Trace-distance correlations for X states and emergence of the pointer basis in Markovian and non-Markovian regimes
aa r X i v : . [ qu a n t - ph ] S e p Trace-distance correlations for X states and emergence of the pointer basis inMarkovian and non-Markovian regimes
Paola C. Obando, ∗ Fagner M. Paula, † and Marcelo S. Sarandy ‡ Instituto de F´ısica, Universidade Federal Fluminense,Avenida General Milton Tavares de Souza s/n, Gragoat´a, 24210-346,Niter´oi, RJ, Brazil Universidade Tecnol´ogica Federal do Paran´a - Rua Cristo rei 19, Vila Becker, 85902-490, Toledo, PR, Brazil (Dated: August 8, 2018)We provide analytical expressions for classical and total trace-norm (Schatten 1-norm) geometriccorrelations in the case of two-qubit X states. As an application, we consider the open-systemdynamical behavior of such correlations under phase and generalized amplitude damping evolutions.Then, we show that geometric classical correlations can characterize the emergence of the pointerbasis of an apparatus subject to decoherence in either Markovian or non-Markovian regimes. Inparticular, as a non-Markovian effect, we obtain a time delay for the information to be retrievedfrom the apparatus by a classical observer. Moreover, we show that the set of initial X statesexhibiting sudden transitions in the geometric classical correlation has nonzero measure.
PACS numbers: 03.65.Ud, 03.67.Mn, 75.10.Jm
I. INTRODUCTION
Correlations are typically behind information-based in-terpretations of physical phenomena [1–3]. In a quan-tum scenario, they appear as key signatures, with op-erational roles e.g. in quantum metrology [4–6], entan-glement activation [7–9], and information encoding anddistribution [10, 11]. In a geometric approach, they canbe defined through a number of distinct formulations,which are based on the relative entropy [12], Hilbert-Schmidt norm [13, 14], trace norm [15, 16], or Buresnorm [17, 18]. All of these distinct versions can be gen-erally described by a unified framework in terms of adistance (or pseudo distance) function. In particular, ithas been shown that the trace norm, which correspondsto the Schatten 1-norm, provides a suitable direction forthe investigation of quantum, classical, and total correla-tions, since it is the only p -norm able to satisfy reasonableaxioms expected to hold for information-based correla-tion functions. Moreover, for the simple case of mixedtwo-qubit systems in Bell-diagonal states, analytical ex-pressions have been found for quantum, classical, andtotal correlations [19–21]. However, for the more generalcase of two-qubit X states, only the quantum contribu-tion for the geometric correlation has been analyticallyderived [22]. Here, our aim is to close this gap, providingclosed analytical expressions for the classical and totalcorrelations of arbitrary two-qubit X states. Remark-ably, they are shown to be as simple to be computed asin the case of Bell-diagonal states.The analytical expressions for the classical correlationof X states can be applied as a powerful resource tocharacterize the open-system dynamics in rather general ∗ [email protected]ff.br † [email protected] ‡ [email protected]ff.br environments. In this direction, we consider a system-apparatus set AS under the effect of X state preservingchannels, with decoherence driving the quantum appa-ratus A to collapse into a possible set of classical statesknown as the pointer basis [23]. We are then able toshow that the geometric classical correlation decays to aconstant value at finite time τ E for decohering processesadmitting a pointer basis. This is exploited in a gen-eral scenario of X states, for either Markovian and non-Markovian evolutions. In particular, we show a delay inthe emergence time τ E in the non-Markovian regime. II. GEOMETRIC CLASSICAL AND TOTALCORRELATIONS: ANALYTICAL EXPRESSIONS
