Fidelity based unitary operation-induced quantum correlation for continuous-variable systems
aa r X i v : . [ qu a n t - ph ] J a n Fidelity based unitary operation-induced quantum correlation for continuous-variablesystems
Liang Liu, Xiaofei Qi,
2, 3 and Jinchuan Hou Institute of Mechanics, Taiyuan University of Technology, Taiyuan 030024, P. R. China ∗ Department of Mathematics, Shanxi University, Taiyuan 030006, P. R. China Institute of Big Data Science and Industry, Shanxi University, Taiyuan 030024, P. R. China † Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. China ‡ We propose a measure of nonclassical correlation N GF in terms of local Gaussian unitary operationsbased on square of the fidelity F for bipartite continuous-variable systems. This quantity is easierto calculate or estimate and is a remedy for the local ancilla problem associated with the geometricmeasurement-induced nonlocality. A simple computation formula of N GF for any (1+1) -mode Gaus-sian states is presented and an estimation of N GF for any ( n + m ) -mode Gaussian states is given. Forany (1 + 1) -mode Gaussian states, N GF does not increase after performing a local Gaussian channelon the unmeasured subsystem. Comparing N GF ( ρ AB ) in scale with other quantum correlations suchas Gaussian geometric discord for two-mode symmetric squeezed thermal states reveals that N GF ismuch better in detecting quantum correlations of Gaussian states. PACS numbers : 03.67.Mn, 03.65.Ud, 03.65.Ta
Keywords : Measurement-induced nonlocality, Gaussian states, Gaussian geometric discord,Gaussian channels, fidelity
INTRODUCTION
The presence of correlations in bipartite quantum systems is one of the main features of quantum me-chanics. The most important among such correlations is surely entanglement [1]. However, much attentionhas been devoted to studying and characterizing the quantum correlations that go beyond the paradigm ofentanglement recently. Non-entangled quantum correlations are also physical resources which play impor-tant roles in various quantum communications and quantum computing tasks.For the last two decades, various methods have been proposed to describe quantum correlations, suchas quantum discord (QD) [2], geometric quantum discord [3–5], measurement-induced nonlocality (MIN)[6] and measurement-induced disturbance (MID) [7] for discrete-variable systems. For continuous-variablesystems, Giorda, Paris [8] and Adesso, Datta [9] independently gave the definition of Gaussian QD for two-mode Gaussian states and discussed its properties. G. Adesso, D. Girolami in [10] proposed the concept ofGaussian geometric discord for Gaussian states. Measurement-induced disturbance of Gaussian states wasstudied in [11]. In [12], the MIN for Gaussian states was discussed. For other related results, see [13–19]and the references therein. Also, many efforts have been made to find simpler methods to quantify thesecorrelations. However, it seems that this is a very difficult task, too. By now, for example, almost all knownquantifications of various correlations, including entanglement measures, for continuous-variable systemsare difficult to evaluate and can only be calculated for (1 + 1) -mode Gaussian states or some special states.Even for finite-dimensional cases, the authors in [20] proved that computing quantum discord is NP-hard.So it makes sense and is important to find more helpful quantifications of quantum correlations.The purpose of this paper is to propose a correlation N GF for bipartite Gaussian systems in terms oflocal Gaussian unitary operations based on square of the fidelity F introduced by Wang, Yu and Yi in [21].This correlation N GF describes the same correlation as Gaussian geometric discord for Gaussian states buthave some remarkable nice properties that the known quantifications are not possed: (1) N GF is a quantumcorrelation without ancilla problem; (2) N GF ( ρ AB ) can be easily estimated for any ( n + m ) -mode Gaussianstates and calculated for any (1 + 1) -mode Gaussian states; (3) N GF is non-increasing after performinglocal Gaussian operations on the unmeasured subsystem. Comparison N GF in scale with other quantumcorrelations for two-mode symmetric squeezed thermal states reveals that N GF is better in detecting thenonclassicality in Gaussian states. GAUSSIAN STATES AND GAUSSIAN UNITARY OPERATIONS
We recall briefly some notions and notations concerning Gaussian states and Gaussian unitary opera-tions. For arbitrary state ρ in a n -mode continuous-variable system with state space H , its characteristicfunction χ ρ is defined as χ ρ ( z ) = tr( ρW ( z )) , where z = ( x , y , · · · , x n , y n ) T ∈ R n , W ( z ) = exp( iR T z ) is the Weyl displacement operator, R =( R , R , · · · , R n ) = ( ˆ Q , ˆ P , · · · , ˆ Q n , ˆ P n ) . As usual, ˆ Q k = ( ˆ a k + ˆ a k † ) / √ and ˆ P k = − i ( ˆ a k − ˆ a k † ) / √ ( k = 1 , , · · · , n ) stand for respectively the position and momentum operators, where ˆ a † k and ˆ a k are thecreation and annihilation operators in the k th mode satisfying the Canonical Commutation Relation (CCR) [ˆ a k , ˆ a † l ] = δ kl I and [ˆ a † k , ˆ a † l ] = [ˆ a k , ˆ a l ] = 0 , k, l = 1 , , · · · , n.ρ is called a Gaussian state if χ ρ ( z ) is of the form χ ρ ( z ) = exp[ − z T Γ z + i d T z ] , where d = ( h ˆ R i , h ˆ R i , . . . , h ˆ R n i ) T = (tr( ρR ) , tr( ρR ) , . . . , tr( ρR n )) T ∈ R n is called the mean or the displacement vector of ρ and Γ = ( γ kl ) ∈ M n ( R ) is the covariance matrix (CM)of ρ defined by γ kl = tr[ ρ (∆ ˆ R k ∆ ˆ R l + ∆ ˆ R l ∆ ˆ R k )] with ∆ ˆ R k = ˆ R k − h ˆ R k i ([22]). Note that Γ is realsymmetric and satisfies the condition Γ + i ∆ ≥ , where ∆ = ⊕ nk =1 ∆ k with ∆ k = − for each k .Here M d ( R ) stands for the algebra of all d × d matrices over the real field R .Now assume that ρ AB is an ( n + m ) -mode Gaussian state with state space H = H A ⊗ H B . Then theCM Γ of ρ AB can be written as Γ = A CC T B , (1)where A ∈ M n ( R ) , B ∈ M m ( R ) and C ∈ M n × m ( R ) . Particularly, if n = m = 1 , by means of localGaussian unitary (symplectic at the CM level) operations, Γ has a standard form: Γ = A C C T B , (2)where A = a a , B = b b , C = c d , a, b ≥ and ab − ≥ c ( d ) .For any unitary operator U acting on H , the unitary operation ρ U ρU † is said to be Gaussian if itsends Gaussian states into Gaussian states, and such U is called a Gaussian unitary operator. It is well-known that a unitary operator U is Gaussian if and only if U † RU = S R + m for some vector m in R n and some S ∈ Sp(2 n, R ) , the symplectic group of all n × n real matrices S that satisfy S ∈ Sp(2 n, R ) ⇔ S ∆ S T = ∆ . Thus, every Gaussian unitary operator U is determined by some affine symplectic map ( S , m ) acting on thephase space, and can be denoted by U = U S , m ([23, 24]).We list some simple facts for Gaussian states and Gaussian unitary operations, and some useful resultsfor matrix theory, which will be used frequently in the present paper. Lemma 1. ([23])
