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Entanglement Criterion for Multi-Mode Gaussian States K. V. S. Shiv Chaitanya∗ Department of Physics, BITS Pilani, Hyderabad Campus, Jawahar Nagar, Shamirpet Mandal, Hyderabad, India 500 078. Sibasish Ghosh† Optics & Quantum Information Group,The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai, India, 600113. V Srinivasan‡ Department of Theoretical Physics, University of Madras, Guindy, Chennai, India, 600025. 5 1 Inthispaper,weextendSimon’scriterion forGaussian statestothemulti-modeGaussian states 0 using the Marchenko-Pastur theorem. We show that the Marchenko-Pastur theorem from random 2 matrix theory as necessary and sufficient condition for separability. n a J Themostimportantquestioninthequantuminforma- The invariant ensembles in random matrix theory is 4 tionishowdoesonesaythatthegivenstateisentangled classified into three class, the Gaussian ensembles these 2 or separable? This question has been be answered by matricesareknownas the Wignermatrices,the Wishart peres[1]inthefinite dimensionalcase. Horodecki[2]has matrices and the two Wishart matrices. For details ref ] shown that this condition is necessary and sufficient for [6]. The Marcenko Pastur’s Quarter-Circle Law is given h p separability in the 2 2 and 2 3 dimensional cases, forthe Wishartmatricesandthey aredefinedasfollows. × × - butfailsforthehigherdimensions. Intheliterature,this Namely,letX1, ,Xn beindependentandidentically nt condition is known as Peres-Horodecki criterion. The distributed random···column vectors of m, with mean 0 R a Peres-Horodecki criterion for the covariance matrix us- and covariance matrix I . Let us consider the m m m u ing Wigner distribution has been given for the Gaussian empirical covariance matrix × q statesbySimon[3]. Ithasbeenshowninthesamepaper [ 1 n 1 that the transpose operation T, which takes every ρˆ to Σ= X XT = YYTwhereY =(X X ). (3) 1 its transpose ρˆT, is equivalent to a mirror reflection in n i i n 1··· n Xi=1 v phase space: 4 Then the distribution of the eigenvalues of Σ are con- 0 ρˆ ρˆT W(q,p) W(q, p). (1) sidered under the different assumptions on the number 0 −→ ⇐⇒ −→ − of variables m and the number of observations n or m 6 In this paper we extend the Simon’s criterion to the is the sample size and n is the dimension of the vectors. 0 ensemble of n-bipartite system of two modes using the Their joint probability distribution function of eigenval- . 1 Merchnko Pasture law in random matrix theory. The ues is given by the Laguerre polynomials [6]. We have 0 Marchenko-PasturlawappearsinthefamousMarchenko- E(1YY) = I and the strong law of large numbers in- 5 Pastur theorem [4]. dicantes that wmith probability one, 1 Inrandommatrixtheory,thedynamicsoftheensemble : 1 v ofaninfinite dimensionalrandommatrix is describedby lim YYT =I . (4) n→∞ m i the probability distribution function [5] n X In the case m grows with n, here m is the number r (λ ,...,λ )dΛ=c e−βH dΛ (2) a P 1 N n of random variables and the n is the dimension of the Hilbert space, one has the following theorem known as where the matrix H is brought into the following form H = N V(λ ) N lnλ λ here V(λ ) is the the Marchenko-Pasturtheorem [4]. − i=1 i − i<1 | i− j| i Theorem: Letusassumefor simplicity that thecompo- potentiaPl and the λi aPre the eigenvalues with i being a nents of X are Gaussian random variables. zero Mean, freeindexrunningfrom1,2..N,then λ λ istheVan- i | i− j| unitvariance, andboundedmomentsthatisthereissome dermonde determinant, dΛ=dλ ...dλ , c is constant 1 N n bound B, independent of m, such that n,E(x k) B. of proportionality and the index β =1,2,4 characterises ∀ | ij| ≤ Then n depends on m in such a way that m/n r 1 the real parameters of the symmetry class of orthogo- → ≤ as n . Under these assumptions, the distribution of nal, unitary and symplectic respectively for the random the e→igen∞values of 1YYT asymptotically approaches the matrix H. n Marcenko-Pastur law as n →∞ (x a)(b x) f(x)= − − . (5) p 2πxr ∗ [email protected] † [email protected] where a = (1 √r)2 and b = (1+√r)2. For the case − ‡ [email protected] r =1 it reduces to a famous quarter circle law. 2 One important question that arises is how does one with nonnegative p ’s, where ρˆ ’s and ρˆ ’s are den- j jAi jBi comparethecovariancematrixconstructedfromtheran- sity operators of the modes of A and B respectively. To dom matrices to a covarience matrix obtained from a check that the given n-bipartite system of two modes is definite state? First thing, if one looks at this two co- entangledornotwewouldperformthatpartialtranspose varince matrices they are Gaussian. Then the ques- operation on the density matrix (8) denoted PT. Then tion of randomness in the quantum system, this comes one calculates the eigenvalues partial transposeddensity from the wave function. This is shown by one of the matrix. If all the eigenvalues are positive then the it is a authors [7], the connection between the action equa- separable density matrix if it has atleast one eigenvalue tion for the stationary points of the random matrix en- then the state is entangled. This is the Peres-Horodecki semble V(x) 1 log(x x ) where V(x) is the separability criterion. − 2 k6=l | k − l| potential and thePquantum momentum function [8, 9] Inthecaseoflargematricesitishardtocheckthepos- p = n −i +Q(x) here the moving poles are sim- itivity of the all eigenvalues, but the Marchenko-Pastur k=1 x−xk ple pPoles with residue i¯h (we take here ¯h = m = 1) theorem in random matrix theory gives a condition for andQ(x)isthesingular−partofthequantummomentum theconvergenceoftheeigenvaluesforthelargematrices. function. This property of the random matrices is used to check Therefore, if one takes this point of view that there is weather a given state is separable or entangled. randomness associated with the wave function then one Inorderto apply the Peres-Horodeckiseparabilitycri- can see that the Wigner function or distribution is like terionfortheGaussianstatesonehastogoovertophase the Wishart distribution. In random matrix theory the space picture and then study the partialtranspose oper- ensembles constructed using covariance matrix are the ation in the Wigner picture, it is convenient to arrange known as the Wishart Matrices and their joint proba- the phase space variables as bility distribution function of eigenvalues is given by the ξ = q q q q p p p p . Laguerre polynomials [6]. The Wigner function for the A1 ··· An B1 ··· Bn A1 ··· An B1 ··· Bn Harmonic oscillator are the Laguerre polynomials [10]. (cid:0) (cid:1) In terms of hermitian canonical operators It is shown in the reference [11] the Wigner function of the squeezed displaced vacuum state gives the Laguerre ξˆ= qˆ qˆ qˆ qˆ pˆ pˆ pˆ pˆ . polynomials. A1 ··· An B1 ··· Bn A1 ··· An B1 ··· Bn (cid:0) (cid:1) It is well known now that the squeezed states are non The