In the general approach introduced in Refs. [1, 21, 24],measures of quantum, classical, and total correlations ofan n -partite system in a state ρ are respectively definedby Q ( ρ ) = K [ ρ, M − ( ρ )] , (1) C ( ρ ) = K [ M + ( ρ ) , M + ( π ρ )] , (2) T ( ρ ) = K [ ρ, π ρ ] , (3)where K [ ρ, τ ] denotes a real and positive function thatvanishes for ρ = τ , M − ( ρ ) is a classical state obtainedthrough a non-selective measurement { M ( i ) − } that min-imizes Q , M + ( ρ ) is a classical state obtained througha non-selective measurement { M ( i )+ } that maximizes C ,and π ρ = ρ ⊗ ... ⊗ ρ n = tr ¯1 ρ ⊗ ... ⊗ tr ¯ n ρ representsthe product of the local marginals of ρ . In order toavoid ambiguities in the correlation measures for Q and C , we take { M ( i ) − } and { M ( i )+ } as independent measure-ment sets [21]. Let us consider correlations based on thetrace norm (Schatten 1-norm) and projective measure-ments operating over one qubit of a two-qubit system,i.e., K [ ρ, τ ] = k ρ − τ k = tr | ρ − τ | and M ± ( ρ ) = Π (1) ± ( ρ ),such that Q G ( ρ ) = tr (cid:12)(cid:12)(cid:12) ρ − Π (1) − ( ρ ) (cid:12)(cid:12)(cid:12) , (4) C G ( ρ ) = tr (cid:12)(cid:12)(cid:12) Π (1)+ ( ρ ) − Π (1)+ ( π ρ ) (cid:12)(cid:12)(cid:12) , (5) T G ( ρ ) = tr | ρ − π ρ | . (6)By adopting the trace norm, Q G is then the Schat-ten 1-norm geometric quantum discord, as introducedin Refs. [15, 16]. In particular, for two-qubit sys-tems, the geometric quantum discord based on Schat-ten 1-norm is equivalent to the negativity of quantum-ness [16] (also referred as the minimum entanglementpotential [25]), which is a measure of nonclassicalityintroduced in Ref. [8] and experimentally discussed inRef. [26]. As a counterpart to Q G , C G is the Schat-ten 1-norm classical correlation. Concerning T G , it is ameasure of total geometric correlation, which vanishes ifthe system is described by a product state. The tracenorm satisfies reasonable criteria expected for correla-tion measures, although these criteria are still source ofdebate [21, 27].We are interested in a two-qubit system as described byan X-shaped mixed state. Two-qubit X states describerather general two-qubit systems. These states generalizethe Bell-diagonal states, which are those whose densitymatrix is diagonal in the Bell basis. An example of aBell-diagonal state (and therefore of an X state) is theWerner state [28], which mixes a singlet (maximally en-tangled) state with the identity (fully classical) state. Incondensed matter physics, X states provide the generalform of reduced density operators of arbitrary quantumspin chains with Z (parity) symmetry (for a review see,e.g., Ref. [3]). For example, both ground and thermalreduced two-spin states of the quantum Ising chain in atranverse magnetic field are described by X states. Thesame holds for other spin chains, such Heisenberg andXXZ models. The density matrix of a two-qubit X statetakes the form ρ X = ρ ρ ∗ ρ ρ ∗ ρ ρ ρ ρ , (7)where computational basis {| i , | i , | i , | i} isadopted. The normalization and the positive semidefi-niteness of state require P i =1 ρ ii = 1, ρ ρ ≥ | ρ | ,and ρ ρ ≥ | ρ | . The diagonal elements are real,whereas the elements ρ and ρ are complex numbersin general. However, they can be brought into real num-bers via local unitary transformations, which preserve the trace distance correlations [22]. By decomposing the Xstate in the Pauli basis, we obtain ρ X = 14 I ⊗ I + X i =1 c i σ i ⊗ σ i + c I ⊗ σ + c σ ⊗ I ! (8)where c = tr( σ ⊗ σ ρ X ) = 2( ρ + ρ ) , (9) c = tr( σ ⊗ σ ρ X ) = 2( ρ − ρ ) , (10) c = tr( σ ⊗ σ ρ X ) = 1 − ρ + ρ ) , (11) c = tr( I ⊗ σ ρ X ) = 2( ρ + ρ ) − , (12) c = tr( σ ⊗ I ρ X ) = 2( ρ + ρ ) − , (13)with all these parameters assuming values in the interval − ≤ c i ≤
