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A statistical portrait of the entanglement decay of two-qubit memories

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A statistical portrait of the entanglement decay of two-qubit memories
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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/232503133 Statistical portrait of the entanglement decay of two-qubit memories  Article   in  Physical Review A · October 2012 DOI: 10.1103/PhysRevA.86.042325 · Source: arXiv CITATIONS 5 READS 18 3 authors , including:K. M. Fonseca-RomeroNational University of Colombia 44   PUBLICATIONS   254   CITATIONS   SEE PROFILE Julián Martínez-RincónUniversity of Rochester 15   PUBLICATIONS   155   CITATIONS   SEE PROFILE All content following this page was uploaded by K. M. Fonseca-Romero on 11 March 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the srcinal documentand are linked to publications on ResearchGate, letting you access and read them immediately.    a  r   X   i  v  :   1   2   1   0 .   5   6   4   5  v   1   [  q  u  a  n   t  -  p   h   ]   2   0   O  c   t   2   0   1   2 A statistical portrait of the entanglement decay of two-qubit memories Karen M. Fonseca-Romero, 1,  ∗  Juli´an Mart´ınez-Rinc´on, 1,2 and Carlos Viviescas 1 1 Universidad Nacional de Colombia - Bogot´ a, Facultad de Ciencias,Departamento de F´ısica, Carrera 30 Calle 45-03, C.P. 111321, Bogot´ a, Colombia  2 Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA (Dated: December 15, 2013)We present a novel approach to the study of entanglement decay, which focuses on collectiveproperties. As an example, we investigate the entanglement decay of a two-qubit system, producedby local identical reservoirs acting on the qubits, for three experimentally and theoretically relevantcases. We study the probability distributions of disentanglement times, a quantity independentof the measure used to quantify entanglement, and the time-dependent probability distribution of concurrence. Analytical results are obtained for initially uniformly distributed pure states. Thecalculation of these probability distributions gives a complete insight on how different decoherencechannels affect the entanglement initially contained in the set of two-qubit pure states. Numericalresults are reported for randomly distributed initial mixed states. Although the paper focusesin Markovian noisy channels, we show that our results also describe non-autonomous and non-Markovian channels. PACS numbers: 03.65.Ud,03.65.Yz,03.67.Mn I. INTRODUCTION. Entanglement is a key resource for quantum informa-tion processing and communication, which can be ma-nipulated, broadcasted, controlled and distributed [1].A better understanding of the effects of decoherence onentanglement could accelerate the development of tech-nological applications of quantum information protocols.Its study under realistic, dissipative conditions, is of fun-damental importance to assess the resilience of quantuminformation processing. The dynamics of quantum coher-ence and entanglement can be strikingly different. Un-der the influence of a noisy environment, coherences usu-ally decay asymptotically in time, while entanglementcan do so in a finite time [2, 3]. Extensive work [4–11], including several experimental demonstrations [12–14], has been devoted to this remarkable phenomenon, some-times called entanglement sudden death (ESD). A com-plex, yet partial picture of the entanglement dynamics,has emerged. Recently, a complete characterization of the dynamics of a two-qubit system in which only oneof the qubits is coupled to a noisy channel [15] has beenpresented and general approaches, based on geometricalarguments [16] and quantum trajectories [17], have been proposed.In this work, adopting a broader perspective, we rendera comprehensive description of the entanglement dynam-ics in two-qubit systems when corrupted by environment-induced decoherence. Our approach relies on a rathernovel tool in the area: the statistical distributions of con-currence and of the ESD times. Obtaining analytic ex-pressions for them is far from being a symple