Classifying directional Gaussian quantum entanglement, EPR steering and discord
CClassifying directional Gaussian quantum entanglement, EPR steering and discord
Q. Y. He , ∗ , Q. H. Gong and M. D. Reid † State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University,Collaborative Innovation Center of Quantum Matter, Beijing, China Centre for Quantum Atom Optics, Swinburne University of Technology, Melbourne, Australia and ∗ qiongyihe @ pku.edu.cn , † mdreid @ swin.edu.au Quantum discord quantifies how much Alice’s system is disrupted after a measurement is per-formed on Bob’s. Conceptually, this behavior acts the same way as quantum steering and we findthat the discord grows with better steering from Bob to Alice. Using Venn diagrams, the relationsbetween the different classes of Gaussian continuous variable entanglement and the links to discordfor the squeezed-thermal states are established. We identify a directional quantum teleportationtask for each class of squeezed-thermal state entanglement, and establish a unified signature forquantum steering, entanglement and discord beyond entanglement. Quantum steering and discordare promising candidates to quantify the potential of the directional quantum tasks where Alice andBob possess asymmetrically noisy channels.
The topic of quantum correlations has received muchattention in modern physics [1, 2]. Entanglement is a dis-tinctive feature of quantum correlations [3] and it is con-sidered that all entangled states are useful for quantuminformation processing (QIP) [4]. Einstein-Podolsky-Rosen (EPR) correlations enable error-free predictionsfor the position − and the momentum − of one particlegiven some type of measurement on another. EPR cor-relations are especially useful [5]. As one example, thefidelity of the quantum teleportation (QT) of a coherentstate is directly related to the strength of EPR correla-tion available in the quantum resource [6].Very recently, there has been an appreciation of the im-portance of asymmetry and direction in quantum correla-tions [7–11]. Entanglement is a property shared betweentwo parties, and measures of it have not been sensitiveto differences between the quantum parties involved [12].Yet, the original EPR argument was expressed asym-metrically between the two systems. The analysis bySchrodinger introduced the asymmetric term “steering”to describe the EPR idea of one party apparently adjust-ing the state of another by way of local measurements[13]. This aspect has been beautifully captured in tworecent alternative definitions for quantum correlations: quantum discord [7, 8] and EPR steering [9, 10]. Be-sides being of intrinsic fundamental interest, these asym-metrical nonlocalities are attracting a great deal of at-tention [14–17] for special tasks in QIP e.g. cloning ofcorrelations [18], quantum metrology [19], quantum statemerging [20], remote state preparation [21] and one-sideddevice-independent quantum key distribution [22]. Sur-prisingly, for mixed states, quantum discord can emergewithout entanglement and recent experiments [23] haveused discord to distribute entanglement using separa-ble states only [24]. Despite the potential value of di-rectional quantum correlation, relatively little is knownabout the quantitative link between discord and steer-ing, and methodologies to characterise quantum statesfor their asymmetrical correlation. Our aim in this Letter is to provide such a characterisa-tion and to explain the link between discord and steeringfor the purpose of continuous variable (CV) QIP. We fo-cus on the subclass of bipartite quantum systems calledGaussian states [9, 26] which have enabled experimentalmilestones such as deterministic QT [27]. Asymmetri-cal Gaussian entanglement and its application to QIP isnot fully understood. To illustrate, it is often interpretedthat CV quantum teleportation (QT) requires a resourcewith the “Duan” [31] symmetric form of entanglement,for which the measures of EPR steering and discord arelargely unaltered if the roles of the two parties are ex-changed [6, 28, 32].Here, we address this gap in