Sudden Birth Versus Sudden Death of Entanglement in Multipartite Systems
aa r X i v : . [ qu a n t - ph ] J u l Sudden Birth Versus Sudden Death of Entanglement in Multipartite Systems
C. E. L´opez , G. Romero , F. Lastra , E. Solano , and J. C. Retamal Departamento de F´ısica, Universidad de Santiago de Chile, USACH, Casilla 307 Correo 2 Santiago, Chile Facultad de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago 22, Chile Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany Departamento de Qu´ımica F´ısica, Universidad del Pa´ıs Vasco - Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain (Dated: October 31, 2018)We study the entanglement dynamics of two cavities interacting with independent reservoirs.Expectedly, as the cavity entanglement is depleted, it is transferred to the reservoir degrees offreedom. We find also that when the cavity entanglement suddenly disappear, the reservoir entan-glement suddenly and necessarily appears. Surprisingly, we show that this entanglement suddenbirth can manifest before, simultaneously, or even after entanglement sudden death . Finally, wepresent an explanatory study of other entanglement partitions and of higher dimensional systems.
PACS numbers: 03.65.Yz, 03.65.Ud, 03.67.Mn
Dynamical behavior of entanglement under the actionof the environment is a central issue in quantum informa-tion [1, 2, 3, 4]. Recently, it has been observed that twoqubits affected by uncorrelated reservoirs can experiencedisentanglement in a finite time despite coherence is lostasymptotically [2, 3, 4, 5, 6]. This phenomenon, calledentanglement sudden death (ESD), has recently deserveda great attention [7, 8, 9, 10, 11, 12, 13], and has beenobserved in the lab for entangled photon pairs [14], andatomic ensembles [15].To our knowledge ESD has been studied mainly in re-lation to bipartite systems, while a deeper understand-ing is associated to the question of where does the lostentanglement finally go. This question would be prop-erly answered by enlarging the system to include reser-voir degrees of freedom. Intuitively, we may think thatthe lost entanglement has to be transferred to the reser-voir degrees of freedom. However, is this entanglementswapped continuously? If the bipartite entanglement suf-fers ESD, what can we say about the transferred entan-glement? Should there be a simultaneous entanglementsudden birth (ESB) on reservoir states, or when wouldthis entanglement be created? In this work, we thor-oughly study the entanglement transfer from the bipar-tite system to their independent reservoirs. We showthat ESD of a bipartite system state is intimately linkedto ESB of entanglement between the reservoirs, thoughtheir apparition times follow counterintuitive rules.To illustrate the problem we have chosen the case ofentangled cavity photons being affected by dissipation, asin the case of two modes inside the same dissipative cavityor single modes in two different ones. The present studycould certainly be extended to other physical systemslike matter qubits. First we study the case of qubits fortwo uncoupled (cavity) modes having up to one photon.Then, we extend our treatment to investigate wether ornot the effect is present in higher dimensions (qudits).Since each mode evolves independently, we can learnhow to characterize the evolution of the overall system from the mode-reservoir dynamics. The interaction be-tween a single cavity mode and an N -mode reservoir isdescribed through the Hamiltonianˆ H = ~ ω ˆ a † ˆ a + ~ N X k =1 ω k ˆ b † ˆ b + ~ N X k =1 g k (cid:16) ˆ a ˆ b † k + ˆ b k ˆ a † (cid:17) . (1)Let us consider the case when a cavity mode is containinga single photon and its corresponding reservoir is in thevacuum state, | φ i = | i c ⊗ | ¯ i r , (2)where, | ¯ i r = Q Nk =1 | k i r . It is not difficult to realize thatthe evolution given by (1) leads to the state | φ t i cr = ξ ( t ) | i c | ¯ i r + N X k =1 λ k ( t ) | i c | k i r , (3)where the state | k i r accounts for the reservoir havingone photon in mode k . The amplitude ξ ( t ) converges to ξ ( t ) = exp ( − κt/
