aa r X i v : . [ qu a n t - ph ] M a y Quantum discord between relatively accelerated observers
Animesh Datta
1, 2, ∗ Institute for Mathematical Sciences, 53 Prince’s Gate, Imperial College, London, SW7 2PG, UK QOLS, The Blackett Laboratory, Prince Consort Road, Imperial College, London, SW7 2BW, UK (Dated: October 25, 2018)We calculate the quantum discord between two free modes of a scalar field which start in a maximally en-tangled state and then undergo a relative, constant acceleration. In a regime where there is no distillable entan-glement due to the Unruh effect, we show that there is a finite amount of quantum discord, which is a measureof purely quantum correlations in a state, over and above quantum entanglement. Even in the limit of infiniteacceleration of the observer detecting one of the modes, we provide evidence for a non-zero amount of purelyquantum correlations, which might be exploited to gain non-trivial quantum advantages.
PACS numbers: 03.67.Mn, 04.62.+vKeywords: Quantum discord, Unruh effect, nonitertial observers
The theory of relativity and quantum theory, together withinformation theory may be said to form the cornerstones oftheoretical physics [1]. The last of these two are being, overthe last decade, amalgamated into the field of quantum infor-mation science that seeks to compute and process informationlimited by the laws of quantum mechanics [2]. The enterpriseof incorporating the principles of the theory of relativity intoquantum information is, in comparison, nascent. Nonethe-less, there have been several studies at the intersection of rel-ativity theory and quantum information science, particularlyin the study of Bell’s inequalities [3, 4, 5, 6, 7], quantumentropy [8, 9], quantum entanglement [10, 11, 12, 13, 14],teleportation [15], and beyond [16]. There have also beenstudies involving the entanglement in fermionic fields [17]and continuous-variable systems in nonintertial frames [18].These have shown that entanglement between some degreesof freedom can be transferred to others, and that the notion ofentanglement is observer dependant.In addition to the investigations into fundamental natureof quantum entanglement in a curved space-time, there havebeen several proposals for detecting relativistic effects in lab-oratory systems like cavity QED [19], ion traps [20] and atomdots in Bose-Einstein condensates [21]. These effects of de-tecting acceleration radiation is a consequence of the Unruheffect [22]. A result from quantum field theory, it states thatuniformly accelerated observers (that is, with constant properacceleration) in Minkowski space-time associate a thermalbath to the vacuum state of the inertial observers. For the iner-tial observer, the Minkowski coordinates ( T, Z ) are appropri-ate, while for a uniformly accelerating observer, Rindler coor-dinates ( τ, ξ ) are more apt. Minkowski space-time is invariantunder the boosts, and this motivates the hyperbolic coordinatetransformations T = 1 a e aξ sinh aτ, Z = 1 a e aξ cosh aτ, | Z | < T, (1) T = − a e aξ sinh aτ, Z = 1 a e aξ cosh aτ, | Z | > T. (2)These two transformations lead to two sets of Rindler coor-dinates, called the right and left Rindler wedges respectively, which together form a complete set of solutions of the Klein-Gordon equation in Minkowski space-time.The solutions of the Klein-Gordon equation in Minkowskispace-time are related to those in the Rindler wedges via aBogoluibov transformation [22]. These transform the vac-uum of the inertial observer into a two-mode squeezed statefor the accelerating observer, the two modes residing in thetwo Rindler wedges. If we probe only one of the wedges,as we are constrained to due to causality, the other mode istraced over, leaving us with a mixed state of free bosons at atemperature proportional to the acceleration. Additionally, ifone starts with a pure entangled state of two free modes of ascalar field shared between two observers, Alice and Bob, andone of them, say Bob accelerates, the result is a mixed state,whose entanglement, as measured by the logarithmic negativ-ity, is degraded from the point of view of Rob (acceleratingBob) [13], while there is no change from the point of view ofAlice.Our endeavor in this paper will be to explore the above phe-nomenon from the perspective of quantum discord [23, 24,25]. It is a measure of purely quantum correlations, and weshow that although the quantum discord suffers