task. Infact, previous analytic work was restricted to averages ∗ kmfonsecar@unal.edu.co [18] and often relied on approximations [19]. Other re- ported explorations of entanglement evolution in bipar-tite [2] and multipartite systems [20], similar in spirit to our approach, are both numerical. Our analytic for-mulation, employed before in a kinematical context [21],allows us to depict a highly detailed image of the entan-glement time evolution.This paper is organized as follows. The dynamical evo-lution of a two-qubit system coupled to local channelsis introduced in section II, where the probability den-sities of the disentanglement times and of concurrenceare defined. We consider three models of decoherencefor a single qubit and assume identical environments foreach qubit. In the next four sections the results for theprobability densities of the disentanglement times and of concurrence are evaluated for the chosen environments,assuming initial pure states uniformly distributed. Theresults for mixed states, a single noisy channel, and non-autonomous systems are briefly presented in sections VII,VIII and IX, respectively. Section X is devoted to the demonstration that, although we concentrate on Marko-vian processes with constant decay rates, our results alsoapply to non-Markovian processes. Some conclusions aredrawn in the last section of this paper. II. DISENTANGLEMENT PROBABILITYDISTRIBUTIONS We investigate the  global   entanglement dynamics of two-qubit systems coupled to local reservoirs (chan-nels), a situation likely to hold for spatially separatedqubits. The qubits are assumed to be quantum mem-ories; they do not follow independent unitary dynam-ics nor do they interact with each other. When-ever the initial system-environment state is separablewith vanishing quantum discord [22, 23], the dynam-  2ical evolution of the state of the two-qubit system isgiven by a completely positive map [24, 25]. Then, ρ ( t ), the system state at the physical time  t , is givenby  ρ ( t ) =   ij  E  (1) i  ⊗ E  (2) j  ρ (0)  E  (1) † i  ⊗ E  (2) † j   =Λ t ( ρ (0)) ,  where the time-dependent Kraus operators E  ( a ) i  =  E  ( a ) i  ( t ), acting on the  a -th qubit, satisfy thetrace-preserving condition   i E  ( a ) † i  E  ( a ) i  =  I ( a ) ,  a  = 1 , 2.These operators can be experimentally determined by aquantum process tomography [26].We chose pairs of identical environments to act on thequbits, and regard the experimentally relevant cases of depolarizing (D), amplitude-damping (AD), and phase-damping (PD) channels. These generally non-Markovianchannels are valid for finite or infinite environments andweak or strong coupling [27]. The Kraus operators forthese noisy environments, in terms of the (reparameter-ized) time  q  , are (0  ≤  q   ≤  1):  E  0 ( q  ) =   1 − 3 q/ 4 I and  E  i ( q  ) =   q/ 4 σ i ,  i  = 1 , 2 , 3 ,  for case D;  E  0  = | 0  0 |  +  √  1 − q  | 1  1 |  and  E  1  =  √  q  | 0  1 |  for case AD;and  E  0  =  √  1 − q  I ,  E  1  =  √  q  | 0  0 | , and  E  2  =  √  q  | 1  1 | for case PD. Here  σ 1 ,σ 2 , and  σ 3  denote, as usual, thethree Pauli matrices. Unless otherwise stated, we focuson the Markovian scenario, with constant decay rates,when  q  ( t ) = 1 − exp( − γt ),  γ   being a positive constant.The motivation behind our selection of channels lies intheir qualitative different long-time behavior,  q   →  1, de-termining the dynamical evolution of entanglement on agross level [16]. All states will suffer ESD if the asymp-totic state belongs to the interior of the set of separablestates  S   (case D). Some states will separate asymptoti-cally if the asymptotic state belongs to  ∂  S  , the borderbetween  S   and  E  , the set of entangled states (case AD).The case PD displays a set of stationary separable states,diagonal in the standard basis, which includes 4 purestates in  ∂  S  .Rather than compute the concurrence (or any otherentanglement measure) for specific initial conditions, wepursue a statistical approach to entanglement dynamicsin open systems. Although randomly chosen normalizedpure states  | ψ   =  ψ 00 | 00  + ψ 01 | 01  + ψ 10 | 10  + ψ 11 | 11  are used for our calculations, other choices might besensible. We choose pure states because they gener-ally offer higher quantum correlations than mixed states,a key point for quantum information protocols. More-over, they are a convenient idealization of the almostpure states that can be produced under experimentalconditions. If minimal prior knowledge about the stateof the system is assumed [28], or invariance under uni-tary transformations is required, it becomes natural touse the uniform (Haar) measure for all possible purestates of the system. Hence, the probability to find astate in a small volume  d 2 ψ  =  ij  dψ ij  dψ ∗ ij  around  ψ  is  p ( ψ ) d 2 ψ  = Γ(4) δ  (1 − ψ | ψ  ) /π 4 d 2 ψ , where Γ( x ) denotes,as usual, the Gamma function.We characterize the disentanglement process throughthe reparameterized separation time probability density,  p ( q  S  ) =    d 2 ψδ  ( q  S   − q  S  ( ψ ))  p ( ψ ) ,  (1)where  q  S  ( ψ ) is the ESD time, i.e., the time at which ρ ( q  ) = Λ q ( | ψ  ψ | ), becomes separable. This distribu-tion is independent of the measure used to quantify theentanglement of the system. However,  p ( q  S  ) may overes-timate entanglement if many states sustain small valuesof entanglement for long times, before undergoing ESD.Therefore, a complete characterization of entanglementdecay also requires its quantification.The global evolution of entanglement is characterizedby using the concurrence probability density,  p ( C  ; q  ) =    d 2 ψδ  ( C   − C  ( q  ; ψ ))  p ( ψ ) ,  (2)where  C  ( q  ; ψ ) is the Wootter’s concurrence [29]of the state  ρ ( q  ) = Λ q ( | ψ  ψ | ). Concurrence isgiven by  C  ( q  ; ψ )  ≡  max { 0 , √  λ 1  −  4 i =2 √  λ i } ,where  λ i  are the eigenvalues of the matrix ρ ( q  )  σ (1)2  ⊗ σ (2)2  ρ ( q  ) ∗  σ (1)2  ⊗ σ (2)2  , being  λ 1  thelargest one. Here  ρ ∗  is the complex conjugate of   ρ  in thestandard basis, and  σ 2  is the second of Pauli matrices.Then, for our chosen ensemble of initial states,  p ( C  ; q  ) dC  is the probability to find a state with concurrence valuein an interval  dC   around  C   at time  q  . In particular, atthe initial time  q   = 0, eq. (2) provides the concurrencedistribution over an ensemble of uniformly distributedpure states,  p ( C  ;0) = 3 C  √  1 − C  2 [21, 30]. The con- tribution of the states with vanishing concurrence tothe distribution  p ( C  ; q  ) can be singled out by writing  p ( C  ; q  ) = ˜  p ( C  ; q  ) +  S  ( q  ) δ  ( C  ), where ˜  p ( C  ; q  ) is theprobability density for strictly positive concurrence and S  ( q  ) = 1  −   C  M  0  dC   ˜  p ( C  ; q  ) is the probability to find aseparable state at a time  q  , where  C  M   =  C  M  ( q  ) is themaximum concurrence at that time.We will show that  p ( C  ; q  ) and  p ( q  S  ) provide a com-plete portrait of the entanglement behavior for all times,enabling us to monitor entanglement degradation, con-centration, and even loss. Despite the dissimilar dynam-ics of the D, AD, and PD channels, in all cases we findanalytical expressions for the disentanglement time andthe concurrence, for initial pure states, and use them toevaluate the probability distributions of separation times  p ( q  s ) and of concurrence  p ( C  ; q  ) as a function of the repa-rameterized time. Our results were checked using purelynumerical methods.  3 III. PROBABILITY DISTRIBUTIONS OF THEDISENTANGLEMENT TIMES. The disentanglement times, q  (D) S   = 1 − 1 /   1 + 2 C  0 ,  (3) q  (AD) S   =  C  0 / 2 | ψ 11 | 2 ,  (4) q  (PD) S   = 1 −   C  20  +  d 2 − C  0  /d  (5)for identical depolarizing, amplitude-damping and phase-damping channels, respectively, are found from the con-dition of separability,  C  ( q  S  ,ψ ) = 0. Here, we haveset  d  = 4 | ψ 00 ψ 01 ψ 10 ψ 11 | 1 / 2 , and  C  0  = 2 | ψ 00 ψ 11  − ψ 01 ψ 10 |  =  C  (0 ,ψ ), and have used superscripts indicat-ing the case/channel we are considering. We use Eq. (1)and these expressions for the separation time to obtainthe following probability distributions of disentanglementtime,  p (D) ( q  S  ) = 3 q  S  (2 − q  S  )   ( q    2 S   − 2 q  S   + 2)(3 q  2 S   − 6 q  S   + 2)4(1 − q  S  ) 7  , p (AD) ( q  S  ) =  q  2 S   − 12 q  S   (1 +  q  2 S  ) 2 +arctan( q  S  )2 q  2 S  +2+  π 8  δ  ( q  S  − 1) , p (PD) ( q  S  ) =    dθdsdrP  J  ( s,r ) δ  ( q  S   − q  (PD) S   ) / 2 π. In the latter case the dependency on the an-gle  θ  comes from the initial concurrence  C  0  = √  2   s 2 +  r 2 + ( s 2 − r 2 )cos( θ ). Here,  P  J   ( s,r ) = 24( s 2 − r 2 ) √  1 −   4 r 2  K   1 − 4 s 2 1 − 4 r 2   stands for the joint probability dis-tribution of   r  and  s , with  s  +  r  = 2 | ψ 00 ψ 11 |  and  s − r  =2 | ψ 01 ψ 10 | . The complete elliptic integral of the first kindand elliptic modulus  k  is denoted by  K  ( k 2 ).The probability distributions of the separation times(Fig. 1) display completely different behaviors. Allpresent a rather conspicuous peak at around  q  S   = 0 . 38, q  S   = 1, and  q  S   = 0 . 59 for cases D, AD, and PD, respec-tively. However, the first is the widest, and the secondis the more narrow and is actually a delta distribution.While in the first case all states become separable before q  S   = 1 − 1 / √  3  ≈  0 . 42, only  6 − π 8  ≈  35 . 73% of the statesexperience ESD in the second case, and a zero-measureset of states separate asymptotically in the third case.The three cases we have considered can be tell apart bytheir disentanglement time probability distributions, andthe fraction of entangled states at a given time can becalculated. However, it is not clear how much entangle-ment remains at a given time. This is especially impor-tant in the AD case, because the entanglement of moststates (64 . 27%) decay asymptotically. In the next threesections, we consider the dynamics of the concurrenceprobability distribution for each case.  0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1Dimensionless Separation Time q S    P  r  o   b  a   b   i   l   i   t  y   d   i  s   t  r   i   b  u   t   i  o  n  p   (  q    S    ) FIG. 1. (Color online) Disentanglement-time probability dis-tributions: solid line (black) for local depolarizing channels(D); dashed line (blue), including a delta function contribu-tion, at  q  s  = 1, for local amplitude damping channels (AD);and dotted line (red) for local phase damping channels (PD). IV. DEPOLARIZING CHANNELS. We start by investigating the case in which each qubitis coupled to a depolarizing channel. As time goes by,the depolarizing channels induce decoherence on the twoqubits and their initial entanglement is lost. The concur-rence at time  q  , C  (D) ( q  ; C  0 ) = max  0 ,C  0 (1 − q  ) 2 −  12 q  (2 − q  )   ,  (6)depends only on (time and) the concurrence of the ini-tial state  C  0 . The loss of entanglement in the system isuniform, and initial maximally entangled states remainas the states with the highest concurrence for all times, C  M  ( q  ) = 1 − 32 q  (2 − q  ). For an ensemble of uniformly dis-tributed initial pure states, the concurrence probabilitydensity at a time  q  ˜  p (D) ( C  ; q  ) = 3 C   +  12 q  (2 − q  )(1 − q  ) 4   1 −  C   +  1   2 q  (2 − q  )(1 − q  ) 2  2 , follows from Eq. (6) and  p ( C  ;0) = 3 C  √  1 − C  2 . In theinset of Fig. 2 we draw ˜  p D ( C  ; q  ) for different values of time,  q  .In Fig. 2, entanglement degradation is apparent. In-deed, due to the loss of entanglement, the whole dis-tribution moves towards zero and the average concur-rence  C  q  =   C  M  0  dC C   ˜  p (D) ( C  ; q  ) decreases. At the sametime entanglement concentrates around its mean valuebecause the standard deviation ( C  2 q  − C  q 2 ) 1 / 2 falls as  q  increases. Finally, entanglement is lost: the probabilityto obtain an entangled state   C  M  0  dC   ˜  p (D) ( C  ; q  ) declineswith time.Entanglement degradation, concentration and loss aredisplayed in all the examples considered in this work,  4  0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5    C  o  n  c  u  r  r  e  n  c  e Dimensionless Reparameterized Time q 0 1 2 3 0 0.2 0.4 0.6 0.8 1 Concurrence C    C  o  n  c  u  r  r  e  n  c  e    d   i  s   t  r   i   b  u   t   i  o  n FIG. 2. (Color online) Identical but independent depolarizingchannels. The solid blue lines represent the average concur-rence plus a standard deviation (darkest line), the averageconcurrence, and the average concurrence minus a standarddeviation (lightest line); maximum concurrence is plotted inred (dashed line), and the value of concurrence at which thedistribution peaks is plotted in green (dotted line). Inset:From right to left, concurrence distributions for  q   = 0 , 0 . 