Quantum discord and measurement-induced disturbance in the background of dilaton black holes
aa r X i v : . [ qu a n t - ph ] A ug Quantum discord and measurement-induced disturbance in the background of dilatonblack holes
Jieci Wang , ∗ , Jiliang Jing and Heng Fan † Department of Physics and Key Laboratory of Low DimensionalQuantum Structures and Quantum Control of Ministry of Education,Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
We study the dynamics of classical correlation, quantum discord and measurement-induced dis-turbance of Dirac fields in the background of a dilaton black hole. We present an alternative physicalinterpretation of single mode approximation for Dirac fields in black hole spacetimes. We show thatthe classical and quantum correlations are degraded as the increase of black hole’s dilaton. We findthat, comparing to the inertial systems, the quantum correlation measured by the one-side measuringdiscord is always not symmetric with respect to the measured subsystems, while the measurement-induced disturbance is symmetric. The symmetry of classical correlation and quantum discord isinfluenced by gravitation produced by the dilaton of the black hole.
PACS numbers: 03.67.-a,03.67.Mn, 04.70.Dy
I. INTRODUCTION
Relativistic quantum information [1], which is the com-bination of general relativity, quantum field theory andquantum information, has been a focus of research forboth conceptual and experimental reasons. Understand-ing quantum effects in a relativistic framework is ulti-mately necessary because the world is essentially nonin-ertial. Also, quantum correlation plays a prominent rolein the study of the thermodynamics and information lossproblem [2, 3] of black holes. It is of a great interest tostudy how the relativistic effects influence the propertiesof entanglement, classical correlation, quantum discord,quantum nonlocality, as well as Fisher information [4–14] in the last few years. Recently, exotic classes of blackholes derived from the string theory, i.e., the dilaton blackholes [15–17] formed by gravitational systems coupled toMaxwell and dilaton fields, have attracted much atten-tion. It is widely believed that the study of dilaton blackholes may lead to a deeper understanding of quantumgravity because it emerges from several fundamental the-ories, such as string theory, black hole physics, and loopquantum gravity.On the other hand, quantum correlation measured bydiscord [18–21] is regarded as a valuable resource forquantum computation and communication in some sit- ∗ [email protected] † [email protected] uations. To calculate the discord of a bipartite state,one makes a one-side measurement on a subsystem A of ρ AB by a complete set of projectors { Π Aj } which yields ρ B | j = T r A (Π Aj ρ AB Π Aj ) /p j with p j = T r AB (Π Aj ρ AB Π Aj ).The mutual information [22] of ρ AB can alternatively bedefined by J { Π Aj } ( A : B ) = S ( ρ B ) − S { Π Aj } ( B | A ) , where S { Π Aj } ( B | A ) = P j p j S ( ρ B | j ) is conditional entropy [23]of the state. This quantity strongly depends on thechoice of the measurements { Π Aj } . One should minimizethe conditional entropy over all possible measurementson A which corresponds to finding the measurementthat disturbs least the overall quantum state [18]. Thequantum discord between parts A and B has the form D ( A : B ) = I ( A : B ) − C ( A : B ), where C ( A : B ) is theclassical correlation C ( A : B ) = max { Π Aj } J { Π Aj } ( A : B )and I ( A : B ) is the quantum mutual information quanti-fying the total correlation. So quantum discord describesthe discrepancy between total correlation and classicalcorrelation, and it thus provides a measure of quantum-ness of correlations. In most situations of inertial sys-tems, the quantum discord is symmetric with respect tothe subsystem to be measured [18, 19, 24]. However,is the symmetry still tenable in the noninertial systems,especially in the curved spacetimes? Besides, does thespacetime background also affects the quantum correla-tions by some other measures such as the measurement-induced disturbance (MID)?In this paper we discuss the properties of classical cor-relation, quantum discord and MID [25] for free modesin the background of a dilaton black hole [15]. The studyof relativistic quantum information on accelerated freemodes