For any ( n + m ) -mode Gaussian state ρ AB , write its CM Γ as in Eq.(1). Then the CMsof the reduced states ρ A = tr B ρ AB and ρ B = tr A ρ AB are matrices A and B , respectively. Denote by S ( H ) the set of all quantum states of the system with state space H . Lemma 2. ([25])
Assume that ρ AB ∈ S ( H A ⊗ H B ) is a ( n + m ) -mode Gaussian state. Then ρ AB is aproduct state, that is, ρ AB = σ A ⊗ σ B for some σ A ∈ S ( H A ) and σ B ∈ S ( H B ) , if and only if Γ = Γ A ⊕ Γ B ,where Γ , Γ A and Γ B are the CMs of ρ AB , σ A and σ B , respectively. Lemma 3. ([23, 24])
Assume that ρ is any n -mode Gaussian state with CM Γ and displacement vector d ,and assume that U S , m is a Gaussian unitary operator. Then the characteristic function of the Gaussianstate σ = U ρU † is of the form exp( − z T Γ σ z + i d T σ z ) , where Γ σ = S Γ S T and d σ = m + Sd . Lemma 4. ([26])
For any quantum states ρ , σ and any numbers a > , we have tr( ρσ ) ≤ (tr ρ a ) a (tr σ b ) b , where b = aa − . Lemma 5. ([27])
Let M = A BC D be a square matrix.(1) If A is invertible, then its determinant det A BC D = (det A )(det( D − CA − B )) .(2) If D is invertible, then its determinant det A BC D = (det D )(det( A − BD − C )) . FIDELITY BASED NONCLASSICALITY OF GAUSSIAN STATES BY GAUSSIAN UNITARYOPERATIONS
Fidelity is a measure of closeness between two arbitrary states ρ and σ , defined as F ( ρ, σ ) =(tr p √ ρσ √ ρ ) [28]. This measure has been explored in various context of quantum information processingsuch as cloning [29], teleportation [30], quantum states tomography [31], quantum chaos [32] and spot-lighting phase transition in physical systems [33]. Though fidelity itself is not a metric, one can define ametric D ( ρ, σ ) = g ( F ( ρ, σ )) , where g is a monotonically decreasing function of distance measure. A fewsuch fidelity induced metrics we mentioned here are Bures angle A ( ρ, σ ) = arccos p F ( ρ, σ ) , Bures metric B ( ρ, σ ) = (2 − p F ( ρ, σ )) and sine metric C ( ρ, σ ) = p − F ( ρ, σ ) [34].Since the computation of fidelity involves square root of density matrix, various forms of fidelity havebeen proposed to simplify the computation. In [21], the authors proposed another form F of fidelity as F ( ρ, σ ) = | tr ρσ | p tr ρ tr σ , (3)In [35], to capture global nonlocal effect of a quantum state of discrete system due to locally invariantprojective measurements, the authors use the fidelity in Eq.(3) to define a metric C ( ρ, σ ) = p − F ( ρ, σ ) for any states ρ and σ . Furthermore, for any finite-dimensional bipartite quantum state ρ AB , a new kind ofMIN in terms of this metric was defined as N F ( ρ AB ) = max Π A C ( ρ AB , Π A ( ρ AB )) , where the maximum is taken over all von Neumann measurements performing on subsystem A that areinvariant at ρ A = tr B ( ρ AB ) , the reduced state of ρ AB . They presented an analytic expression of thisversion of MIN for pure bipartite states and × n dimensional mixed states.In the present paper, motivated by the work of [35], we propose a quantum nonclassicality N GF forcontinuous-variable systems by local Gaussian unitary operations for ( n + m ) -mode states using the samemetric based on the fidelity Eq.(3). Definition 1.
For any ( n + m ) -mode state ρ AB ∈ S ( H A ⊗ H B ) , the quantity N GF ( ρ AB ) is defined by N GF ( ρ AB ) = sup U C ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I )) = sup U { − (tr ρ AB ( U ⊗ I ) ρ AB ( U † ⊗ I )) tr( ρ AB )tr(( U ⊗ I ) ρ AB ( U † ⊗ I )) } , (4) where the supremum is taken over all Gaussian unitary operators U on H A satisfying U ρ A U † = ρ A . Remark 1.
For any Gaussian state ρ AB , there are many nontrivial Gaussian unitary operators U (otherthan the identity I ) satisfying U ρ A U † = ρ A [16], and hence Definition 1 makes sense. Different from [16],in which a quantum nonclassicality N is proposed by Gaussian unitary operations based on the Hilbert-Schmidt norm, the quantity N GF ( ρ AB ) measures the global nonlocal effect of a quantum state due to locallyinvariant Gaussian unitary operations by the metric C ( ρ, σ ) = 1 − F ( ρ, σ ) with the fidelity F as inEq.(3). Recall that, the MIN [6] is defined as the square of Hilbert-Schmidt norm k · k ( k A k = p tr( A † A ) )of difference of pre- and post-measurement states. i.e., N ( ρ AB ) = max Π A k ρ AB − (Π A ⊗ I ) ρ AB (Π A ⊗ I ) † k , where the maximum is taken over all von Neumann measurements which maintain the reduced state ρ A invariant corresponding to part A. In [16], a kind of quantum correlation N for ( n + m ) -mode continuous-variable systems is defined as the square of Hilbert-Schmidt norm of difference of pre- and post-transformstates N ( ρ AB ) = 12 sup U k ρ AB − ( U ⊗ I ) ρ AB ( U ⊗ I ) † k , where the supremum is taken over all unitary operators which maintain ρ A invariant corresponding to partyA. There are other quantum correlations defined by Hilbert-Schmidt norm, for example, the Gaussian ge-ometric discord and the quantum correlation proposed respectively in [10, 13]. These kinds of quantitydefined by Hilbert-Schmidt norm mentioned above may change rather wildly through some trivial and un-correlated actions on the unoperated party B. For example, if we append an uncorrelated ancilla C, andregarding the state ρ ABC = ρ AB ⊗ ρ C as a bipartite state with the partition A:BC. After some straight-forward calculations, one gets N ( ρ ABC ) = N ( ρ AB )tr ρ C , which means that the quantity N differs arbitrarily due to local ancilla C as long as ρ C is mixed. While thisproblem can be avoided if one employs N GF as in Definition 1 since F ( ρ ABC , ( U ⊗ I ⊗ I ) ρ ABC ) = F ( ρ AB ⊗ ρ C , ( U ⊗ I ) ρ AB ⊗ Iρ C )= F ( ρ AB , ( U ⊗ I ) ρ AB ) · F ( ρ C , ρ C ) = F ( ρ AB , ( U ⊗ I ) ρ AB ) , according to the multiplicativity of the fidelity [21]. Thus, we reach the following conclusion. Theorem 1. N GF is a quantum nonclassicality without ancilla problem. We explore further the properties of N GF below. Denote by B ( H ) the algebra of all bounded linearoperators acting on H . Theorem 2. N GF is locally Gaussian unitary invariant, that is, for any ( n + m ) -mode Gaussian state ρ AB ∈ S ( H A ⊗ H B ) and any Gaussian unitary operators W ∈ B ( H A ) and V ∈ B ( H B ) , we have N GF (( W ⊗ V ) ρ AB ( W † ⊗ V † )) = N GF ( ρ AB ) . Proof.