some degra-dation, there is a finite amount of quantum discord betweenAlice and Rob at accelerations at which the distillable en-tanglement has gone to zero. The use of quantum discordis firstly motivated by the fact that noninertial observers in-evitably encounter mixed states, for which there is a lack ofuniversally accepted, easily computable measures of entan-glement. Quantum discord is ideally suited for applicationto mixed states. Secondly, quantum discord is a measure ofpurely quantum correlations, over and above entanglement,although for pure states, they coincide. Finally, the quan-tum discord has been presented as a possible resource for cer-tain quantum advantages [26], and the presence of nonzeroamounts of quantum discord as perceived by the noninter-tial observer might allow him to achieve nontrivial quantumadvantage beyond a point where the distillable entanglementtouches zero. We will also show that a ‘symmetrized’ formof quantum discord, called the MID (measurement induceddisturbance) measure [27, 28], and defined as the differencebetween the entropy of a quantum state, and that obtained bymeasuring both the subsystems in their reduced eigenbases,has a finite value at accelerations at which the logarithmicnegativity is zero. This is comparatively easier to calculatethan the quantum discord, and is an upper bound on it. Whatboth these measures however show, is, that starting with aninitially entangled state shared between Alice and Bob, therewill persist quantum correlations between them when Bob ac-celerates, beyond accelerations at which the distillable entan-glement has fallen to zero.For concreteness, we start with the maximally entangledstate between Alice and Bob, of two Minkowski modes s and k | Ψ i M = 1 √ | s i M | k i M + | s i M | k i M ) . (3)When Bob accelerates with respect to Alice with a constantacceleration, the Minkowski vacuum can be expressed as atwo-mode squeezed state of the Rindler vacuum [22] | k i M = 1cosh r ∞ X n =0 tanh n r | n k i | n k i (4)with tanh r = e − π | k | c/a ≡ t, (5)and | n k i and | n k i refer to the two modes, correspondingto the left and right Rindler wedges. An excitation in theMinkowski mode can be easily represented as | k i M = 1cosh r ∞ X n =0 tanh n r √ n + 1 | ( n + 1) k i | n k i . (6)As only one of the modes is accessible to Rob due to thecausality constraint, modes in one of the Rindler wedges (saymode 2) need to be traced over. Using the above expressions,the maximally entangled state in Eq. (3) is now transformedinto ρ AR = 12 cosh r ∞ X n =0 tanh n rρ n , (7)where ρ n = | , n ih , n | + √ n + 1cosh r | , n ih , n + 1 | + √ n + 1cosh r | , n + 1 ih , n | + n + 1cosh r | , n + 1 ih , n + 1 | with | m, n i ≡ | m s i M | n k i . The entanglement in the stateEq. (7) shared by Alice and Rob has been calculated inRef [13]. Our aim in this paper will be to calculate the quan-tum discord in this state.Quantum discord aims at capturing all quantum correla-tions in a state, including entanglement. The quantum mutualinformation is generally taken to be the measure of total cor-relations, classical and quantum, in a quantum state. For twosystems, A and R , it is defined as I ( A : R ) = H ( A ) + H ( R ) − H ( A, R ) , (8) where H ( · ) stands for the von Neumann entropy, H ( ρ ) ≡− Tr( ρ log ρ ) . In our paper, all logarithms are taken to base 2.For a classical probability distribution, Bayes’ rule leads to anequivalent definition of the mutual information as I ( A : R ) = H ( R ) − H ( R | A ) , where the conditional entropy H ( R | A ) isan average of the Shannon entropies of R, conditioned on thealternatives of A. It captures the ignorance in R once the stateof A has been determined. For a quantum system, this de-pends on the measurements that are made on A. If we restrictto projective measurements described by a complete set ofprojectors { Π i } , corresponding to the measurement outcome i, the state of R after the measurement is given by ρ R | i = Tr A (Π i ρ AR Π i ) /p i , p i = Tr A,R (Π i ρ AR Π i ) . (9)A quantum analogue of the conditional entropy can then bedefined as ˜ H { Π i } ( R | A ) ≡ P i p i H ( ρ R | i ) , and an alternativeversion of the quantum mutual information can now be de-fined as J { Π i } ( A : R ) = H ( R ) − ˜ H { Π i } ( R | A ) . (10)The above quantity depends on the chosen set of measure-ments { Π i } . To capture all the classical correlations presentin ρ AR , we maximize J { Π i } ( A : R ) over all { Π i } , arriv-ing at a measurement independent quantity J ( A : R ) =max { Π i } ( H ( R ) − ˜ H { Π i } ( R | A )) ≡ H ( R ) − ˜ H ( R | A ) , where ˜ H ( R | A ) = min { Π i } ˜ H { Π i } ( R | A ) . The quantum discord isfinally defined as D ( A : R ) = I ( A : R ) − J ( A : R ) (11) = H ( A ) − H ( A : R ) + min { Π i } ˜ H { Π i } ( R | A ) . As a first step towards the calculation of quantum discord,we begin by rewriting the state ρ AR in a more conducive form,as ρ AR = 1 − t (cid:16) | ih | ⊗ M + | ih | ⊗ M + | ih | ⊗ M + | ih | ⊗ M (cid:17) , (12)where M = ∞ X n =0 t n | n ih n | ,M = (1 − t ) ∞ X n =0 ( n + 1) t n | n + 1 ih n + 1 | ,M = p − t ∞ X n =0 √ n + 1 t n | n ih n + 1 | ,M = M † . (13)This form of the state suggests a natural bipartite split acrosswhich to calculate the quantum discord. We have, in effect, a × ∞ dimensional system, and we will make our measure-ment on the 2 dimensional subsystem, which in our case, willbe Alice’s side. It is now easy to obtain the reduced state ofthe measured subsystem as ρ A = Tr R ( ρ AR ) = 12 (cid:18) (cid:19) , (14)whereby H ( A ) = 1 . The spectrum of the complete state ρ AR is given by λ ( ρ AR ) = (cid:26) − t t n (1 + ( n + 1)(1 − t )) (cid:27) ∞ n =0 , (15)whereby H ( A : R ) = − − t ∞ X n =0 t n (1 + ( n + 1)(1 − t )) (16) × log (cid:18) − t t n (1 + ( n + 1)(1 − t )) (cid:19) . The evaluation of the quantum conditional entropy, requiresa minimization over all one-qubit projective measurements,which are of the form Π ± = I ± x . σ (17)with x . x = x + x + x = 1 , and I is the one-qubit, × identity matrix. The post-measurement state is then given by ρ R |± = 1 − t p ± (cid:16) (1 ± x ) M + (1 ∓ x ) M ± ( x + ix ) M ± ( x − ix ) M (cid:17) , (18)with outcome probabilities p ± = 1 − t ± x )Tr[ M ] + (1 ∓ x )Tr[ M ]) = 12 . The density matrices ρ R |± are tridiagonal, whose eigenval-ues can be obtained easily numerically, in particular, by usingthe parameterization x = sin θ cos φ, x = sin θ sin φ, and x = cos θ. It is immediately realized that the eigenvalues ofthese states, that are used to calculate the conditional quantumentropy, are independent of φ . This is because the initial statein Eq. (12) is azimuthally invariant, and the final state whosespectrum is to be evaluated reduces to ρ R |± = 1 − t (cid:16) (1 ± cos θ ) M + (1 ∓ cos θ ) M ± sin θM ± sin θM (cid:17) , (19)having spectra λ ± . Then, following Eq. (11), the expressionfor quantum discord in the state ρ AR , as a function of the pa-rameter θ, is given by D θ = 1 + 1 − t ∞ X n =0 t n (1 + ( n + 1)(1 − t )) × log (cid:18) − t t n (1 + ( n + 1)(1 − t )) (cid:19) − X i = ± Tr( λ i log λ i ) , (20) Θ D Θ FIG. 1: (Color online) The plot of the quantum discord, D θ , Eq. (20),as a function of acceleration parameter t = tanh r and θ. In blackdots are shown the minima for different values of t, which can beseen to be attained for θ = π/ . They corresponds to the solid greenline in Fig. (2). and is plotted in Fig. (1) as a function of θ and t . Realiz-ing that the minimum is obtained for θ = π/ , we obtainedthe final value of the quantum discord for the state ρ AR as D = D θ = π/ . This value is plotted in Fig. (2). This is themain result of our paper. To put our result in perspective, wealso plot the logarithmic negativity of the same state [13]. Thisshows that, in the range of accelerations where the state hasno distillable entanglement, as shown by the vanishing loga-rithmic negativity, the state indeed has finite quantum discord,In the calculation of quantum discord, as per Eq. (11),one maximizes over one-dimensional projective measure-ments on one of the subsystems, in our case Alice. For themeasurement-induced disturbance (MID) measure, one per-forms measurements on both the subsystems, with the mea-surements being given by projectors onto the eigenvectors ofthe reduced subsystems. This can be thought of as a bidirec-tional form of discord, which actually depends on the partymaking the measurement [29]. The MID measure of quantumcorrelations for a quantum state ρ AR is given by [27] M ( ρ AR ) := I ( ρ AR ) − I ( P ( ρ AR )) (21)where P ( ρ AR ) := M X i =1 N X j =1 (Π Ai ⊗ Π Rj ) ρ AR (Π Ai ⊗ Π Rj ) . (22)Here { Π Ai } , { Π Rj } denote rank one projections onto the eigen-bases of ρ A and ρ R , respectively. I ( σ ) is the quantum mutualinformation, which is considered to the measure of total, clas-sical and quantum, correlations in the quantum state σ . Sinceno optimizations are involved in this measure, it is much easierto calculate in practice than the quantum discord. The mea-surement induced by the spectral resolution leaves the entropy D,M