commutation relations take the compact form[14] classical states and shows the property of entanglement [3]. Theconditionforsqueezingischaracterizedinterms [ξˆ ,ξˆ ] = iΩ , α,β =1,2,3,4; α β αβ of covariance matrix using Wigner function reduces to a J 0 uncertainty relationship [12–14] ··· · 0 1 Ω =  · · · · , J = , (9) i (cid:18) 1 0 (cid:19) V + 2Ω≥0, (6)  0· · · J·  −  ··· ·  where V is the covariance matrix and is obtained by here Ω is a 4n 4n matrix. In the two mode case it arranging the phase space variables and the hermitian × has been shown in ref [3] under partial transpose opera- canonicaloperatorsintofour-dimensionalcolumnvectors tiononthebipartitedensityoperatortranscribeshasthe following transformation on the Wigner distribution: ξ = q p q p , ξˆ= qˆ pˆ qˆ pˆ . 1 1 2 2 1 1 2 2 (cid:0) (cid:1) (cid:0) (cid:1) PT : W(q ,p ,q ,p ) W(q ,p ,q , p ). (10) Then the vector ξ satisfies the following commutation 1 1 2 2 −→ 1 1 2 − 2 relationship [12] It is clear from the above that the sign momentum of [ξˆ ,ξˆ ] = iΩ , α,β =1,2,3,4; the secondis changing. Thereforewhen we apply partial α β αβ transpositiononmulti-mode systemis equivalentto flip- J 0 0 1 pingthesignofmomentumvariableofsecondmodethat Ω = , J = . (7) (cid:18) 0 J (cid:19) (cid:18) 1 0 (cid:19) is p . − Bi The Peres-Horodecki separability criterion for Gaus- In particular, the condition for the multimode squeezing sian state is defined in ref [3] as reads: if ρˆ is separable, is also defined in terms of covariance matrix in ref [14]. then its Wigner distribution necessarily goes over into a Let us consider a ensemble n-bipartite system of Wigner distribution under the phase space mirror reflec- two modes described by annihilation operators aˆ = i tion Λ. W(Λξ), like W(ξ), should possess the “Wigner n (qˆ +ipˆ )/√2 and ˆb = n (qˆ +ipˆ )/√2. i=1 Ai Ai i i=1 Bi Bi quality”, for any separable bipartite state. PBy definition, a quantum state ρˆPof the bipartite system In multimode case, the covariance matrix will be of is separableif and only if ρˆcan be expressedin the form dimension 4n 4n that is dimension 2n from mode A × n and another dimension 2n from mode B. By following ρˆ= p ρˆ ρˆ , (8) the procedure suggested in the ref [3] and arranging the j jAi ⊗ jBi AiX,Bi=1Xj uncertainties or variances into a 4n ×4n real variance 3 matrix V, defined through V = ∆ξˆ ,∆ξˆ , where where Λ = (1, ,1,1, 1, ,1, 1), there are total 4n αβ α β h{ }i ··· − ··· − ∆ξˆ= ξˆ ξˆ, here ξˆα = trξˆαρˆ. Then the uncertainty ones, the first 2n the +1 corresponds to for the qAi,qBi principle−[3h,i13, 14].h i in the second 2n the +1 corresponds to the pAi and the 1correspondstothe p . Inthe covariancematrixthis Bi − i is equivalent to transposing each σ . Hence one has V + Ω 0. (11) ij 2 ≥ σT σT Note that (6) implies, in particular, that V > 0, for de- 11 ··· · 1n tails ref [3, 13, 14]. V˜ = · · · · , (16) Nowbyusingareallineartransformationonξˆspecified  · · · ·  σT σT  by 4n 4n real matrix S(r)  n1 ··· · nn  × ξˆ ξˆ′ =S(r)ξˆ (12) Given a Gaussian states say ρˆconstructed form Hilbert → space H H , under partial transposition of B, ρˆ A B ⊗ Thistransformationpreservesthecommutationrelations has to be a bonifide density matrix. Then the Peres- [14]. We use S(r) in such a way that the Horodecki criterion is imposed on the covariance matrix constructedfromthe phasespacevariables[3]. Thenthe ξˆ′ = qˆ qˆ qˆ qˆ pˆ pˆ pˆ pˆ . Peres-Horodecki criterion for the multimode