has its own advantages other than that of localmodes [13, 14] in the understanding of quantum effectsin curved spacetimes in the sense that there are no provedfeasible localized detector models inside the event horizonof a black hole. We assume that two observers, Alice andBob, measure their local state respectively. After shar-ing an entangled initial state, Alice stays stationary at anasymptotically flat region, while Bob moves with uniformacceleration and hovers near the event horizon of the dila-ton black hole. We calculate the classical correlation andquantum discord by making one-side measurements on asubsystem of the bipartite system, and then get the MIDmeasurement correlations by measuring both of the twosubsystems. We are interested in how the dilaton chargewill influence the classical correlation, quantum discord,and MID, as well as if these correlations are symmetricunder the effect of gravitation produced by the dilatonof the black hole.The paper is organized as follows. In the next sectionwe discuss the quantization of Dirac fields in the back-ground of the dilaton black hole beyond single mode ap-proximation [26, 27]. In Sec. III we study the propertiesof classical correlation, quantum discord, and MID forDirac fields in the dilaton spacetime. We will summarizeand discuss our conclusions in the last section. II. QUANTIZATION OF DIRAC FIELDS INDILATON BLACK HOLE SPACETIMES
The massless Dirac equation in a general backgroundspacetime can be written as [28][ γ a e aµ ( ∂ µ + Γ µ )]Ψ = 0 , (1)where γ a are the Dirac matrices, the four-vectors e aµ represent the inverse of the tetrad e aµ defined by g µν = η ab e aµ e bν with η ab = diag( − , , , µ = [ γ a , γ b ] e aν e bν ; µ are the spin connection coefficients.The metric for a Garfinkle-Horowitz-Strominger dila-ton black hole spacetime can be expressed as [15] ds = − (cid:18) r − Mr − α (cid:19) dt + (cid:18) r − Mr − α (cid:19) − dr + r ( r − α ) d Ω , (2)where M and α are the mass of the black hole and thedilaton. Throughout this paper we set G = c = ~ = κ B = 1. This black hole has two singular points at r =2 M and r = 2 α . Besides, the dilaton α and the mass M of the black hole should satisfy α < M . In order toseparate the Dirac equation, we adopt a tetrad as e aµ = diag (cid:18)p f , √ f , √ r ˜ r, √ r ˜ r sin θ (cid:19) , (3)where f = ( r − M ) / ˜ r and ˜ r = r − α . Then Eq. (1)in the Garfinkle-Horowitz-Strominger dilaton black-hole spacetime becomes − γ √ f ∂ Ψ ∂t + p f γ (cid:20) ∂∂r + r − αr ˜ r + 14 f dfdr (cid:21) Ψ (4)+ γ √ r ˜ r ( ∂∂θ + 12 cot θ )Ψ+ γ √ r ˜ r sin θ ∂ Ψ ∂ϕ = 0 . (5)If we rescale Ψ as Ψ = f − Φ and use an ansatz for theDirac spinor similar to Ref. [29], we can solving the Diracequation near the event horizon. For the outside andinside region of the event horizon, we obtain the positivefrequency outgoing solutions [30, 31]Ψ + out, k = G e − iω U , (6)Ψ + in, k = G e iω U , (7)where U = t − r ∗ and G is a four-component Dirac spinor, k is the wavevector we used to label the modes hereafterand for massless Dirac field ω = | k | . In terms of thesebasis the Dirac field Ψ can be expanded asΨ = Z d k [ˆ a out k Ψ + out, k + ˆ b out †− k Ψ − out, k + ˆ a in k Ψ + in, k + ˆ b in †− k Ψ − in, k ] , (8)where ˆ a out k and ˆ b out † k are the fermion annihilation andantifermion creation operators acting on the state of theexterior region, and ˆ a in k and ˆ b in † k are the fermion annihi-lation and antifermion creation operators acting on theinterior vacuum of the black hole respectively. The anni-hilation operator ˆ a out k and creation operator ˆ a out k satisfythe canonical anticommutation { ˆ a out k , ˆ a out k ′ } = δ kk ′ and { ˆ a out k , ˆ a out † k ′ } = { ˆ a out † k , ˆ a out † k ′ } = 0, where { ., . } denotesthe anticommutator. Clearly, two fermionic Fock are an-tisymmetric with respect to the exchange of the modelabels k and k ′ due to the anticommutation relations.We therefore define || k k ′ ii = ˆ a † k , ˆ b † k ′ || ii = − ˆ b † k ′ , ˆ a † k || ii , (9)where the states in the antisymmetric fermionic Fockspace are denoted by double-lined Dirac notation || . ii [32] rather than the single-lined notations.Making analytic continuation for Eqs. (6) and (7),we find a complete basis for positive energy modes,i.e., the Kruskal modes, according to the suggestion