Assume that ρ AB ∈ S ( H A ⊗ H B ) is an ( n + m ) -mode Gaussian state. For given Gaussianunitary operators W ∈ B ( H A ) and V ∈ B ( H B ) , let σ AB = ( W ⊗ V ) ρ AB ( W † ⊗ V † ) . Denote U G ( H A ) theset of all Gaussian unitary operators acting on H A . Since N GF ( ρ AB ) = sup U ∈U G ( H A ) , Uρ A U † = ρ A C ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I ))= sup U ∈U G ( H A ) , Uρ A U † = ρ A { − F ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I )) } =1 − inf U ∈U G ( H A ) , Uρ A U † = ρ A F ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I )) , to demonstrate that N GF is locally Gaussian unitary invariant, it is sufficient to prove inf U ∈U G ( H A ) , Uρ A U † = ρ A F ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I )) (5) = inf U ′ ∈U G ( H A ) , U ′ σ A U ′† = σ A F ( σ AB , ( U ′ ⊗ I ) σ AB ( U ′† ⊗ I )) , where σ AB = ( W ⊗ V ) ρ AB ( W † ⊗ V † ) , W and V are given Gaussian unitary operators acting on Hilbertspaces H A and H B , respectively.Note that σ A = W ρ A W † . For any Gaussian unitary operator U ∈ B ( H A ) satisfying U ρ A U † = ρ A , let U ′ = W U W † . Then U ′ is a Gaussian unitary operator satisfing U ′ σ A U ′† = W U W † W ρ A W † W U † W † = σ A . Conversely, if U ′ σ A U ′† = σ A , U = W † U ′ W will satisfy U ρ A U † = ρ A . By Eq.(3), we have F ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I )) = (tr ρ AB ( U ⊗ I ) ρ AB ( U † ⊗ I )) tr ρ AB tr(( U ⊗ I ) ρ AB ( U † ⊗ I )) = (tr( W † ⊗ V † ) σ AB ( W ⊗ V )( U ⊗ I )( W † ⊗ V † ) σ AB ( W ⊗ V )( U † ⊗ I )) tr(( W † ⊗ V † ) σ AB ( W ⊗ V )) tr(( U ⊗ I )( W † ⊗ V † ) σ AB ( W ⊗ V )( U † ⊗ I )) = (tr σ AB ( U ′ ⊗ I ) σ AB ( U ′† ⊗ I )) tr σ AB tr(( U ′ ⊗ I ) σ AB ( U ′† ⊗ I )) = F ( σ AB , ( U ′ ⊗ I ) σ AB ( U ′† ⊗ I )) . Therefore, Eq.(5) holds, as desired. (cid:3)
Notice that, for any ( n + m ) -mode product quantum state ρ AB , one must have N GF ( ρ AB ) = 0 by thedefinition. But for Gaussian states, the converse is also true. Hence, when restricted to Gaussian states,the correlation N GF describes the same nonclassicality as that described by Gaussian QD (two-mode) [8, 9],Gaussian geometric discord [10], the correlations Q , Q P discussed in [13] and the correlation N discussedin [16]. Theorem 3.
For any ( n + m ) -mode Gaussian state ρ AB ∈ S ( H A ⊗ H B ) , N GF ( ρ AB ) = 0 if and only if ρ AB is a product state. Proof.
By Definition 1, the “if” part is apparent. Let us check the “only if” part. Since the mean of anyGaussian state can be transformed to zero under some local Gaussian unitary operation, by Theorem 2, it issufficient to consider the Gaussian states whose mean are zero.Assume that ρ AB is an ( n + m ) -mode Gaussian state with CM Γ = A CC T B as in Eq.(1) andzero mean such that N GF ( ρ AB ) = 0 . By Lemma 1, the CM of ρ A is A . According to the WilliamsonTheorem, there exists a symplectic matrix S such that S A S T0 = ⊕ ni =1 v i I and U ρ A U † = ⊗ ni =1 ρ i , where U = U S , and ρ i s are some thermal states. Write σ AB = ( U ⊗ I ) ρ AB ( U † ⊗ I ) . It follows from Theorem2 that N GF ( σ AB ) = N GF ( ρ AB ) = 0 . Obviously, σ AB has the CM Γ ′ = ⊕ ni v i I C ′ C ′ T B ′ and the mean 0.By Lemma 3 and [16], for any Gaussian unitary operator U S , m ∈ B ( H A ) so that m = 0 and S = ⊕ ni =1 S θ i with S θ i = cos θ i sin θ i − sin θ i cos θ i for some θ i ∈ [0 , π ] , we have U S , m σ A U † S , m = σ A = tr B ( σ AB ) . Then, by the definition Eq.(4), N GF ( σ AB ) = 0 entails (tr σ AB (U S , m ⊗ I) σ AB (U † S , m ⊗ I)) = tr σ tr((U S , m ⊗ I) σ AB (U † S , m ⊗ I)) . Since the Holder’s inequality (Lemma 4) asserts that tr( ρσ ) ≤ tr ρ tr σ and clearly, the equality holds ifand only if σ = ρ , we must have σ AB = ( U S , m ⊗ I ) σ AB ( U † S , m ⊗ I ) . Hence σ AB and ( U S , m ⊗ I ) σ AB ( U † S , m ⊗ I ) have the same CMs, that is, ⊕ ni =1 v i I C ′ C ′ T B ′ = ⊕ ni =1 v i I S C ′ C ′ T S T B ′ . If we take θ i ∈ (0 , π ) for each i , then I − S is an invertible matrix, which forces C ′ = 0 . So σ AB is aproduct state by Lemma 2. It follows that ρ AB = ( U † ⊗ I ) σ AB ( U ⊗ I ) is also a product state. (cid:3) In the rest of this paper, we mainly consider the case when the states ρ AB are Gaussian.A remarkable virtue of N GF is that it can be evaluated easily. For any two-mode Gaussian state ρ AB , wecan give an analytic computation formula. Theorem 4.