knowledge by introduc-ing a classification of the space of Gaussian states intodistinct sets of directional entanglement classes. We es-tablish the strict relations between the classes and thelinks to quantum discord, for the experimentally relevantsubclass of squeezed-thermal (STS) states. Moreover, werelate each of these classes to a special directional QTtask, showing that the whole subclass of STS Gaussianentangled states including those with asymmetric quan-tum correlation can be used for QT. By introducing anEPR steering parameter, we establish an experimentalsignature to distinguish the states of different classes,whether EPR steering, entanglement, or discord beyondentanglement. Finally, we show how one can manipulatethe two-mode squeezed EPR state to cross between thedifferent classes of quantum correlation, by adding asym-metric amounts of thermal noise to each sub-system.Our method connects three Gaussian measures ofquantum correlation: Simon’s positive partial transpose(PPT) condition for entanglement [29], the criterion ofRef. [30] for
EPR steering , and the measure of Giordaand Paris for discord [26]. We explain how the PPTcondition is equivalent to a condition on an EPR-typevariance. The condition works efficiently for all Gaus-sian states due to the introduction of a gain factor g sym which we show gives information about the symmetry of a r X i v : . [ qu a n t - ph ] J un D B | A ≥ D A | B ≥ II III E B | A < E A | B < PPT I VI IV V ∅∅ Δ ent < < ent Δ Figure 1. (Color online) The Venn diagram relations classify-ing the different types of quantum correlation for the subclassof Gaussian states. The larger blue circle II contains statessatisfying the Duan criterion for entanglement ∆ ent < . Theinner blue circle I contains states with the symmetric EPRsteering correlation given by ∆ ent < . . The set of all entan-gled states quantified by the PPT criterion Ent
PPT < arecontained in the larger green ellipse V I . The smaller orange IV and yellow V ellipses enclose states that display one-waysteering E A | B < and E B | A < , respectively. Their inter-section (colored yellow) is the set of two-way steerable states,which is a strict superset of the states in I . All two-way steer-able states are a subset of the entangled states quantified bythe Duan condition ∆ ent < . One-way steering states are astrict subset of the PPT entangled states, and are strictly notcontained in the Duan circle ∆ ent < . The outer ellipse IIIcontains the set of Gaussian states with non-zero quantum A and B discord. All Gaussian states except product states arecontained in III , which is a strict superset of all Gaussianentangled states [26]. the quantum correlation, and how the resource can beutilised for QT. Entanglement can be quantified by thesteering measure for each party, and by g sym . Interest-ingly, we find that “quantum A(B) discord” grows withbetter steering from Bob (Alice) to Alice (Bob). We willsee that the steering from B to A and quantum A ( B ) dis-cord are asymmetrically sensitive to the thermal noise onthe two systems. In fact, steering can be created “one-way” using thermal manipulation. We then show thatwhile resources with symmetic quantum correlation areuseful for QT via traditional protocols, those with asym-metric correlation require asymmetric protocols.All Gaussian properties can be determined from thesymplectic form of the covariance matrix (CM) definedas C ij = (cid:104) ( X i X j + X j X i ) (cid:105) / − (cid:104) X i (cid:105)(cid:104) X j (cid:105) where X ≡ ( X A , P A , X B , P B ) is the vector of the field quadratures: C = n c n c c m c m (1)The symplectic invariants are defined by I = n , I = m , I = c c , I ≡ det ( C ) = ( nm − c )( nm − c ) , and the symplectic eigenvalues d ± = (cid:114)(cid:16) ∆ ± (cid:112) ∆ − det ( C ) (cid:17) / with ∆ = I + I + 2 I [26, 32]. Our classification will be exemplified by theGaussian two-mode squeezed thermal state (STS) forwhich c = − c = c . We thus follow [26] and focuson this subclass of Gaussian states for the remainder ofthe paper. The covariance matrix elements in the STScase are n = (2 n A + 1) cosh ( r ) + (2 n B + 1) sinh ( r ) , m = (2 n B + 1) cosh ( r ) + (2 n A + 1) sinh ( r ) , c = ( n A + n B + 1) sinh (2 r ) , where n A , n B are the average num-ber of thermal photons for each system and r denotesthe squeezing parameter. Here, we normalise the vac-uum fluctuations so that ∆ X ∆ P ≥ . We can specifySimon’s PPT criterion for entanglement as [32] Ent