1. If c = c = 0, we obtain the Bell-diagonalstate: ρ X = ρ B ( c = c = 0) . (14)In terms of the parameters { c i } , the Schatten 1-normquantum correlation can be written as [22] Q G ( ρ X ) = r ac − bda − b + c − d , (15)where a = max { c , d + c } , b = min { c, c } , c =max { c , c } , and d = min { c , c } . Now, let us calculatethe corresponding classical and total correlations. First,by computing the marginal density operators, we get ρ = tr ¯1 ρ = ( I + c σ ) / ρ = tr ¯2 ρ = ( I + c σ ) / π ρ X = ρ ⊗ ρ reads π ρ X = 14 ( I ⊗ I + c I ⊗ σ + c σ ⊗ I + c c σ ⊗ σ ) . (16)From Eq. (5), we observe that Π (1)+ ( ρ ) − Π (1)+ ( π ρ ) =Π (1)+ ( ρ − π ρ ). Then, by using Eqs. (8) and (16), wecan observe that the difference of X-states ρ X − π ρ X ismathematically equivalent to a difference between Bell-diagonal states. Indeed, we can rewrite the difference ρ X − π ρ X [also appearing in Eq. (6)] in terms of effectiveBell-diagonal states ˜ ρ B and π ˜ ρ B , i.e., ρ X − π ρ X = ˜ ρ B − π ˜ ρ B , (17)where ˜ ρ B = 14 " I ⊗ I + X i =1 ˜ c i σ i ⊗ σ i (18)and π ˜ ρ B = 14 ( I ⊗ I ) , (19)with (˜ c , ˜ c , ˜ c ) = ( c , c , c − c c ) . (20)In this case, we can directly apply the analytical expres-sions of C G and T G already obtained for the Bell-diagonalstate [19, 21]. This procedure implies in the correlationmeasures for X states obtained in this work, which read C G ( ρ X ) = C G (˜ ρ B ) = ˜ c + (21)and T G ( ρ X ) = T G (˜ ρ B ) = 12 [˜ c + + max { ˜ c + , ˜ c + ˜ c − } ] , (22)where ˜ c − = min {| ˜ c | , | ˜ c | , | ˜ c |} , ˜ c = int {| ˜ c | , | ˜ c | , | ˜ c |} ,and ˜ c + = max {| ˜ c | , | ˜ c | , | ˜ c |} represent the minimum, in-termediate, and the maximum of the absolute values ofthe parameters ˜ c i ( i = 1 , , III. APPLICATIONS
We illustrate the applicability of the geometric mea-sure of classical correlations by considering the decohere-ing dynamics of the quantum systems. We will take thesystem as a two qubit state coupled independently withweak sources of noise [29] (either phase or generalizedamplitude damping). This scenario appears in many situ-ations, such as optical quantum systems [30] and nuclearmagnetic resonance (NMR) setups [31].
A. Markovian dynamics
Let us consider a Markovian process as described bythe operator-sum representation formalism [29]. In thisscenario, the evolution of a quantum state ρ is governedby a trace-preserving quantum operation ε ( ρ ), which isgiven by ε ( ρ ) = X i,j (cid:0) E Ai ⊗ E Bj (cid:1) ρ (cid:0) E Ai ⊗ E Bj (cid:1) † , (23)where { E sk } is the set of Kraus operators associated witha decohering process of a single qubit, with the trace-preserving condition reading P k E s † k E sk = I . We providein Table I the Kraus operators for phase damping (PD)and generalized amplitude damping (GAD), which arethe channels considered in this work.Both the PD and GAD decoherence processes preservethe X form of the density operator. As a next step, wehave to find out the evolved parameters ˜ c i ( t ), as definedby Eq.(20). In this direction, we use Eq. (8) into Eq. (23).Remarkably, the parameters ˜ c i ( t ) turn out to be indepen-dent of λ s . Since the evolution is Markovian, we furthertake the decoherence probability p s = 1 − exp( − t γ s ) for Kraus operatorsPD E s = p − p s / I, E s = p p s / σ GAD E s = √ λ s (cid:18) √ − p s (cid:19) , E s = √ − λ s (cid:18) √ − p s
00 1 (cid:19) E s = √ λ s (cid:18) √ p s (cid:19) , E s = √ − λ s (cid:18) √ p s (cid:19) TABLE I. Kraus operators for phase damping (PD) and gen-eralized amplitude damping (GAD), where p s and λ s are thedecoherence probabilities for the qubit s . both PD and GAD channels. In turn, the evolution is de-scribed by the parameters displayed in Table II in termsof the decoherence time τ D = 1 γ A + γ B . (24) TABLE II. Correlation parameters ˜ c i ( t ) ( i = 1 , ,
3) for PDand GAD channels.Channel ˜ c ( t ) ˜ c ( t ) ˜ c ( t )PD c exp [ − t/τ D ] c exp [ − t/τ D ] ( c − c c )GAD c exp [ − t/ τ D ] c exp [ − t/ τ D ] ( c − c c ) exp [ − t/τ D ] C l a ss i c a l C o rr e l a t i o n *1 | | | | | | C o rr e l a t i o n F un c t i o n s c ccc FIG. 1. (Color online) Classical correlation as a function of τ = ( γ A + γ B ) t for a two-qubit system under the GAD chan-nel. The initial state is in the X form, where the values for c i are selected to show the behavior of the sudden transition,with c = 0 . c = 0 . c = 0 . c = 0 .