continuous A1 B1 An Bn A1 B1 An Bn ··· ··· variables or the Simon’s criterion for multimode reduces (cid:0) (cid:1) In terms of annihilation and creation operators which is to the following theorem. equivalent to Theorem : It is necessary and sufficient condition for separability for a given multimode Gaussian state if the ξˆ′ = aˆ ˆb aˆ ˆb aˆ† ˆb† aˆ† ˆb† . eigenvalue distribution of the partially transposed covari- (cid:0) A1 B1 ··· An Bn A1 B1 ··· An Bn (cid:1) ance matrix V˜ satisfies Marcenko-Pastur law . Thus the covariance matrix is written as Proof: Let us assume that the partial transposed covariance matrix V˜ is a bonifide covariance matrix of σ σ 11 1n ··· · Gaussian states and the eigenvalue distribution of this V = · · · · , (13) covariance matrix satisfy’s the Merchnko Pasture law. σ· · · σ·  Thus by identifying this covariance matrix of Gaussian  n1 ··· · nn  states with the Gaussian random covariance matrix we where σ is a 2 2 matrix constructed from expec- getthatm=2nandn=4n. Onecanclearlyseethatas ij tisataionbovnailfiudees tohfe(ac×Aovi,abrBiain,ace†Aim,ba†Btir,ix) eolfemGeanutsss.iaSninsctaeteVs dmis→trib∞utaionndonf→the∞etigheenrvaatliuoems o/fnV==1/2ξˆ.′ξˆT′Thearseyfomrpe,tothtie- then the eigenvalue distribution of this covariance ma- cally approaches the Marcenko-Pasturlaw as m,n →∞ trix should follow Merchnko Pasture law. By identifying this covariance matrix Gaussian states with the Gaus- (x a)(b x) f(x)= − − . (17) sian random covariance matrix we get that m = 2n and p πx n=4n. One canclearly see that as m andn →∞ →∞ the rationm/n=1/2. Therefore, the distribution of the where a = (1 1)2 and b = (1+ 1)2. This gives a eigenvalues of V = ξˆ′ξˆ′T asymptotically approaches the −q2 q2 condition that the eigenvalues x is bounded above and Marcenko-Pasturlaw as m,n →∞ below by 2√2 < x 3 < 2√2. Within this range the − − states are separable and outside they are entangled. (x a)(b x) f(x)= − − . (14) In Conclusion, we extend Simon’s criterion for Gaus- p πx sian states to the multi-mode Gaussian states using the Marchenko-Pastur theorem from random matrix theory where a = (1 1)2 and b = (1+ 1)2. It is evident −q2 q2 as necessary and sufficient condition for separability. fromeq(10)thatunderthepartialtransposetheWigner distributionundergoesmirrorreflection,andthatreflects at the covariance matrix level by the following transfor- mationV V˜ =ΛVΛ. Thenthenewdistributionfunc- Acknowledgments → tion W(Λξ) has to be a Wigner distribution if the state under consideration is separable, then one has Authors thank KVSSC acknowledges the Department of Science and technology, Govt of India (fast track scheme(D. O.No: SR/FTP/PS-139/2012))for financial i V˜ + Ω 0, V˜ =ΛVΛ, (15) support. 2 ≥ 4 [1] A.Peres, Phys.Rev. Lett.77, 1413 (1996). Phys. Lett. A,12, 295 (1997). [2] P.Horodecki, Phys.Lett. A 232, 333 (1997). [10] M.Hillery,R.OConnell,M.Scully,andE.Wigner,Phys. [3] R.Simon, Phys. Rev.Lett. 84, 2726 (2000) Rep. 106, 121-167 (1984). [4] Mat.SbornikMath.Tom72(114),No.4,(1967),USSR- [11] AbdAl-KaderGM2003J.Opt.B:QuantumSemiclass. Sbornik Vol. 1 ,No. 4,(1967). Opt.5S228, VenkataSatyanarayanaM1985 Phys.Rev. [5] Mehta,M.L.(2004).RandomMatrices,3rdEdition,Pure D 32 400. and Applied Mathematics (Amsterdam), 142, Amster- [12] R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. dam, Netherlands: Elsevier/Academic Press. Rev. A 37, 3028 (1988). 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