ofDomour-Ruffini [30]. Then we can quantize the masslessDirac field in black hole and Kruskal modes respectively[29, 31], from which we can easily get the Bogoliubovtransformations [33] between the creation and annihila-tion operators in different coordinates [29]. Consideringthat it is more interesting to quantize the Dirac field be-yond the single mode approximation [26, 27]. We con-struct a different set of creation operators that are linearcombinations of creation operators in the inside and out-side regions [26, 27]˜ c k ,R = cos r ˆ a out k − sin r ˆ b in †− k , ˜ c k ,L = cos r ˆ a in k − sin r ˆ b out †− k , ˜ c † k ,R = cos r ˆ a out † k − sin r ˆ b in − k , ˜ c † k ,L = cos r ˆ a in † k − sin r ˆ b out − k , (10)where cos r = [ e − πω ( M − α ) + 1] − and sin r =[ e πω ( M − α ) + 1] − . We name the modes (or operators)with subscripts L and R by “left” and “right” modes(or operators), respectively. After properly normaliz-ing the state vector, the Kruskal vacuum is found tobe || ii K = N k || k ii K = N k || k ii R ⊗ || k ii L , where ||| k ii R and ||| k ii L are annihilated by the annihilationoperators ˜ c k ,R and ˜ c k ,L . The vacuum state || k ii K formode k is given by || k ii K = cos r || ii − sin r || ii + sin r cos r || ii − sin r || ii , (11)where || mnm ′ n ′ ii = || m k ii + out || n − k ii − in || m ′− k ii − out || n ′ k ii + in , {|| n − k ii − in } and {|| n k ii + out } are the orthonormal basesfor the inside and outside region of the dilaton black holerespectively, and the { + , −} superscript on the kets isused to indicate the fermion and antifermion vacua. Forthe observer Bob who travels outside the event horizon,the Hawking radiation spectrum from the view of hisdetector can be obtained by N ω = K h | ˆ a out † k ˆ a out k | i K = 1 e ω/T + 1 , where T = π ( M − α ) is Hawking temperature [34] of theblack hole. This equation shows that the observer inthe exterior of the Garfinkle-Horowitz-Strominger dila-ton black hole detects a thermal Fermi-Dirac distributionof particles. Because of the Pauli exclusion principle, onlythe first excited state for each fermion mode || k ii + K isallowed, and similarly for antifermions. The first excitedstate for the fermion mode is given by || k ii + K = h q R (˜ c † k ,R ⊗ I L ) + q L ( I R ⊗ ˜ c † k ,L ) i || k ii K = q R [cos r || ii − sin r || ii ]+ q L [sin r || ii + cos r || ii ] , (12)with | q R | + | q L | = 1. The study of fermionic quan-tum information beyond the single mode approximation,which was proposed in [26] and widely adopted recently,has a lack of physical interpretation so far. Here wepresent an alternative physical interpretation on the op-erators and states that obtained beyond such an approx-imation in black hole spacetimes. The operator ˜ c † k ,R inEq.(10) indicates the creation of two particles, i.e., afermion in the exterior vacuum and an antifermion in theinterior vacuum of the black hole. Similarly, the create operator ˜ c † k ,L means that an antifermion and an fermionare created outside and inside the event horizon, respec-tively. Hawking radiation comes from spontaneous cre-ation of paired particles and antiparticles by quantumfluctuations near the event horizon. The particles andantiparticles can radiate toward the inside and outsideregions randomly from the event horizon with the totalprobability | q R | + | q L | = 1. Thus, | q R | = 1 meansthat all the particles move toward the black hole exteri-ors while all the antiparticles move to the inside region,i.e., only particles can be detected as Hawking radiation.Similarly, | q L | = 1 indicates that only antiparticles es-capes from the event horizon. Therefore, the single modeapproximation (either | q R | = 1 or | q L | = 1) is a specialcase when either only particles or only antiparticles aredetected. III. QUANTUM DISCORD AND MID INDILATON BLACK HOLE SPACETIMES
