derived a complete set of 共global or local兲 conditions that a generic symmetric matrix must satisfy in order to represent the CM of a separable 共or entangled兲 Gaussian state of two bosonic modes. In Sec. VI, some of our results have been specified for positive-definite matrices and a comparison with the previous results by Simon has been thoroughly pre-sented. The rigourous agreement with all the conditions here considered should constitute a constant reference in both the theoretical practice and the analysis of experimental data in-volving quantum systems of two canonical degrees of free-dom.

ACKNOWLEDGMENTS

S.P. was supported by a Marie Curie Action of the Euro-pean Community. S.L. was supported by the W.M. Keck foundation center for extreme quantum information theory 共xQIT兲.

APPENDIX A: SIMPLE PROOF OF WILLIAMSON’S THEOREM

Let us construct the diagonalizing symplectic according to the decomposition

S = W1/2RV−1/2, 共A1兲 with a suitable R苸SO共2n兲. In fact

SVST=共W1/2RV−1/2兲V共V−1/2RTW1/2兲 = W1/2RIRTW1/2

= W1/2IW1/2= W. 共A2兲

Notice that Eq. 共A1兲 is well defined since V and W are

positive-definite共therefore, nonsingular兲. However, the rota-tion R in Eq. 共A1兲 is not arbitrary but must be chosen in

order to make S symplectic.

Let us apply Eq. 共A1兲 to the symplectic condition S⍀ST =⍀. Then, we have 共W1/2_RV−1/2_兲⍀共V−1/2_RT_W1/2_{兲 = ⍀} ⇔ R共V−1/2_⍀V−1/2_兲RT_{= W}−1/2_⍀W−1/2 ⇔ RXRT = Y, 共A3兲 where Xª V−1/2_⍀V−1/2_{, Yª W}−1/2_⍀W−1/2 _共A4兲 are antisymmetric 共because V and W are symmetric, while ⍀ is antisymmetric兲. In particular, we have

Y =

丣

k=1 n

冉

0 ␯k −1 −␯k−1 0

冊

. 共A5兲

Now the existence of R in Eq.共A3兲 is assured by the

follow-ing theorem on the block-diagonalization of real antisymmet-ric matantisymmet-rices 共specialized to even dimensions兲 关29兴

Theorem 11. For every A = −AT_{苸M共2n,R兲, there exists a} 共unique兲 O苸SO共2n兲 such that

OAOT=

丣

k=1 n

ak␻ª A˜ , 共A6兲 where the共unique兲 block diagonal form A˜ has ak⬎0.

APPENDIX B: FINDING THE DIAGONALIZING SYMPLECTIC MATRIX

Let us show a possible procedure for deriving the proper rotation O that block-diagonalizes a generic antisymmetric matrix A as in theorem 11. We can easily prove the following connection between the block-diagonalization of A and its unitary diagonalization

Theorem 12. The proper rotation O performing the

block-diagonalization of Eq.共A6兲 is given by

O =⌫U†_{, 共B1兲} where ⌫ =

_冑

1 2

丣

k=1 n ␥, ␥ª

_冑

1 2

冉

i − i 1 1

冊

, 共B2兲 and U is an arbitrary unitary performing the diagonalization of A, i.e., U†AU =

丣

k=1 n iak

冉

− 1 1

冊

ª AD. 共B3兲

Proof. First, let us prove how A can be transformed into

the diagonal form ADof Eq.共B3兲 by a unitary matrix. From

Eq. 共B2兲, we have that

␥†_{共ak␻兲␥= iak}

冉

− 1

1

冊

. 共B4兲

As a consequence, by applying⌫ to Eq. 共A6兲, we get

⌫†_OAOT_{⌫ = ⌫}†_{A˜ ⌫ =}

丣

k=1 n iak

冉

− 1 1

冊

= AD. 共B5兲 In other words, there exists a unitary OT_{⌫ that diagonalizes}

A according to Eq.共B3兲.

Then, let us prove that, for every unitary U diagonalizing

A according to Eq. 共B3兲, we can write Eq. 共B1兲 where O

performs the block diagonalization of Eq.共A6兲. For proving

this, let us consider the orthonormal eigenvectors 兵u1,¯ ,u2n其 of A

Au₁= − ia₁u₁, Au₂= ia₁u₂, 共B6兲 ]

Au_2n−1= − ianu2n−1, Au2n= + ianu2n, 共B7兲 more compactly denoted by 兵u2k−1, u2k其_k=1n with

Au2k−1= − iaku2k−1, Au2k= iaku2k. 共B8兲 These vectors are unique up to phase factors ␸ ª兵␸1,¯ ,␸2n其, i.e., up to the replacements

u2k−1→ u2k−1ei␸2k−1, u2k→ u2kei␸2k. 共B9兲 This means that, for every choice of␸, we have an equiva-lent unitary matrix U = U共␸兲 in the diagonalization of A. Now, by conjugating Eq.共B8兲, one easily checks that

u_2k−1= u_2kⴱ . 共B10兲 As a consequence, the most general unitary matrix that di-agonalizes A has the specific form