P P T = ( nm − c ) + 1 − (cid:0) n + m + 2 c (cid:1) < , (2)which becomes a necessary and sufficient condition forGaussian states [29]. According to the PPT criterion(2), a two-mode STS is entangled iff r exceeds the fol-lowing threshold value: cosh ( r ent ) = ( n A +1)( n B +1) n A + n B +1 [15].The complete set of PPT entangled states is depicted ascontained within the green ellipse of Fig. 1. This set isnot exhaustive for Gaussian states as seen by the valuesfor Ent
P P T versus the thermal noises n A and n B shownin Fig. 2a [26].Entanglement can also be determined using an EPR-type correlation [31, 35]. On defining the weighted dif-ference variance ∆ ( X A − gX B ) = n − gc + g m =∆ ( P A + gP B ) , entanglement between modes A and B isconfirmed when Ent A | Bg =∆ ( X A − gX B ) / (1 + g ) < . (3)Here g is an arbitrary real constant but which can beoptimally chosen to minimise the value of Ent A | Bg . Forthe restricted subclass of Gaussian EPR resources, thereis symmetry between the X and P moments so that asingle g suffices. With the optimal choice of g = g A | Bsym ≡ (cid:16) n − m + (cid:112) ( n − m ) + 4 c (cid:17) / c , it is straightforward toshow that the entanglement bounds of Ent g < and Ent
P P T < ( ˜ d − = 1 , obtained by replacing I → − I in the formula for d − ) are equivalent. Note that theentanglement between modes A and B can be also con-firmed when Ent B | Ag (cid:48) = ∆ ( X B − g (cid:48) X A ) / (1 + g (cid:48) ) < , which is the same threshold as for Ent A | Bg but with g (cid:48) = g B | Asym ≡ (cid:16) m − n + (cid:112) ( m − n ) + 4 c (cid:17) / c = 1 /g A | Bsym .This is to be expected: Entanglement is by definition aquantity shared between two systems, and its PPT mea-sure does not account for the directional properties asso-ciated with quantum correlation.Where one has complete symmetry between the sys-tems, n = m and g A | Bsym = 1 . The PPT criterion (3) forentanglement then reduces to the measure of “Duan en-tanglement” [31, 32] ∆ ent = (cid:2) ∆ ( X A − X B ) + ∆ ( P A + P B ) (cid:3) / < . (4)Resources with the property (4) are required for theCV quantum teleportation (QT) of a coherent state, asachieved using the standard protocol of Braunstein andKimble [6]. The STS squeezing threshold for Duan en-tanglement is r > r QT,duan = ln √ n A + n B + 1 . Thesestates are depicted as enclosed within the dark blue cir-cle II of Fig. 1. Sufficiently asymmetric systems (where n (cid:29) m ) may arise for example when coupling massiveobjects to laser pulses, and may require the full PPT en-tanglement test (outside the blue circle II , but withinthe green ellipse) as illustrated in Fig. 1 [36].Quantum discord is by definition a measure of asym-metric quantum correlation between the two subsystems[7]. The “quantum A discord” that considers the condi-tional information for Alice’s system A based on mea-surements on system B by Bob, has been derived for aGaussian state by Giorda and Paris as [26] D A | B = f ( m ) − f ( d + ) − f ( d − ) + f ( z ) , (5)where z = n + mn − c m +1 and f ( x ) = ( x +12 ) ln ( x +12 ) − ( x − ) ln ( x − ) . With the exchanging m ↔ n and hence I ↔ I , we obtain the result for the B discord D B | A .Quantum A discord is obtained for all bipartite Gaus-sian states that are not product states, although there arenon-entangled states that have nonzero discord [26]. Thequantum discord is the difference between two classically-equivalent definitions of conditional entropy [7, 8, 26].Denoting the von Neumann entropy of the quantum state ρ by S ( ρ ) , the first S ( ρ A | B ) ∼ f ( d + )+ f ( d − ) − f ( m ) arisesfrom using the definition of mutual information based onthe bipartite state ρ AB . The