2) in the limit of N → ∞ for a reservoirwith a flat spectrum. The right-hand term of the lastequation can be rewritten in terms of a collective stateof the reservoir modes as | φ t i = ξ ( t ) | i c | ¯ i r + χ ( t ) | i c | ¯1 i r . (4)Here, we defined the normalized collective state with oneexcitation in the reservoir as | ¯1 i r = 1 χ ( t ) N X k =1 λ k ( t ) | k i r , (5)and the amplitude χ ( t ) in Eq. (4) converge to the ex-pression χ ( t ) = (1 − exp ( − κt )) / in the large N limit.Described in this way the cavity and reservoir evolve asan effective two-qubit system [16].Let us now study the joint evolution of two qubits withtheir corresponding reservoirs initially in the global state | Φ i = ( α | i c | i c + β | i c | i c ) | ¯ i r | ¯ i r . (6)According to Eq.(4), the evolution of the overall systemwill be given by | Φ t i = α | i c | ¯ i r | i c | ¯ i r + β | φ t i c r | φ t i c r . (7)We observe that the overall state evolves as a four-qubitsystem. By tracing out the reservoir states, the reducedtwo-cavity reduced density matrix reads ρ c c = α + β χ αβξ β ξ χ β ξ χ αβξ β ξ . (8)This reduced state ρ c c has the structure of an X matrixand exhibits ESD for α < β [5, 6]. On the other hand,when tracing out cavity modes we are led to the reducedreservoir state ρ r r = α + β ξ αβχ β χ ξ β ξ χ αβχ β χ , (9)whose structure also corresponds to an X state. Whenreplacing ξ ( t ) ↔ χ ( t ), this state is complementary to thestate in Eq. (8). If ρ c c is exhibiting ESD, what happensthen with ρ r r ? To answer this question we calculate the concurrence [17] for ρ c c , which for the particular stateis given by the simple expression C ( t ) = max { , − λ } , (10)with λ being the negative eigenvalue of the density matrixpartial transpose. For reduced states ρ c c and ρ r r thesenegative eigenvalues are given by λ c c = e − κt (cid:2) β (1 − e − κt ) − | αβ | (cid:3) , (11) λ r r = (1 − e − κt ) (cid:2) β e − κt − | αβ | (cid:3) . (12)Figure 1 shows the evolution of concurrence between thetwo cavities (solid line) and the two reservoirs (dashedline). Despite the entanglement between the two cavi-ties suddenly disappears, sudden birth of entanglementarises between the two reservoirs. Note that the entangle-ment contained initially in the cavity-cavity subsystem istransferred to the bipartite reservoir system. The timefor which ESD and the entanglement sudden birth (ESB)occur can be calculated from Eqs. (11) and (12), lookingfor the time where λ c c becomes positive for ESD andthe time for which λ r r becomes negative for ESB, t ESD = − κ ln (cid:16) − αβ (cid:17) , t ESB = κ ln βα . (13) κ t C on c u rr en c e FIG. 1: Evolution of two-qubit concurrence C c c (solid line)and C r r (dashed line), for the initial state of Eq. (6) with α = 1 / √ β = 2 / √ κ t C on c u rr en c e FIG. 2: Evolution of two-qubit concurrence for different parti-tions: C c c (solid line), C r r (dashed line), C c r (dot-dashedline), C c r (dotted line), for the initial state of Eq. (6) with α = 1 / √
10 and β = 3 / √ From these expressions we learn that ESB occurs for β > α , as is the case for ESD. In other words, the pres-ence of ESD implies necessarily the apparition of ESBand, consequently, asymptotic decay of entanglement be-tween cavities implies an asymptotic birth and growingof entanglement between reservoirs.For the situation in Fig. 1 we have t ESB < t