10, and c = 0 . C G occurs at τ ∗ = 0 .
37. In the inset,we show the correlation parameters | ˜ c | , | ˜ c | , and | ˜ c | . Then, we can directly obtain the dynamics of classicalcorrelations C G ( ρ X ( t )), as given by Eq. (21). It can beobserved from Table II that both | c ( t ) | and | c ( t ) | dis-play the same decay rate, which means that they do notcross as functions of time. Therefore only the crossingsallowed are for | c ( t ) | = | c ( t ) | and | c ( t ) | = | c ( t ) | , im-plying at most a single nonanalyticity (sudden change)in the geometric classical correlation. This conclusionholds for both PD and GAD channels. Indeed, a nec-essary and sufficient condition for sudden change in thecase of PD and GAD channels are ˜ c − = | ˜ c | 6 = 0 and˜ c + = | ˜ c | 6 = 0, respectively. Therefore, the generaliza-tion of the initial state to an X state does not allow forfurther sudden changes in the classical correlation. Thissustains the result that double sudden changes is an ex-clusive feature of quantum correlations, as discussed forBell-diagonal states in Ref. [31, 32]. We illustrate thisbehavior in Fig. 1, where we plot C G as a function ofthe dimensionless time τ = ( γ A + γ B ) t for a mixed Xstate under the GAD channel. It can be observed that asingle sudden transition occurs at τ ∗ = 0 .
37, which canbe determined from the correlation parameters c i ( t ) inTable II. B. Pointer basis for Markovian dynamics
Let us now apply the classical correlation C G for Xstates to investigate the emergence of the pointer basisof a quantum apparatus A subject to decoherence in aMarkovian regime. The apparatus A measuring a sys-tem S suffers decoherence through the contact with theenvironment, which implies in its relaxation to a possibleset of classical states known as the pointer basis [23]. Asa consequence, the information about S turns out to beaccessible to a classical observer through the pointer ba-sis associated with the apparatus. The emergence of thepointer basis occurs for an instant of time τ E at whichthe classical correlation between A and S becomes con-stant [31–33]. Therefore, we will consider a compositesystem AS under decoherence described by the densityoperator given by Eq. (8). The classical correlation canbe used to characterize the time τ E when the pointerstates emerges, which exactly corresponds to the instantof time at which C G ( t ) shows a sudden transition to aconstant function.For the GAD channel, there is no emergence of pointerbasis at a finite time, since no decay of C G to a constantfunction of time is possible. On the other hand, for thePD channel, we can analytically determine τ E . Indeed,from Table II, C G ( t ) gets constant after a sudden transi-tion at finite time given by τ E = τ D ln (cid:20) ˜ c + | ˜ c | (cid:21) . (25)Comparing τ E with the decoherence time scale τ D , wecan observe that the pointer basis may emerge at a timesmaller or larger than τ D . This generalizes the resultobtained in Refs. [31–33] for Bell-diagonal states. Toillustrate the emergence of the pointer basis, we plot in Fig. 2 the decay of the classical correlation as a functionof τ = ( γ A + γ B ) t under the PD channel for an initialstate in the X form. The emergence of the pointer basisthrough the behavior of C G occurs then at τ ∗ = 0 . τ E = 0 . τ D . C l a ss i c a l C o rr e l a t i o n *1 ccc | | | | | | C o rr e l a t i o n F un c t i o n s FIG. 2. (Color online) Classical correlation as a function of τ = ( γ A + γ B ) t for a two-qubit system under the PD channel.The initial state is in the X form, where c = 0 . c = 0 . c = 0 . c = 0 .