second arises from quan-tisation of the expressions for the conditional entropy: H ( ρ A | B ) = (cid:80) k p B ( k ) S ( ρ A | k ) ∼ f ( √ z ) where p B ( k ) isthe probability of result k for a measurement at B , and S ( ρ A | k ) = (cid:80) i p ( i | k ) S ( ρ i | k ) where p ( i | k ) is the condi-tional probability of outcome i at A given the result k at B. The discord (5) is obtained by minimising the mis-match over all Gaussian measurements. The terms inthe quantum A discord H quantify the available infor-mation for the conditional state of A after measurementon B , and also reflect uncertainty in measurements ofAlice when Bob’s outcome k is known.Interestingly, this reminds us of the other asymmetricnonlocality, EPR steering from B to A [1, 9, 10], whichis realized for Gaussian systems iff [9, 30] E A | B = ∆ inf X A | B ∆ inf P A | B < (6)Here ∆ inf X A | B = (cid:80) k p B ( k )∆ ( X A | k ) where ∆ ( X A | k ) is the variance of the conditional distribution for Al-ice’s “position” X A conditional on the result k . Themeasurement at B is selected to minimise the quantity ∆ inf X A | B . The ∆ inf P A | B = (cid:80) k (cid:48) p B ( k (cid:48) )∆ ( P A | k (cid:48) ) is de-fined similarly, for the momentum P A . The states withthe property (6) are depicted by the small orange ellipseof Fig. 1. For Gaussian states, we can write ∆ inf X A | B = (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) a (cid:76) Ent
PPT (cid:62) n A n B H b L D A B > n A n B E A B (cid:61) E A B (cid:61)
PPT (cid:61)
PPT (cid:61) (cid:72) c (cid:76) E A B (cid:61) m (cid:43) n (cid:45) m g (cid:61) n A n B H b - L H H Ρ A B L n A n B un (cid:45) physcial Ent
PPT (cid:68) ent E A B
IV III D (cid:62) (cid:72) d (cid:76) II VI
UP VI m H b - L H Ρ A B L n A n B Figure 2. (Color online) Contour plots show the effect ofasymmetric noises n A and n B on quantum correlation, for thetwo-mode STS with r = 0 . : (a) entanglement measured by Ent
PPT , (b) discord measured by D A | B , and (c) the steeringparameter E A | B . In (c), the states can be used as quantumresources with EPR steering (below red curve), entanglement(below green curve), or discord beyond entanglement (abovegreen curve) as explained in text. (d) shows the different re-gions certified by criteria of steering, entanglement, discord,and unphysical CMs (light gray area UP ) versus n and m .The terms contributing to discord, S ( ρ A | B ) and H ( ρ A | B ) , areshown in (b-1) and (b-2), and are discussed in the Supple-mental Materials. ∆ ( X A − gX B ) and ∆ inf P A | B = ∆ ( P A + gP B ) where g isa real constant [30, 33], noting that for the restricted sub-class E A | B ( g ) = ∆ inf X A | B = ∆ inf P A | B = n + g m − gc .The optimal measurement is defined by the optimal g .The quantity E A | B ( g ) is minimized to E A | B = n − c /m by the optimal factor g = c/m [30, 32], and its small-ness gives a measure of the degree of nonlocal correla-tions. Ideally, it becomes zero in the limit of large r .As with discord, we obtain the result for the steeringfrom A to B by interchanging parameters: E B | A ( g (cid:48) ) = ∆ inf X B | A ∆ inf P B | A = m − c /n where g (cid:48) = c/n (smallyellow ellipse of Fig. 1).Two-way steering is confirmed when both E A | B ( g s ) < and E B | A ( g (cid:48) s ) < , given by the yellow intersectionof the two smaller ellipses of Fig. 1. For g s = 1 , wehave E A | B = E B | A = n + m − c = 2∆ ent and hence ∆ ent < . is a criterion sufficient to confirm two-wayEPR steering. This is also the Grosshans and Grang-ier condition required of an EPR resource for the se-cure teleportation (ST) of a coherent state [37]. In thatcase, a teleportation fidelity F = 1 / (1 + ∆ ent ) > / ,as opposed to F > / for QT, is needed. To sat-isfy E A | B ( g ) < or E B | A ( g ) < requires the squeez-ing r to exceed the threshold value given by r A | B and r B | A respectively, where cosh (cid:0) r A | B (cid:1) = (2 n A +1)( n B +1)1+ n B + n A or cosh (cid:0) r B | A (cid:1) = ( n A +1)(2 n B +1)1+ n B + n A . The two-mode STSwith r > { r A | B , r B | A } max can be used to produce two-way steering, which is only possible for sufficient sym-metry given by | n A − n B | < / . The states satisfyingthe strongly symmetric EPR correlation ∆ ent < . aredepicted by the centre light blue circle I of Fig. 1. Thisrequires the squeezing r to exceed the threshold value r > r ST,duan = ln (cid:112) n A + n B + 1) , and ∆ ent < . isnot therefore a necessary condition for two-way steering.Two-way steering is possible when { r A | B , r B | A } max