For any (1 + 1) -mode Gaussian state ρ AB whose CM has the standard form Γ = A C C T0 B = a c a dc b d b , we have N GF ( ρ AB ) = 1 − ( ab − c )( ab − d )( ab − c / ab − d / . Particularly, the value of N GF ( ρ AB ) is independent of the mean of the state ρ AB . Proof.
For any (1 + 1) -mode Gaussian state ρ AB with CM Γ ′ and mean ( d ′ A , d ′ B ) , we can always findtwo Gaussian operators U and V so that the CM Γ of σ AB = ( U ⊗ V ) ρ AB ( U † ⊗ V † ) is of the standardform Γ = A C C T0 B = a c a dc b d b . Denote the mean of σ AB by ( d A , d B ) . Since N GF is locally Gaussian unitary invariant, one has N GF ( ρ AB ) = N GF ( σ AB ) . Hence, we may assume that the CM of ρ AB is Γ and the mean of ρ AB is ( d A , d B ) . For anyGaussian unitary operator U S , m such that U S , m ρ A U † S , m = ρ A , we see that S and m meet the conditions S A S T = A and Sd A + m = d A . As A = aI , we have SS T = I . It follows from S ∆ S T = ∆ that there exists some θ ∈ [0 , π ] such that S = S θ = cos θ sin θ − sin θ cos θ . So the CM of Gaussian state ( U S , m ⊗ I ) ρ AB ( U † S , m ⊗ I ) is Γ θ = a c cos θ d sin θ a − c sin θ d cos θc cos θ − c sin θ b d sin θ d cos θ b , and the mean of ( U S , m ⊗ I ) ρ AB ( U † S , m ⊗ I ) is ( S ⊕ I )( d A ⊕ d B ) + m ⊕ Sd A + m ) ⊕ d B = d A ⊕ d B = ( d A , d B ) as Sd A + m = d A . Conversely, for any S θ , taking m = d A − S θ d A , we have U S θ , m satisfies the condition U S θ , m ρ A U † S θ , m = ρ A .Also, notice that, for any n -mode Gaussian states ρ, σ with CMs V ρ , V σ and means d ρ , d σ , respectively,it is shown in [36] thatTr ( ρσ ) = 1 p det[( V ρ + V σ ) /
2] exp[ − δ h d i T det[( V ρ + V σ ) / − δ h d i ] , (6)where δ h d i = d ρ − d σ .0Hence, by Eq.(4) and Eq.(6) as well as the fact that det Γ θ = det Γ = ( ab − c )( ab − d ) , one obtains N GF ( ρ AB ) = sup U ∈U G ( H A ) , Uρ A U † = ρ A C ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I ))= sup U ∈U G ( H A ) , Uρ A U † = ρ A { − ( tr ρ AB ( U ⊗ I ) ρ AB ( U † ⊗ I )) tr ( ρ AB ) tr (( U ⊗ I ) ρ AB ( U † ⊗ I )) } = sup θ ∈ [0 , π ] { − √ det Γ det Γ θ det((Γ + Γ θ ) / } = max θ ∈ [0 , π ] { − ( ab − c )( ab − d )[ ab − c (1 + cos θ ) / ab − d (1 + cos θ ) / } =1 − ( ab − c )( ab − d )( ab − c / ab − d / , and, this quantity is independent of the mean of ρ AB , completing the proof. (cid:3) Next, we are going to give an estimate of N GF for any ( n + m ) -mode Gaussian state ρ AB . Theorem 5.
For any ( n + m ) -mode Gaussian state ρ AB with CM Γ = A CC T B , N GF ( ρ AB ) is indepen-dent of the mean of ρ AB and ≤ N GF ( ρ AB ) ≤ − det( B − C T A − C )det B < . Furthermore, the upper bound is tight. Proof.
Let ρ AB be any ( n + m ) -mode Gaussian state with CM Γ = A CC T B and mean d =( d A , d B ) . Note that, by Lemma 1, the CM of ρ A is A . Write σ AB = ( U S , m ⊗ I ) ρ AB ( U † S , m ⊗ I ) , where U S , m is any Gaussian unitary operator of the subsystem A . Clearly, U S , m ρ A U † S , m = ρ A if and only if thesymplectic matrix S satisfies S A S T = A and the vector m = d A − Sd A . In this case σ AB has the CM Γ S = A S CC T S T B and the mean d S = ( S ⊕ I )( d A ⊕ d B ) + m ⊕ Sd A + m ) ⊕ d B = d A ⊕ d B = ( d A , d B ) = d .Denote by S (2 n ) = Sp(2 n, R ) , the set of all n × n symplectic matrices. Then, by Eq.(6), N GF ( ρ AB ) = sup U ∈U G ( H A ) , Uρ A U † = ρ A C ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I ))= sup U ∈U G ( H A ) , Uρ A U † = ρ A { − ( tr ρ AB ( U ⊗ I ) ρ AB ( U † ⊗ I )) tr ( ρ AB ) tr (( U ⊗ I ) ρ AB ( U † ⊗ I )) } = sup S ∈S (2 n ) , S A S T = A { − S ) / √ det Γ 1 √ det Γ S } = sup S ∈S (2 n ) , S A S T = A { − √ det Γ det Γ S det((Γ + Γ S ) / } . N GF ( ρ AB ) = sup S ∈S (2 n ) , S A S T = A { − √ det Γ det Γ S det((Γ + Γ S ) / } . (7)Obviously, N GF ( ρ AB ) is independent of the mean d .It is easy to verify that det Γ = det Γ S . Since Γ = A CC T B > , by Lemma 4, we have < det Γ = det A det( B − C T A − C ) = det Γ S , which implies that det( B − C T A − C ) > . In addition, as Γ+Γ S = A C + S C C T + C T S T B and A , B are positive-definite, by Fischer’s inequality ([27, pp.506]), we have det Γ+Γ S ≤ det A det B . Hence, byEq.(7), we get ≤ N GF ( ρ AB ) ≤ − det A det( B − C T A − C )det A det B = 1 − det( B − C T A − C )det B < . We claim that the upper bound is tight, that is, we have sup ρ AB N GF ( ρ AB ) = 1 . (8)To see this, consider a two-mode squeezed vacuum state ρ ( r ) = S ( r ) | ih | S † ( r ) , where S ( r ) =exp( − r ˆ a ˆ a + r ˆ a † ˆ a † ) is a two-mode squeezing operator with squeezed number r ≥ and | i is thevacuum state ([37]). The CM of ρ ( r ) is A r C r C Tr B r , where A r = B r = exp( − r ) + exp(2 r ) 00 exp( − r ) + exp(2 r ) and C r = C Tr = − exp( − r ) + exp(2 r ) 00 exp( − r ) − exp(2 r ) . By Theorem 4, it is easily checked that N GF ( ρ ( r )) = 1 − exp( − r )+exp(4 r )2 + 3) . Clearly, N GF ( ρ ( r )) → as r → ∞ . So sup r N GF ( ρ ( r )) = 1 and Eq.(8) is true. (cid:3) Suppose that ρ AB is an ( n + m ) -mode Gaussian state with CM Γ = A CC T B as in Eq.(1). One canalways perform a local Gaussian unitary operation on the state ρ AB , say σ AB = ( U S A ⊗ V S B ) ρ AB ( U † S A ⊗2