U =共u₂ⴱ u2 ¯ u2nⴱ u2n兲. 共B11兲 Let us explicitly compute the matrix product⌫U†. By apply-ing ⌫ =

_冑

1 2

冢

i − i 1 1 0
0 i − i 1 1

冣

共B12兲

to the conjugate matrix

U†=

冢

u₂T u2† ] u_2nT u_2n†

冣

=

冢

u_2,1 u_2,2 u_2,2n−1 u_2,2n u_2,1ⴱ u_2,2ⴱ u_2,2n−1ⴱ u_2,2nⴱ
u2n,1 u2n,2 u2n,2n−1 u2n,2n u_2n,1ⴱ u_2n,2ⴱ u_2n,2n−1ⴱ u_2n,2nⴱ

冣

, 共B13兲 one explicitly gets

⌫U†₌ 1

冑

2

冢

␣1,1 ␣1,2 ␣1,2n−1 ␣1,2n ␤1,1 ␤1,2 ␤1,2n−1 ␤1,2n
␣n,1 ␣n,2 ␣n,2n−1 ␣n,2n ␤n,1 ␤n,2 ␤n,2n−1 ␤n,2n

冣

, 共B14兲 where ␣k,jª − 2 Im共u2k,j兲, ␤k,jª 2 Re共u2k,j兲. 共B15兲 From Eqs. 共B14兲 and 共B15兲, we have that ⌫U† _{is real for} every choice of U in Eq. 共B3兲, i.e., for every choice of the

phases ␸ in the corresponding eigenvectors. More strongly, we have ⌫U†_{苸SO共2n兲 共since ⌫U}† _{real implies ⌫U}† or-thogonal with det= + 1兲. Then, from Eq. 共B3兲, we easily get

⌫U†_AU⌫†_=⌫AD⌫†_{= A˜ . 共B16兲} In conclusion, for every diagonalizing unitary U, the proper rotation ⌫U† corresponds to the unique proper rotation O that performs the block-diagonalization of Eq.共A6兲. 䊏

Both theorem 11 and theorem 12 can be applied to the Eq. 共A3兲, by setting O=R, A=X and A˜ =Y. These theorems

al-low to reduce the computation of the rotation R in Eq.共A3兲

to a unitary diagonalization. In fact, we have just to find a unitary U that diagonalizes X, i.e.,

U†_{XU =}

丣

k=1 n i␯k−1

冉

1 − 1

冊

, 共B17兲 and then construct

R =⌫U†_{. 共B18兲}

Once that we have R, we use Eq.共A1兲 to get the symplectic

S. Here is the complete algorithm:

共1兲 Find the symplectic spectrum of V, i.e., its Williamson form W.

共2兲 Compute the matrices W1/2 _{共immediate兲 and V}−1/2 共needs orthogonal diagonalization兲.

共3兲 Construct the matrix XªV−1/2_⍀V−1/2_.

共4兲 Find the eigenvectors of X and construct the corre-sponding unitary U.

共5兲 Compute R=⌫U†_. 共6兲 Compute S=W1/2_RV−1/2_.

By construction, this algorithm reduces the determination of S to unitary diagonalizations. Actually, this task can be achieved via faster methods when the symplectic spectrum is nondegenerate. In general, the determination of S is equiva-lent to the construction of a symplectic basis关30兴.

CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS PHYSICAL REVIEW A 79, 052327共2009兲

关1兴 This is actually true only when a denumerable number of bosonic degrees of freedom is considered共usually the case in quantum optics兲 and Stone-von Neumann theorem applies. 关2兴 S. L. Braunstein and A. K. Pati, Quantum Information Theory

with Continuous Variables 共Kluwer Academic, Dordrecht,

2003兲; S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513共2005兲.

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关23兴 For every M苸M共n,R兲 there always exists a pair of proper rotations R1, R2苸SO共n兲 such that M=R1DR2

T

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关28兴 Notice that Ref. 关6兴 adopts the commutation relations 关xˆ_l, xˆ_m兴 = i⍀lm, so that the variance of the vacuum noise is equal to 1/2. In this notation, our Eq. 共85兲 becomes det A det B

+关共1/4兲−det C兴2_{− I}

4ⱖ共det A+det B兲/4, which is exactly the

Eq.共17兲 of Ref. 关6兴.

关29兴 This theorem is a specialization of the orthogonal block diago-nalization which is valid for all the normal matrices关see, e.g., R. A. Horn and C. R. Johnson, Matrix Analysis 共Cambridge University Press, New York, 2006兲, Chap. 2.5兴. Clearly, the uniqueness of A˜ and O holds up to permutation in the spec-trum.

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