10, and c = 0 .
20, with these values chosento illustrate the emergence of the pointer basis. This occursat τ ∗ = 0 .
92, i.e., τ E = 0 . τ D . In the inset, we detail theevolution of the correlation parameters | ˜ c | , | ˜ c | , and | ˜ c | . C. Non-Markovian dynamics
We now consider the classical correlations for X-statesin a non-Markovian open quantum system under the PDchannel. Non-Markovian dynamics describes many phys-ical situations, e.g. single flourescent systems hosted incomplex environments, superconducting qubits, dephas-ing in atomic and molecular physics, among others [34–36]. For this work the non-Markovianity of the evolutionwill be handled in the local time framework developed inRef. [37]. In this scenario, we start by supposing a quan-tum process governed by a Markovian master equation dρdt = L [ ρ ( t )] , (26)where the generator L is given by L [ • ] = − i [ H, • ] + X i γ i (cid:18) A i • A † i − n A † i A i , • o(cid:19) , (27)with H denoting the effective system Hamiltonian, A i the Lindblad operators, and γ i ≥ ρ s ( t ) of thesystem is written as ρ S ( t ) = R max X R =1 ρ R ( t ) , (28)where each auxiliary (unnormalized) operator ρ R definesthe system dynamics given that the reservoir is in theR-configurational bath state, with R max the number ofconfigurational states of the environment. The probabil-ity P R ( t ) that the environment is in a given state at time t reads P R ( t ) = tr[ ρ R ( t )] . (29)We note that the set of states { ρ R ( t ) } encodes both thesystem dynamics and the fluctuations of the environ-ment [37, 39]. When the transitions between the con-figurational states do not depend on the system state,the fluctuations between the configurational states aregoverned by a classical master equation [40], with a struc-ture following from Eq. (29). This kind of environmentalfluctuations are called self-fluctuating environments. Forour work, we restrict our attention to a two-qubit system A and B interacting with a self-fluctuating environment.Then, we model the environment as being characterizedby a two-dimensional configurational space (R max = 2),which only affects the decay rates of the system. Eachstate follows by itself a Markovian master equation dρ ( t ) dt = − i [ H , ρ ( t )] + γ A ( L A [ ρ ( t )]) + γ B ( L B [ ρ ( t )]) − φ ρ ( t ) + φ ρ ( t ) , (30) dρ ( t ) dt = − i [ H , ρ ( t )] + γ A ( L A [ ρ ( t )]) + γ B ( L B [ ρ ( t )]) − φ ρ ( t ) + φ ρ ( t ) , (31)where the structure of the superoperator L for the PDchannel is given by L A,B [ • ] = ( σ A,Bz • σ A,Bz − • ) . (32)The first line of Eqs. (30) and (31) defines the unitaryand dissipative dynamics for the two-qubit system, giventhat the bath is in the configurational state 1 or 2, re-spectively. The constants { γ A , , γ B , } are the natural de-cay rates of the system associated with each reservoirstate [29]. The positivity of the density matrix will beensured as long as these decoherence coefficients obey γ A,Bi ≥ φ and φ ) [41].For a matter of simplicity, the decay rates associated witheach subsystem will be chosen to be the same, namely, γ A = γ B ≡ γ and γ A = γ B ≡ γ . Moreover, we definethe characteristic dimensionless parameters ǫ = γ γ + γ , ǫ ∈ [0 , , (33) η = φ φ + φ , η ∈ [0 , , (34) v = φ + φ γ + γ , v ∈ [0 , ∞ ) . (35) *3*2 v 0.001 v 5 v 100 C l a ss i c a l C o rr e l a t i o n *1 FIG. 3. (Color online) Classical correlation as a functionof τ = ( γ + γ ) t for a two-qubit system under the non-Markovian PD channel. The initial state is in the X form,with c = 0 . c = 0 . c = 0 . c = 0 .