Let ρ AB be any ( n + m ) -mode Gaussian state with CM Γ = A CC T B and mean d =( d A , d B ) . Note that, by Lemma 1, the CM of ρ A is A . Write σ AB = ( U S , m ⊗ I ) ρ AB ( U † S , m ⊗ I ) , where U S , m is any Gaussian unitary operator of the subsystem A . Clearly, U S , m ρ A U † S , m = ρ A if and only if thesymplectic matrix S satisfies S A S T = A and the vector m = d A − Sd A . In this case σ AB has the CM Γ S = A S CC T S T B and the mean d S = ( S ⊕ I )( d A ⊕ d B ) + m ⊕ Sd A + m ) ⊕ d B = d A ⊕ d B = ( d A , d B ) = d .Denote by S (2 n ) = Sp(2 n, R ) , the set of all n × n symplectic matrices. Then, by Eq.(6), N GF ( ρ AB ) = sup U ∈U G ( H A ) , Uρ A U † = ρ A C ( ρ AB , ( U ⊗ I ) ρ AB ( U † ⊗ I ))= sup U ∈U G ( H A ) , Uρ A U † = ρ A { − ( tr ρ AB ( U ⊗ I ) ρ AB ( U † ⊗ I )) tr ( ρ AB ) tr (( U ⊗ I ) ρ AB ( U † ⊗ I )) } = sup S ∈S (2 n ) , S A S T = A { − S ) / √ det Γ 1 √ det Γ S } = sup S ∈S (2 n ) , S A S T = A { − √ det Γ det Γ S det((Γ + Γ S ) / } . N GF ( ρ AB ) = sup S ∈S (2 n ) , S A S T = A { − √ det Γ det Γ S det((Γ + Γ S ) / } . (7)Obviously, N GF ( ρ AB ) is independent of the mean d .It is easy to verify that det Γ = det Γ S . Since Γ = A CC T B > , by Lemma 4, we have < det Γ = det A det( B − C T A − C ) = det Γ S , which implies that det( B − C T A − C ) > . In addition, as Γ+Γ S = A C + S C C T + C T S T B and A , B are positive-definite, by Fischer’s inequality ([27, pp.506]), we have det Γ+Γ S ≤ det A det B . Hence, byEq.(7), we get ≤ N GF ( ρ AB ) ≤ − det A det( B − C T A − C )det A det B = 1 − det( B − C T A − C )det B < . We claim that the upper bound is tight, that is, we have sup ρ AB N GF ( ρ AB ) = 1 . (8)To see this, consider a two-mode squeezed vacuum state ρ ( r ) = S ( r ) | ih | S † ( r ) , where S ( r ) =exp( − r ˆ a ˆ a + r ˆ a † ˆ a † ) is a two-mode squeezing operator with squeezed number r ≥ and | i is thevacuum state ([37]). The CM of ρ ( r ) is A r C r C Tr B r , where A r = B r = exp( − r ) + exp(2 r ) 00 exp( − r ) + exp(2 r ) and C r = C Tr = − exp( − r ) + exp(2 r ) 00 exp( − r ) − exp(2 r ) . By Theorem 4, it is easily checked that N GF ( ρ ( r )) = 1 − exp( − r )+exp(4 r )2 + 3) . Clearly, N GF ( ρ ( r )) → as r → ∞ . So sup r N GF ( ρ ( r )) = 1 and Eq.(8) is true. (cid:3) Suppose that ρ AB is an ( n + m ) -mode Gaussian state with CM Γ = A CC T B as in Eq.(1). One canalways perform a local Gaussian unitary operation on the state ρ AB , say σ AB = ( U S A ⊗ V S B ) ρ AB ( U † S A ⊗2 V † S B ) , such that the corresponding CM of σ AB is of the form Γ ′ = ⊕ ni v i I C ′ C ′ T ⊕ mi s i I , where v i s and s i sare the symplectic roots of ρ A and ρ B respectively, C ′ = S A CS T B . By Theorem 2, N GF ( σ AB ) = N GF ( ρ AB ) .This gives an estimation of N GF ( ρ AB ) for ( n + m ) -mode Gaussian state ρ AB in terms of symplectic rootsof the CMs of the reduced states ρ A and ρ B : ≤ N GF ( ρ AB ) ≤ − det( ⊕ mi s i I − S B C T S T A ( ⊕ ni /v i I ) S A CS T B ) Q mi =1 s i < . NONLOCALITY CONNECTED TO GAUSSIAN CHANNELS
In this section we intend to investigate the fidelity based nonlocality connected to a Gaussian quantumchannel. Here we mainly consider the (1 + 1) -mode Gaussian states whose CM are of the standard form.Since a Gaussian state ρ is described by its CM Γ and displacement vector d , we can denote it as ρ = ρ (Γ , d ) . Recall that a Gaussian channel is a quantum channel that transforms Gaussian states intoGaussian states. Assume that Φ is a Gaussian channel of n -mode Gaussian systems. Then, there exist realmatrices M, K ∈ M n ( R ) satisfying M = M T ≥ and det M ≥ (detK − , and a vector d ∈ R n , suchthat, for any n -mode Gaussian state ρ = ρ (Γ , d ) , we have Φ( ρ (Γ , d )) = ρ (Γ ′ , d ′ ) with d ′ = K d + d and Γ ′ = K Γ K T + M. So we can parameterize the Gaussian channel Φ as Φ = Φ(
K, M, d ) . Theorem 6.