We assume that Alice and Bob share a maximally en-tangled state || Φ ii AB = 1 √ || ii A || ii B + || ii + A || ii + B ) , (13)at the same point in the asymptotically flat region of thedilaton black hole. Then Alice stays stationary at theasymptotically flat region, while Bob moves with uni-form acceleration and hovers near the event horizon ofthe dilaton black hole. Bob will detects a thermal Fermi-Dirac distribution of particles and his detector is foundto be excited. Using Eqs. (11) and (12), we can rewriteEq. (13). Since Bob is causally disconnected from theregion inside the event horizon we should trace over thestate of the inside region and obtain ̺ AB out = 12 h C || ii hh | | + S C ( || ii hh | | + || ii hh | | ) + S || ii hh | | + | q R | ( C || ii hh | | + S || ii hh | | )+ | q L | ( S || ii hh | | + C || ii hh | | )+ q ∗ R ( C || ii hh | | + S C || ii hh | | ) − q ∗ L ( C S || ii hh | | + S || ii hh | | ) − q R q ∗ L SC || ii hh | | i + (H.c.) non-diag. (14)where || lmn ii = || l ii A || m k ii + out || n − k ii − out , S = sin r and C = cos r . We assume that Bob has a detector sen-sitive only to the particle modes, which means that anantifermion cannot be excited in a single detector when afermion was detected. Thus, we should also trace out theantifermion mode || n − k ii − out in the outside region [27] ̺ A ˜ B = 12 (cid:20) C || iihh | + q ∗ R C || iihh || + q R C || iihh || ) + q L C || iihh || + S || iihh || + χ || iihh || (cid:21) , (15)where || lm ii = || l ii A || m k ii + out and χ = | q R | + | q L | S .Hereafter we call the mode k| m k ii + out as ˜ B . Now oursystem has two subsystems, i.e., the inertial subsystem A and accelerated subsystem ˜ B . We can easily obtainthe von Neumann entropy S ( ρ A, ˜ B ) of this state, S ( ρ A )for the reduced density matrix of the mode A and S ( ρ ˜ B )for the mode ˜ B , respectively. A. Measurements on subsystem A Now let us first make measurements on the subsystem A , the projectors are defined as [7, 18, 20]Π A + = I + n · σ ⊗ I , Π A − = I − n · σ ⊗ I , (16)where n = sin θ cos ϕ, n = sin θ sin ϕ, n = cos θ and σ i are Pauli matrices. The measurement operators in Eq.(16) include a two-outcome projective measurement op-erator I ± n · σ on subsystem A and an identity operatoron subsystem B . These operators are orthogonal projec-tors spanning the qubit Hilbert space and can thereforebe parameterized by the unit vector n = ( n , n , n ).For simplicity, here we can take the measurements onthe particle-number degree of freedom, i.e., to measurewhether or not a fermion with wave vector k is excitedin the particle detector. After the measurement of Π A + ,the quantum state ρ A, ˜ B changes to ρ MA + = T r A (Π A + ρ A, ˜ B Π A + ) /p A + = 12 (cid:20) ς e iϕ q ∗ R C sin θe − iϕ q R C sin θ χ (cid:21) , (17)where p A + = T r (Π A + ρ A, ˜ B Π A + ) = 1 / ς = C [1 + cos θ + q L (1 − cos θ )] and χ = χ (1 − cos θ ) + (1 + cos θ ) S . Thesame method is used to compute the state after measure-ment Π A − , then we have ρ MA − = 14 p A − (cid:20) ς − e iϕ q ∗ R C sin θ − e − iϕ q R C sin θ χ (cid:21) , (18)where p A − = T r (Π A − ρ A, ˜ B Π A − ) = 1 / ς = C [1 − cos θ + q L (1 + cos θ )] and χ = χ (1 + cos θ ) + (1 − cos θ ) S . Nowwe can obtain the conditional entropy S { Π Aj } ( ˜ B | A ) ≡ P j p j S ( ˜ B | j ). The classical correlation in this case is C ( ˜ B | A ) = S ( ρ ˜ B ) − min Π Aj S { Π Aj } ( ˜ B | A ) , and the quantum discord is D ( ˜ B | A ) = S ( ρ ˜ A ) − S ( ρ A, ˜ B ) + min Π Aj S { Π Aj } ( ˜ B | A ) . Note that the conditional entropy has to be numeri-cally evaluated by optimizing over the angles θ and φ .Thus we should minimize it over all possible measure-ments on A [18]. We find that the condition entropy isindependent of ϕ and its minimum can be obtained when θ = π/ B. Measurements on subsystem ˜ B Then let us make our measurements on the subsystem˜ B ; the projectors are defined asΠ ˜ B + = I ⊗ I + n · σ , Π ˜ B − = I ⊗ I − n · σ . (19)After the measurement of Π ˜ B + , the state ρ A, ˜ B changes to ρ M ˜ B + = 14 p ˜ B + (cid:20) ς e iϕ q ∗ R sin r sin θe − iϕ q R sin r sin θ χ (cid:21) , where p ˜ B + = (1 − cos 2 r cos θ )(1 + q ) + q (1 + cos θ ), ς = (1 + cos θ ) C + (1 − cos θ ) S and χ = q C (1 − cos θ )+ χ (1+cos θ ). Similarly, we can calculate the stateafter Π ˜ B − and get the classical correlation C ( A | ˜ B ) and thequantum discord D ( A | ˜ B ) respectively. Α CC H A È B Ž L C H B Ž È A L FIG. 1: (Color online) The classical correlation C ( ˜ B | A ) ob-tained by measuring the subsystem A and the classical corre-lation C ( A | ˜ B ) (dashed line) as a function of the dilaton α . Weset M = ω = 1 and q R = 1. Figure 1 shows how the dilaton α of the black holeinfluences the classical correlations of the system whenwe obtain them by measuring the subsystem A and ˜ B ,respectively. From which we can see that for all the twocases the classical correlations decrease with increasingdilaton α . The classical correlation C ( ˜ B | A ) obtained bythe one-side measurements on subsystem A , is alwaysnot equal to C ( A | ˜ B ) for any dilaton value. In contrast,the classical correlations satisfy C ( B | A ) = C ( A | B ) = 1for the initial state, Eq.