10, and c = 0 . ǫ = 0 .
92 and η = 0 .
10. The emergencetimes τ E are associated with τ ∗ = 3 . τ ∗ = 4 .
3, and τ ∗ = 7 . Then, we characterize the evolutions given by Eqs. (30)and (31) by observing that the non-Markovian PD pro-cess preserves the X state form. Similarly as we havedone in the Markovian case, we can directly obtain thedynamics of the classical correlations from Eqs. (9)-(13)and from the definition of C G in Eq. (21). We will an-alyze the system in the limit of either fast or slow envi-ronmental fluctuations. The fast limit of environmentalfluctuations occurs when the reservoir fluctuations aremuch faster than the average decay rates of the system,namely, { φ R ′ R } ≫ { γ R } , which implies that the systemexhibits Markovian behavior. Then, from Eq. (35), wetake v ≫
1. On the other hand, when the bath fluc-tuations are much slower than the average decay rate,namely, { φ R ′ R } ≪ { γ R } , the system is in the limit ofslow environmental fluctuations. Then, from Eq. (35),we take v ≪
1. Let us now investigate the emergenceof the pointer basis for the case of the non-MarkovianPD channel, given by Eqs. (30)-(32). In this scenario,the classical correlation can witness the emergence time τ E , which is illustrated in Fig. 3. Moreover, we observethat, for { φ R ′ R } ≪ { γ R } , the classical correlation dis-plays a bi-exponential decay. On the other hand, for { φ R ′ R } ≫ { γ R } , the classical correlation shows a singleexponential decay, such as expected for a Markovian be-havior. In addition, we can observe the emergence of thepointer basis for any v through the sudden transitions,with τ E greater for slower environmental fluctuations. *3*2 C l a ss i c a l C o rr e l a t i o n *1 FIG. 4. (Color online) Classical correlation as a function of τ = ( γ + γ ) t for for a two-qubit system under the non-Markovian PD channel in the limit of slow fluctuations, with v = 0 .
001 and η = 0 .
70. The initial state is in the X form,with c = 0 . c = 0 . c = 0 . c = 0 . c = 0 . τ E are associated with τ ∗ = 1 . τ ∗ =5 . τ ∗ = 7 . By focusing attention on the slow configurational tran-sitions, we show in Fig. 4 that τ E strongly depends on ǫ , i.e., on the ratio of decay rates γ and γ . The short-est emergence time occurs for the central value ǫ = 0 . γ = γ . As we move away from ǫ = 0 .
5, the emergence of the pointer basis is delayed. Inparticular, for the limit cases ǫ = 0 or ǫ = 1, the systemshows a soft decay, with no sudden transition at finite time. IV. CONCLUSIONS
In summary, we have analytically evaluated the trace-distance classical correlations for the case of two-qubitsystems described by X states. In addition, we haveshown the applicability of such correlations to investi-gate the dynamics of open quantum systems through thecharacterization of the pointer basis of an apparatus suf-fering either Markovian or non-Markovian decoherence.Since the non-Markovianity brings a flow of informationfrom the environment back to the system during its evo-lution, the pointer basis has been found to emerge ina delayed time in comparison with the Markovian be-havior. The experimental characterization of such delayin the emergence time can be achieved by a similar ap-proach as used in Refs. [31, 33] for Markovian evolutions.It is also remarkable to observe that, differently fromthe case of Bell-diagonal states, sudden transitions of en-tropic correlations for X states have been conjecturedto display zero measure [42], which may compromise aprecise characterization of the pointer basis. Our geo-metric approach avoids this obstacle, since actual suddenchanges are shown to be typical for general X states. Thismay have further implications in the characterization ofquantum phase transitions through geometric classicalcorrelations.
ACKNOWLEDGMENTS
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