ESD . How-ever, as can be easily seen from Eqs. (13), when β = 2 α , t ESB = t ESD , that is, ESB and ESD happen simulta-neously. Furthermore, when β > α , ESB occurs afterESD. Although this is clear from Eqs. (13), it is not nec-essarily intuitive. In fact, this condition yields a timewindow where neither the cavity-cavity nor the reservoir-reservoir subsystems have entanglement.To have an idea of how the entanglement is sharedamong the parties, we study the entanglement presentin different partitions. We start considering all bipartitepartitions of two qubits, namely: c ⊗ c , r ⊗ r , c ⊗ r and c ⊗ r , as shown in Fig. 2. In particular for partition κ t (b) (e) (f)(d)(a) (c) FIG. 3: Evolution of entanglement for different partitions: (a) c ⊗ r ⊗ c ⊗ r ; (b) ( c ⊗ r ) ⊗ ( c ⊗ r ); (c) ( c ⊗ c ) ⊗ ( r ⊗ r );(d) ( c ⊗ r ) ⊗ ( c ⊗ r ); (e) c ⊗ ( r ⊗ c ⊗ r ); (f) r ⊗ ( c ⊗ c ⊗ r )for the initial state of Eq. (6) with α = 1 / √
10 and β = 3 / √ c ⊗ r , the entanglement is given by C c r ( t ) = 2 β p (1 − e − κt ) e − κt . (14)In the region where there is no entanglement, that is, C c c = C r r = 0, entanglement between a cavity andits corresponding reservoir C c r ( t ) reaches its maximumvalue. This fact is independent of the initial probabilityamplitudes α and β and occurs for a time t = κ − ln (2)which corresponds also to the time when t ESD = t ESB .Entanglement of other bipartite partitions is shown inFig. 3(b)-(f), and the multipartite entanglement betweenthe four effective qubits in Fig. 3(a). Such entanglementis described by the multipartite concurrence C N [18]. Forpartitions (b)-(f) the entanglement is obtained throughthe square root of the tangle [19] which in the pure two-qubit case coincides with the concurrence. Note that C N has the same value at t = 0 and t → ∞ , showing completeentanglement transfer from cavities to reservoirs.Although entanglement transfer from cavities to reser-voirs is mediated only by the interaction of each cav-ity and its corresponding reservoir, entanglement mayalso flow through other parties. Figure 3(b) shows thatthe partition ( c ⊗ r ) ⊗ ( c ⊗ r ) has constant entangle-ment. However, Fig. 2 shows that entanglement in thetwo-qubit partition c ⊗ r is created along the evolution,implying that entanglement flows also to the noninter-acting partitions. This fact can be visualized as follows:initially the entanglement is contained in the partition c ⊗ c . Then, due to the interaction between cavities andreservoirs, for example c and r , the information aboutthe quantum state of c begins to be mapped into thequantum state of r . Therefore, some of the quantum in-formation contained in the joint-system of the cavities [5]is now present in the joint-system of c ⊗ r , producingentanglement in this partition. It is interesting to investigate whether the features wehave analyzed so far are present for higher dimensionalsystems. For example, we consider the case of qutrit cav-ity states. Following similar steps used to obtain Eq. (4),it is not difficult to calculate the evolution of a singlecavity mode, initially in a two-photon | i c state, inter-acting with the reservoir initially in the vacuum state.The initial state | φ (2)0 i = | i ⊗ | ¯0 i evolves according with | φ (2) t i = ξ ( t ) | i c | ¯ i r + √ ξ ( t ) χ ( t ) | i c | ¯1 i r + ϑ ( t ) | i c | ¯2 i r , (15)where, | ¯2 i r = 1 ϑ ( t ) (cid:18) N X k =1 | λ k ( t ) | | k i (16)+ √ N X k = q =1 λ k ( t ) λ q ( t ) | k . . . q i r (cid:19) , and ϑ ( t ) = p − ξ ( t ) − ξ ( t ) χ ( t ). We can now studythe entanglement when the initial state is given by | Φ i = ( α | i c | i c + β | i c | i c + γ | i c | i c ) ⊗| ¯ i r | ¯ i r . (17)As no entanglement monotone exists for an arbitraryhigher dimensional state, we focus on the analytical ex-pression for a lower bound of entanglement (LBOE)found by Chen, et. al. [20], based on the PPT [21, 22]and realignment criterion [23, 24]. The LBOE monotoneof a bipartite system (A and B) denoted Λ is given byΛ = max (cid:0)(cid:13)(cid:13) ρ T A (cid:13)(cid:13) , k R ( ρ ) k (cid:1) , where the trace norm k·k