(13), in the asymptotically flatregion. Comparing to the flat spacetime, the classicalcorrelation (of course the related quantum correlation) is not symmetrical in the dilaton black hole spacetime. Thisasymmetry is due to the effect of gravitation produced bythe black hole. We also note that C ( ˜ B | A ) is larger than C ( A | ˜ B ) when the dilaton is smaller than a fixed value( α ≃ . C ( A | ˜ B ) when thedilaton is larger than this value. C. Symmetric measurement of the correlations
From the foregoing discussion, we see that the clas-sical and quantum correlations in the curved spacetimedepend on the measurement process. At the same time, asymmetric measurement of the quantum correlation wasproposed recently. The MID measurement [25], which isobtained by a complete set of projective measurementsover both partitions of a bipartite state, is given by [25] D ( M ID ) = I ( ρ A, ˜ B ) − I ( η A, ˜ B ) (20)with η A, ˜ B = m X i =1 n X j =1 π Ai ⊗ π ˜ Bj ρ A, ˜ B π Ai ⊗ π ˜ Bj . (21)where π Ai = || i iihh i || and π ˜ Bj = || j iihh j || are one-dimensional orthogonal projections for parties A and ˜ B ,respectively. Such a symmetrized version of the quan-tum correlation was recently discussed theoretically [25]and experimentally measured by anuclear magnetic res-onance setup at room temperature [35]. Besides, MIDrequires only the local measurement strategy rather thanthe cumbersome optimization required by the derivationof discord [36]. Here we can easily obtain η A, ˜ B and findthe MID measure of classical correlation C ( M ID ) andquantum correlation D ( M ID ). Α CC H MID L D H A È B Ž L D H B Ž È A L FIG. 2: (Color online) The quantum discord D ( ˜ B | A )(solidline), D ( A | ˜ B ) (dashed line), and D ( MID ) (dotted line) of ρ A, ˜ B as a function of the dilaton α . We set M = ω = 1 and q R = 1. Figure 2 shows how the dilaton of the black hole affectsthe quantum correlations (discord and MID) that ob-tained by different measuring methods. Both the quan-tum discord and the MID decrease as the increasing of α ,which means the quantum correlations degraded as theincrease of dilaton α . It is shown that the discord D ( A | ˜ B )is not equal to D ( ˜ B | A ) for any α , which is extremely dif-ferent from D ( B | A ) = D ( A | B ) = 1 in the initial stateEq. (13). Comparing to the flat spacetime, the quan-tum discord is not symmetric for any α in the dilatonblack hole spacetime. The symmetry of quantum discordis truly influenced by the dilaton of the black hole. It isalso noted that the quantum correlation obtained via theMID measurement is always larger than that obtained bythe one-side measurement. IV. SUMMARY
The effect of black hole’s dilaton on the symmetry ofclassical correlation, quantum discord, and MID for theDirac fields is investigated. We give a physical interpre-tation of the single mode approximation in the curvedspacetime, i.e., such an approximation is a special casewhen either only particles or only antiparticles are de-tected. It is shown that the classical and quantum cor-relations decrease monotonously as increasing dilaton,which means all type of correlations are degraded due tothe effect of gravitation generated by the dilaton of theblack hole. We find that both the one-side measured clas-sical and quantum correlations are not symmetric withrespect to the subsystem being measured. So both thequantification and the symmetry of classical correlationand quantum discord are influenced by the gravitationwhen taking the one-side measurement. This is a sharpcomparison between the inertial systems and the systemin the curved spacetime. The results obtained here arenot only helpful to understanding the symmetric proper-ties of classical and quantum correlations in the presenceof strong gravitation but also give a better insight intoquantum properties of dilaton black holes.
Acknowledgments
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