2and 0 . 4. and seem typical for open markovian systems, whoseasymptotic states belong to  S   and  ∂  S  . This asymp-totic behavior srcinates also entanglement concentrationdue to the shrinking average distance between the statesthat remain entangled. Entanglement is lost when statescross the border between entangled and separable states.Thus, the concurrence probability distribution providesa complete picture of the process of entanglement de-cay. However, the most salient features of this processare contained in the evolution of the concurrence maxi-mum, its average, its deviation and the separation-timeprobability distribution. V. AMPLITUDE-DAMPING CHANNELS. In our second example each qubit experiences dissi-pation under the action of an amplitude-damping chan-nel. For an initial two-qubit pure state, the concurrenceevolves in time as [17] C  (AD) ( q  ; C  0 ,ψ 11 ) = max  0 , (1 − q  )  C  0  − 2 | ψ 11 | 2 q    . (7)We notice that the set of initially entangled states sep-arates into two classes: those states for which  C  0  < 2 | ψ 11 | 2 , becoming separable at a finite time; while allthe others states, for which  C  0  >  2 | ψ 11 | 2 , reaching sepa-rability only asymptotically. Hence, contrary to the pre-vious case, the loss of concurrence is not uniform: whencomparing states with the same initial concurrence, thosewith smaller | ψ 11 | are more robust. At a given time  q  , themaximum concurrence  C  M   = 1 − q   corresponds to initialstates of the form ( | 01   + exp( iφ ) | 10  ) / √  2, with arbi-trary phase  φ . Since  C  (AD) ( q  ) depends on both the ini-  0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1    C  o  n  c  u  r  r  e  n  c  e Dimensionless Reparameterized Time q 0 2 4 6 0 0.2 0.4 0.6 0.8 1    C  o  n  c  u  r  r  e  n  c  e    d   i  s   t  r   i   b  u   t   i  o  n Concurrence C FIG. 3. (Color online) Identical but independent amplitude-damping channels. The solid blue lines represent the aver-age concurrence plus a standard deviation (darkest line), theaverage concurrence, and the average concurrence minus astandard deviation (lightest line); maximum concurrence isplotted in red (dashed line), and the value of concurrenceat which the distribution peaks is plotted in green (dottedline). Inset: From right to left, concurrence distributions for q   = 0 , 0 . 4 , 0 . 8 and 0 . 9. tial concurrence and the state component in the  | 11   di-rection, the joint probability distribution  P  J  ( | ψ 11 | 2 ,C  0 )over all two-qubit pure states is calculated first in orderto evaluate the concurrence distribution ˜  p (AD) ( C  ). This joint probability either vanish or is equal to 3 C  0  log((1+   1 − C    20 ) /z ), where  z  = max(1 −   1 − C  20 ,  2 | ψ 11 | 2 ) for2 | ψ 11 | 2 ≤  1+   1 − C    20 . The concurrence probability dis-tribution ˜  p (AD) ( C  ; q  ) =    d | ψ 11 | 2 dC  0  δ  ( f  ) P  J  ( | ψ 11 | 2 ,C  0 )(where  f   =  C  − C  (AD) ( q  ; C  0 ,ψ 11 )) can be calculated an-alytically, but the expression is cumbersome and not il-luminating.The concurrence probability distribution is depicted inthe inset of Figure 3, for some values of time  q  . As in theprevious case the degradation of concurrence (the disen-tanglement) is more important for smaller (larger) times.The separation between these processes is more conspicu-ous here because most states disentangle asymptotically.The asymptotic massive disentanglement is not evidentin the graphic of average concurrence. VI. PHASE-DAMPING CHANNELS. In our last example each qubit evolves under the actionof a dephasing channel, which washes away the coher-ences in the  σ 3  representation. The concurrence at time q   of an initial pure two-qubit state under the influence of 
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