Consider the (1+1) -mode continuous-variable system AB. Let
Φ = Φ(
K, M, d ) be a Gaussianchannel performed on the subsystem B with K = k k k k and M = m m m m . Assume that ρ AB ∈ S ( H A ⊗ H B ) is any (1 + 1) -mode Gaussian state with CM Γ = a c a dc b d b . Then N GF (( I ⊗ Φ) ρ AB ) = 1 − ( ab − c )( ab − d ) n + a ( ab − c ) n + a ( ab − d ) n + a n ( ab − c / ab − d / n + a ( ab − c / n + a ( ab − d / n + a n , where n = (1 + cos θ ) / , n = k k + k k − k k k k , n = m k + m k − m k k , n = m k + m k − m k k and n = m m − m . Proof.
Suppose that the (1 + 1) -mode Gaussian state ρ AB has CM Γ = a c a dc b d b and the mean ( d A , d B ) . Then the CM Γ ′ and the mean d ′ of σ AB = ( I ⊗ Φ) ρ AB are respectively Γ ′ = I K A C C T B I K T + M = A C K T KC T KB K T + M and d ′ = ( I ⊕ K )( d A ⊕ d B ) + 0 ⊕ d = d A ⊕ ( K d B + d ) . After a local invariant Gaussian unitary operation on the subsystem A, one has ( U ⊗ I ) σ AB ( U † ⊗ I ) = σ ′ AB . Remind that U ρ A U † = ρ A , which forces that, at the symplectic transformation level, U = U S , m with m = 0 and S = S θ = cos θ sin θ − sin θ cos θ for some θ ∈ [0 , π ] . Hence the CM and the mean of σ ′ AB arerespectively Γ ′ S = S I A C K T KC T KB K T + M S T I = A S C K T KC T S T KB K T + M and d ′ S = ( S ⊕ I )( d A ⊕ ( K d B + d )) + m ⊕ Sd A + m ) ⊕ ( K d B + d ) = d A ⊕ ( K d B + d ) . After some straight-forward calculations, one can immediately get N GF (( I ⊗ Φ) ρ AB ) = N GF ( σ AB )= sup U ∈U G ( H A ) , Uσ A U † = σ A C ( σ AB , ( U ⊗ I ) σ AB ( U † ⊗ I ))= sup θ ∈ [0 , π ] { − q det Γ ′ det Γ ′ S θ det((Γ ′ + Γ ′ S θ ) / } . By the fact that det Γ ′ = det Γ ′ S = det A det( KB K T + M − KC T A − C K T ) , the above formula can4rewritten as the following N GF (( I ⊗ Φ) ρ AB ) = sup θ ∈ [0 , π ] { − det A C K T KC T KB K T + M det A I + S θ ) C K T KC T ( I + S Tθ )2 KB K T + M } = sup θ ∈ [0 , π ] { − det A det( KB K T + M − KC T A − C K T )det A det( KB K T + M − KC T ( I + S Tθ )2 A −
10 ( I + S θ ) C K T ) } = sup θ ∈ [0 , π ] { − det( K ( B − C T A − C ) K T + M )det( K ( B − C T ( I + S Tθ )2 A −
10 ( I + S θ ) C ) K T + M ) } . Clearly, the quantity N GF (( I ⊗ Φ) ρ AB ) is independent of the parameter d . Notice that K , M can not be zerosimultaneously, substituting S θ = cos θ sin θ − sin θ cos θ into the above equation, after tedious calculations,one has N GF (( I ⊗ Φ) ρ AB )= sup θ ∈ [0 , π ] { − det( K ( B − C T A − C ) K T + M )det( K ( B − C T ( I + S Tθ )2 A −
10 ( I + S θ ) C ) K T + M ) } = sup θ ∈ [0 , π ] { − ( ab − c )( ab − d ) n + a ( ab − c ) n + a ( ab − d ) n + a n ( ab − c n )( ab − d n ) n + a ( ab − c n ) n + a ( ab − d n ) n + a n } = 1 − ( ab − c )( ab − d ) n + a ( ab − c ) n + a ( ab − d ) n + a n ( ab − c / ab − d / n + a ( ab − c / n + a ( ab − d / n + a n , where n =(1 + cos θ ) / ,n = k k + k k − k k k k , n = m k + m k − m k k ,n = m k + m k − m k k , n = m m − m . The proof is completed. (cid:3)
Remark 2. If K = 0 , then det M ≥ , and we have N GF (( I ⊗ Φ(0 , M, d )) ρ AB ) = { − det M det M } = 0 . In fact, in this case, the Gaussian channel I ⊗ Φ(0 , M, d ) maps any Gaussian state ρ AB to a product state.Thus, by Theorem 3, we always have N GF (( I ⊗ Φ(0 , M, d )) ρ AB ) = 0 .5 Remark 3. If M = 0 , then det K = 1 = det K T , and N GF (( I ⊗ Φ( K, , d )) ρ AB ) = sup θ ∈ [0 , π ] { − det( K ( B − C T A − C ) K T )det( K ( B − C T ( I + S Tθ )2 A −
10 ( I + S θ ) C ) K T ) } = sup θ ∈ [0 , π ] { − det( B − C T A − C )det( B − C T ( I + S Tθ )2 A −
10 ( I + S θ ) C ) } = sup θ ∈ [0 , π ] { − det A C C T B det A I + S θ ) C C T ( I + S Tθ )2 B } = N GF ( ρ AB ) . In this case, one can conclude that, after performing the Gaussian operation I ⊗ Φ( K, , d ) , the quantity N GF remains the same for those (1 + 1) -mode Gaussian states whose CM are of the standard form.The following result gives a kind of local Gaussian operation non-increasing property of N GF , which wasnot discussed for other known similar correlations such as the Gaussian QD (two-mode) [8, 9], Gaussiangeometric discord [10], the correlations Q , Q P discussed in [13] and the correlation N discussed in [16]. Theorem 7.
Let ρ AB be a (1 + 1) -mode Gaussian state. Then, for any Gaussian channel Φ performed onthe subsystem B , we have ≤ N GF (( I ⊗ Φ) ρ AB ) ≤ N GF ( ρ AB ) . Proof.