Alice X A + X V V B A P A − P V Bob g g n A n B g α g α α m x m p Figure 3. Scheme for quantum teleportation with directiondistinguished. Here we depict CV quantum teleportation ofVictor’s coherent state | α (cid:105) at V to an amplified coherent state | ¯ gα (cid:105) at Bob’s location, which can then be attenuated by factor g = 1 / ¯ g . The Bell measurement is defined at A as in Ref. [6].The scheme uses a resource with the directional entanglementspecified by g B | Asym = ¯ g ≥ . The resource is generated byadding asymmetric noise ( n B > n A ) to the output channelsof the EPR source. The maximum fidelity of the scheme is F g = 1 / ¯ g and is achieved when E B | A (¯ g ) = ¯ g − ; QT isachieved for F > . which is satisfied iff Ent
PPT < . the presence of asymmetric noises creates the possibilityof asymmetric steering/ discord, making steering/ dis-turbance from A to B more difficult than that from B to A . Entanglement is absent for Ent
P P T ≥ , the re-gion above the green curve in Fig. 2a. All regions show“quantum A discord”, given by D A | B > (Fig. 2b) [26].Thermal noises tend to suppress entanglement, for whichthe dependence on n A and n B is symmetric. However,the effect on the discord is more complex and asymmet-rical. We can see that D A | B is maximised when most ofthermal noise is placed on the unmeasured system A.Figure 2 (c) shows the behaviour of the steering param-eter E A | B . The sensitivity to the noises is asymmetricaland “one-way steering” (the states contained in the small-est left ellipse of Fig. 1 but exclusive of the right one)is evident. The value of E A | B is minimised (and steeringincreased) when most of thermal noise is placed on thesystem B , since E A | B < m + n − m ∼ when m (cid:29) n .The behavior of discord is strongly related to steering(Fig. 2). We note the similarity between the conditionalentropy H ( ρ A | B ) (Fig. 2(b-1)) and E A | B (Fig. 2c). Assteering increases (so that E A | B → ) the variances ofthe conditional distribution are reduced [30, 33]. We findthat for better steering of Alice by Bob, quantum discordbecomes larger (see Supplemental Materials [34]). Thisis consistent with the picture that more of the EPR-typedisturbances happen to Alice’s system because of Bob’smeasurements.Finally, we emphasize potential applications of asym-metric correlation. We show in the Supplementary Mate-rials [34] that the directional entangled states are usefulas a resource for the quantum teleportation of a coherentstate from Alice to Bob (if g A | Bsym ≤ ), or from Bob to Al-ice (if g A | Bsym ≥ ). This is achieved using the asymmetricprotocol of Fig. 3. We leave open the question of whetherthe asymmetric value of discord may also produce direc-tional quantum tasks only successful either from Alice toBob, or Bob to Alice.In conclusion, we have established classes of CV Gaus-sian quantum correlation, determined how to signify andgenerate states of a given class, and shown how the statesof each entanglement class can be utilised for a quantumteleportation task. We explored the relation between twoasymmetric nonclassical correlations, steering and dis-cord. Our results suggest asymmetric correlations suchas EPR steering and discord to be promising candidatesfor quantum tasks requiring a directional operation.We thank the Australian Research Council for fund-ing via Discovery and DECRA grants. Q. Y. H thankssupport from the National Natural Science Foundationof China under grants No.11121091 No.11274025. [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] J. S. Bell, Physics , 195 (1965).[3] E. 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