FIG. 2: (Color online) The solid green line is the quantum discordin the state ρ AR for a measurement made on Alice’s side. The reddashed line is the logarithmic negativity in the same state, as inRef. [13]. The blue dotted line is the MID measure for the samestate ρ AR . of the reduced states invariant and is, in a certain sense, theleast disturbing. Actually, this choice of measurement evenleaves the reduced states invariant [27]. For pure states, boththe quantum discord and the MID measure reduce to the von-Neumann entropy of the reduced density matrix, which is ameasure of bipartite entanglement.Starting from the expression of ρ AR in Eq. (12), we have ρ R = Tr A ( ρ AR ) = 1 − t M + M ) , (23)which, being diagonal, leads to { Π Rj } = { E j } where [ E j ] kl = δ kj δ lj , j, k, l = 1 , · · · , ∞ . From Eq. (14), { Π Aj } = { E j } where [ E j ] kl = δ kj δ lj , j, k, l = 1 , . Given these, P ( ρ AR ) = diag ( ρ AR ) and, H ( P ( ρ AR )) = − − t ∞ X n =0 t n log (cid:18) − t t n (cid:19) − (1 − t ) × ∞ X n =0 ( n + 1) t n log (cid:18) ( n + 1) (1 − t ) t n (cid:19) = 1 − t − t log( t ) −
32 log(1 − t ) − (1 − t ) S , (24)where S = P ∞ n =0 t n ( n + 1) log( n + 1) . The MID measurecan now be calculated as M ( ρ AR ) = H ( P ( ρ AR )) − H ( ρ AR ) , (25)the latter of which can be obtained from Eq. (15). A plot ofthis measure is shown in Fig. (2), as the blue dotted line.We have shown the existence of purely quantum correla-tions between two, initially entangled, free modes of a scalarfield, when the party detecting one of the modes undergoes a constant acceleration, while the other is inertial. In thisregime, there there is no distillable entanglement betweenthem as a consequence of the Unruh effect. In particular,we provide evidence that there is a finite amount of quan-tum discord in such a state in the limit of infinite acceleration.As quantum discord captures nonclassical correlations beyondentanglement, it might be possible to use these correlations toattain nontrivial quantum advantage.AD was supported in part by the EPSRC (Grant No.EP/C546237/1), EPSRC QIP-IRC and the EU IntegratedProject (QAP). ∗ Electronic address: [email protected][1] A. Peres and D. R. Terno, Rev. Mod. Phys. , 93 (2004).[2] M. A. Nielsen and I. L. Chuang, Quantum Computation andQuantum Information (Cambridge Univ. Press, 2000).[3] M. Czachor, Phys. Rev. A , 72 (1997).[4] P. Caban, J. Rembieli´nski, and M. Wlodarczyk, Phys. Rev. A , 014102 (2009).[5] W. T. Kim and E. J. Son, Phys. Rev. A , 014102 (2005).[6] S. Massar and P. Spindel, Phys. Rev. D , 085031 (2006).[7] H. Terashima and M. Ueda, Int. J. Quant. Inf. , 93 (2002).[8] A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. ,230402 (2002).[9] M. Czachor, Phys. Rev. Lett. , 078901 (2005).[10] R. M. Gingrich, A. J. Bergou, and C. Adami, Phys. Rev. A ,042102 (2003).[11] R. M. Gingrich and C. Adami, Phys. Rev. Lett. , 270402(2002).[12] Y. Shi, Phys. Rev. D , 105001 (2004).[13] I. Fuentes-Schuller and R. B. Mann, Phys. Rev. Lett , 120404(2005).[14] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A ,022101 (2007).[15] P. M. Alsing and G. J. Milburn, Phys. Rev. Lett. , 180404(2003).[16] A. Bermudez and M. A. Martin-Delgado, J. Phys. A , 485302(2008).[17] P. M. Alsing, I. Fuentes-Schuller, R. B. Mann, and T. E. Tessier,Phys. Rev. A , 032326 (2006).[18] G. Adesso, I. Fuentes-Schuller, and M. Ericsson, Phys. Rev. A , 062112 (2007).[19] M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and F. Ca-passo, Phys. Rev. Lett. , 243004 (2003).[20] P. M. Alsing, J. P. Dowling, and G. J. Milburn, Phys. Rev. Lett. , 220401 (2005).[21] A. Retzker, J. I. Cirac, M. B. Plenio, and B. Reznik, Phys. Rev.Lett. , 110402 (2008).[22] L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev. Mod.Phys. , 787 (2008).[23] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. , 017901(2002).[24] L. Henderson and V. Vedral, J. Phys. A , 6899 (2001).[25] A. Datta, Ph.D. thesis, University of New Mexico,arxiv:0807.4490 (2008).[26] A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett ,050502 (2008).[27] S. Luo, Phys. Rev. A , 022301 (2008).[28] A. Datta and S. Gharibian, Phys. Rev. A79