We first consider the case that the (1 + 1) -mode Gaussian states ρ AB whose CM Γ are of thestandard form, that is, Γ = a c a dc b d b . Let
Φ = Φ(
K, M, d ) be any Gaussian channel performed onsubsystem B with K = k k k k and M = m m m m . We have to show that N GF (( I ⊗ Φ) ρ AB ) ≤ N GF ( ρ AB ) .If N GF ( ρ AB ) = 0 , then, by Theorem 3, ρ AB is a product state. So ( I ⊗ Φ) ρ AB is a product state, andhence N GF (( I ⊗ Φ) ρ AB ) = 0 = N GF ( ρ AB ) .Assume that N GF ( ρ AB ) = 0 . Then N GF (( I ⊗ Φ) ρ AB ) ≤ N GF ( ρ AB ) holds if and only if N GF (( I ⊗ Φ) ρ AB ) N GF ( ρ AB ) ≤ . Let α = ( ab − c )( ab − d ) , β = ( ab − c / ab − d / , γ = a ( ab − c ) n + a ( ab − d ) n + a n and δ = a ( ab − c / n + a ( ab − d / n + a n with n , n , n as in Theorem 6. Then, according to6Theorem 6, we have N GF (( I ⊗ Φ) ρ AB ) N GF ( ρ AB ) ≤ ⇔ − αn + γβn + δ − αβ ≤ ⇔ αn + γβn + δ ≥ αβ ⇔ γβ ≥ αδ. Therefore, it suffices to prove that γβ − αδ ≥ . By some computations, one sees that γβ = [ a ( ab − c ) n + a ( ab − d ) n + a n ]( ab − c ab − d a ( ab − c )( ab − c ab − d n + a ( ab − d )( ab − c ab − d n + a ( ab − c ab − d n and αδ = a ( ab − c )( ab − c ab − d ) n + a ( ab − d )( ab − c )( ab − d n + a ( ab − c )( ab − d ) n . Note that n = k k + k k − k k k k = ( k k − k k ) ≥ and n = m m − m =det M ≥ . Since m k + m k ≥ √ m √ m k k ≥ m k k , we have n ≥ . One canverify n ≥ by the same way. Also note that a, b ≥ and ab ≥ c ( d ) by the constraint condition of theparameters in the definition of the Gaussian state. Now it is clear that γβ − αδ = a ( ab − c )( ab − c d n + a ( ab − d )( ab − d c n + a c d n ≥ , as desired. To this end, we come to the conclusion that N GF (( I ⊗ Φ) ρ AB ) ≤ N GF ( ρ AB ) , and the equalityholds if M = 0 (See Remark 3 after the proof of Theorem 6).Now let us consider the general case. Let U ⊗ V be a local Gaussian unitary operation, that is, forsome Gaussian unitary operators U and V on the subsystem A and B respectively, so that ( U ⊗ V )( ρ AB ) =( U ⊗ V ) ρ AB ( U † ⊗ V † ) for each state ρ AB . Then, ( I ⊗ Φ) ◦ ( U ⊗ V ) =
U ⊗ (Φ ◦ V ) = ( U ⊗ I ) ◦ ( I ⊗ (Φ ◦ V )) . Note that, Φ ◦ V is still a Gaussian channel which sends ρ B to Φ( V ρ B V † ) . Keep this in mind and let ρ AB be any (1 + 1) -mode Gaussian state. Then there exists a local Gaussian unitary operation U ⊗ V such that σ AB = ( U † ⊗ V † ) ρ AB ( U ⊗ V ) has CM of the standard form. By what we have proved above and Theorem2, we see that N GF (( I ⊗ Φ) ρ AB ) = N GF (( I ⊗ Φ)(( U ⊗ V ) σ AB ( U † ⊗ V † )))= N GF (( I ⊗ Φ) ◦ ( U ⊗ V ) σ AB ) = N GF (( U ⊗ I ) ◦ ( I ⊗ (Φ ◦ V )) σ AB )= N GF (( I ⊗ (Φ ◦ V )) σ AB ) ≤ N GF ( σ AB ) = N GF ( ρ AB ) , as desired, which completes the proof. (cid:3) COMPARISON BETWEEN N GF AND OTHER QUANTIFICATIONS OF THE GAUSSIAN QUANTUMCORRELATIONS N GF , D G and Q describe the same quantum nonclassicality when they are restricted to Gaussian statesbecause they take value 0 at a Gaussian state ρ AB if and only if ρ AB is a product state. In this section, wecalculate N GF ( ρ AB ) for all two-mode symmetric squeezed thermal states ρ AB and compare it with Gaussiangeometric discord D G ( ρ AB ) and Q ( ρ AB ) in scale. Our result reveals that N GF is bigger and thus is easier todetect the correlation in states. Since the known computation formula of D G ( ρ AB ) is only for symmetricsqueezed thermal states ρ AB , we compare them on such states. Symmetric squeezed thermal states:
Assume that ρ AB is any two-mode Gaussian state; then its standardCM has the form as in Eq.(3). Recall that the symmetric squeezed thermal states (SSTSs) are Gaussian stateswhose CMs are parameterized by ¯ n and µ such that a = b = 1 + 2¯ n and c = − d = 2 µ p ¯ n (1 + ¯ n ) , where ¯ n is the mean photon number for each part and µ is the mixing parameter with ≤ µ ≤ (ref. [38]). Thusevery SSTS may be denoted by ρ AB (¯ n, µ ) .Thus by Theorem 4, for any SSTS ρ AB (¯ n, µ ) , we have N GF ( ρ AB (¯ n, µ )) = 1 − ((1 + 2¯ n ) − µ ¯ n (1 + ¯ n )) ((1 + 2¯ n ) − µ ¯ n (1 + ¯ n )) . (9)For any two-mode Gaussian state ρ AB , recall that the Gaussian geometric discord of ρ AB ([10]) isdefined as D G ( ρ AB ) = inf Π A k ρ AB − Π A ( ρ AB ) k , where Π A = Π A ( α ) runs over all Gaussian positive operator valued measurements of subsystem A, Π A ( ρ AB ) = R (Π A ( α ) ⊗ I ) ρ AB (Π A ( α ) ⊗ I ) d α . According to the analytical formula of D G ( ρ AB ) provided in [10], for any SSTS ρ AB with parameters ¯ n and µ , one has D G ( ρ AB (¯ n, µ )) = 1(1 + 2¯ n ) − µ ¯ n (1 + ¯ n ) − p n ) − µ ¯ n (1 + ¯ n ) + (1 + 2¯ n )] . (10)By Eqs.(9)-(10), it is clear that lim ¯ n →∞ N GF ( ρ AB (¯ n, µ )) = 1 − (1 − µ ) (1 − µ ) > µ ∈ (0 , , while lim ¯ n →∞ D G ( ρ AB (¯ n, µ )) = 0 for µ ∈ (0 , . This shows that, for the case µ = 0 , , N GF is able to recognize well the quantum correlation in the stateswith large mean photon number but D G is not. It is clear that µ = 0 if and only if ρ AB is a product SSTS,8and in this case, N GF ( ρ AB (¯ n, D G ( ρ AB (¯ n, . When µ = 1 , we have N GF ( ρ AB (¯ n, − n + 2¯ n ) and D G ( ρ AB (¯ n, − n + 2 √ n + ¯ n ] , which reveals that we always have N GF ( ρ AB (¯ n, > D G ( ρ AB (¯ n, . Moreover, we randomly chose 100000 pairs of (¯ n, µ ) with ¯ n ∈ (0 , and µ ∈ (0 , ,numerical results show that N GF ( ρ AB (¯ n, µ )) > D G ( ρ AB (¯ n, µ )) . On the other hand, the numerical methodsuggests that N GF is better than D G in detecting the quantum correlation contained in any SSTS because wealways have N GF ( ρ AB (¯ n, µ )) > D G ( ρ AB (¯ n, µ )) for all SSTSs ρ AB (¯ n, µ ) with µ = 0 .In Fig.1, we compare N GF ( ρ AB ) with D G ( ρ AB ) for SSTSs ρ AB by considering N GF ( ρ AB ) − D G ( ρ AB ) for ¯ n ≤ . Fig.1 shows that N GF ( ρ AB ) − D G ( ρ AB ) ≥ and N GF ( ρ AB ) ≫ D G ( ρ AB ) for SSTSs ρ AB with µ near 1. For example, considering the state ρ AB with ¯ n = 49 and µ = 0 . , we have D G ( ρ AB ) ≈ . , which is very close to 0 and difficult to judge weather or not ρ AB contains thecorrelation. However, N GF ( ρ AB ) ≈ . ≫ , which guarantees that ρ AB does contain the quantumcorrelation. For large mean photon number, for example, ¯ n = 10000 , taking µ = 0 . , we have N GF ( ρ AB ) ≈ . ≫ , but D G ( ρ AB ) ≈ . × − . Furthermore, Fig.2 shows that N GF ( ρ AB ) − D G ( ρ AB ) ≥ holds as well for ¯ n ∈ (100000 , and µ ∈ (0 , . Q is a quantum correlation for ( m + n ) -mode continuous-variable systems defined in terms of averagedistance between the reduced states under the local Gaussian positive operator valued measurements [13]: Q ( ρ AB ) := sup Π A Z p ( α ) k ρ B − ρ ( α ) B k d m α, where Π A = Π A ( α ) runs over all Gaussian positive operator valued measurements of subsystem A, Π A = { Π A ( α ) } on the subsystem H A , ρ B = Tr A ( ρ AB ) , p ( α ) = Tr[(Π A ( α ) ⊗ I B ) ρ AB ] and ρ ( α ) B = p ( α ) Tr A [(Π A ( α ) ⊗ I B ) ρ AB (Π A ( α ) ⊗ I B ) ] .9FIG. 1: z= N GF ( ρ AB ) − D G ( ρ AB ) with SSTSs, and ≤ µ ≤ , ≤ ¯ n ≤ .FIG. 2: z= N GF ( ρ AB ) − D G ( ρ AB ) with SSTSs, and ≤ µ ≤ , ≤ ¯ n ≤ .For any SSTS ρ AB with parameters ¯ n and µ , by [13], Q ( ρ AB (¯ n, µ )) = 11 + 2¯ n (1 − µ ) −
11 + 2¯ n . (11)Obviously, lim ¯ n →∞ Q ( ρ AB (¯ n, µ )) = 0 , for µ ∈ (0 , . which reveals that Q is not valid for those states with µ ∈ (0 , and large mean photon number. For thecase µ = 1 , we have Q ( ρ AB (¯ n, −
11 + 2¯ n < N GF ( ρ AB (¯ n, for any ¯ n . Also, we always have N GF ( ρ AB ) > Q ( ρ AB ) for all SSTSs with µ = 0 . For random pairs (¯ n, µ ) with ¯ n ∈ (0 , and µ ∈ (0 , , 100000numerical results illustrate that N GF ( ρ AB (¯ n, µ )) > Q ( ρ AB (¯ n, µ )) .0FIG. 3: z= N GF ( ρ AB ) − Q ( ρ AB ) with SSTSs, and ≤ µ ≤ , ≤ ¯ n ≤ .FIG. 4: z= N GF ( ρ AB ) − D G ( ρ AB ) with SSTSs, and ≤ µ ≤ , ≤ ¯ n ≤ .The deference of N GF ( ρ AB ) and Q ( ρ AB ) for SSTSs is showed in Fig.3 for ¯ n ≤ . It reveals that N GF ( ρ AB ) ≫ Q ( ρ AB ) for those SSTSs ρ AB with large mean photon number ¯ n and larger mixing parameter µ . Consider the states ρ AB with respectively (¯ n, µ ) = (49 , . and (¯ n, µ ) = (10000 , . , the sameexamples as above. We have respectively Q ( ρ AB ) ≈ . < N GF ( ρ AB ) ≈ . and Q ( ρ AB ) ≈ . ≪ N GF ( ρ AB ) ≈ . , which means that applying N GF is much more easier than Q to guaranteethat ρ AB contains the quantum correlation. Fig.4 demonstrates that N GF ( ρ AB ) − Q ( ρ AB ) ≥ also holdsfor these ¯ n ∈ (100000 , and µ ∈ (0 , . CONCLUSION
In this paper, based on fidelity F ( ρ, σ ) = | tr ρσ | √ tr ρ tr σ and the distance C ( ρ, σ ) = 1 − F ( ρ, σ ) , wehave proposed a new kind of quantum nonclassicality N GF by local Guassian unitary operations for anystates in ( n + m ) -mode continuous-variable systems. Though, when restricted to the Gaussian states, N GF describes the same nonclassical correlation as several known correlations such as Gaussian QD, Gaussiangeometric discord D G and the nonlocality Q , it is comparatively much easier to be computed and estimated.Furthermore, N GF has several nice properties that other known quantifications of such correlation do not1possess: N GF is a quantum correlation without ancilla problem; N GF (( I ⊗ Φ) ρ AB ) ≤ N GF ( ρ AB ) holds forany (1 + 1) -mode Gaussian state ρ AB and any Gaussian channel Φ , that is, undergoing a local Gaussianchannel performed on the unmeasured part, the quantity we proposed will not increase. We guess thatthis nice property is still valid for ( n + m ) -mode systems. We give a computation formula of N GF for any (1 + 1) -mode Gaussian states and an upper bound for any ( n + m ) -mode Gaussian states, which are simpleand easily calculated. Furthermore, by comparing N GF ( ρ AB ) with D G ( ρ AB ) and Q ( ρ AB ) for two-modesymmetric squeezed thermal states, we find that N GF is greater than D G and Q , and so is better in detectingquantum correlation in Gaussian states. Acknowledgement.
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