isdefined as k G k = tr ( GG † ) . The matrix ρ T A is the par-tial transpose with respect to the subsystem A , that is, ρ T A ik,jl = ρ jk,il , and the matrix R ( ρ ) is realignment matrixdefined as R ( ρ ) ij,kl = ρ ik,jl . The values of Λ ranges from1 (separable state) to d (maximally entangled), where d is the dimension of the lower dimensional subsystem.In Fig. (4), the evolution of Λ c c ( t ) and Λ r r ( t ) isshown. We observe that the sudden death of the cavity-cavity entanglement is accompanied by sudden birth ofreservoir-reservoir entanglement as in the two-qubit case.Moreover, the LBOE dynamics between the reservoirs ex-hibits abrupt changes as the LBOE between cavities [13].The times for the ESD and the ESB to appear are t c c ESD = − κ ln − (cid:18) αγ (cid:19) ! , (18) t r r ESB = 12 κ ln γα . (19)As for the two-qubit case, the time for wich ESD andESB occur simultaneously results to be t = κ − ln (2).In general, for a d ⊗ d -dimensional bipartite system,each one coupled to an independent reservoir, and ini-tially prepared in a state of the form | Ψ i = d X k =0 α k | k i c | k i c ⊗ | ¯ i r | ¯ i r , (20) κ t Λ ( t ) FIG. 4: Evolution of two-qutrit LBOE for different partitions:Λ c c (solid line), Λ r r (dashed line), Λ c r (dot-dashed line),Λ c r (dotted line), for the initial state of Eq. (17) with α =1 / √
38, and β = 1 / √
38 and γ = 6 / √ κ t Λ ( t ) FIG. 5: Evolution of the LBOE Λ c c and Λ r r (solid lines)and Λ c r (dashed lines) for initial state of Eq. (20) with d =2 ,
3, and 4. Probability amplitudes α k with k = 0 , , .., d − / qP dk =0 | α k | and α d = 2 d − α . we have numerically observed that the time when t ESD = t ESB does not depend on the dimension of the sys-tems. As can be seen from Fig. 5 the time for which t ESD = t ESD = κ − ln (2). The necessary condition forthese times to be equal is α d /α = 2 d − . Although thiscondition does not depend on the remaining probabilityamplitudes α k with k = 0 , d , the condition α k < α d mustbe satisfied to ensure the presence of ESD and ESB.In conclusion, we have shown that ESD in a bipartitesystem independently coupled to two reservoirs is nec-essarily related to the ESB between the environments.The loss of entanglement is related to the birth of en-tanglement between the reservoirs and other partitions.We analytically demonstrate that ESD and ESB occur attimes depending on the amplitudes of the initial entan-gled state. We found that ESB occur before, together,or even after ESD. In the latter case, when neither cavi-ties nor reservoirs have entanglement, we have analyzed how the entanglement flows to other partitions. Finally,we showed that the simultaneous occurrence of ESD andESB is independent of the system dimension.C.E.L. acknowledges financial support from Fonde-cyt 11070244, DICYT USACH and PBCT-CONICYTPSD54, F.L. from Fondecyt 3085030, G.R. from CONI-CYT grants, J.C.R. from Fondecyt 1070157 and MilenioICM P06-067, E.S. from SFB 631, EU EuroSQIP, andIkerbasque Foundation. [1] T. S. Cubitt, F. Verstraete, W. D¨ur, and J. I. Cirac ,Phys. Rev. Lett. , 037902 (2003); T. S. Cubitt, F.Verstraete, and J. I. Cirac , Phys. Rev. A , 052308(2005).[2] L. Di´osi, Lect. Notes Phys. 622, 157-163 (2003).[3] P. J. Dodd and J. J. Halliwell, Phys. Rev. A, , 052105(2004).[4] A. R. R. Carvalho, F. Mintert, S. Palzer, and A. Buch-leitner, Eur. Phys. J. D , 425-432 (2007).[5] Ting Yu and J. H. Eberly, Phys. Rev. Lett. , 140404(2004); ibid. , 140403 (2006).[6] M.F. Santos, P. Milman, L. Davidovich, and N. Zagury,Phys. Rev. A. , 040305(R), 2006.[7] Anna Jamr´oz, J. Phys. A , 7727 (2006).[8] Z. Ficek and R. Tana´s, Phys. Rev. A , 024304 (2006).[9] M. O. Terra Cunha, New J. Phys. , 237 (2007).[10] A. Al-Qasimi and D. F. V. James, Phys. Rev. A ,012117 (2008); Ting Yu and J. H. Eberly, arXiv:quant-ph/0707.3215; L. Roa, R. Pozo-Gonz´alez, M. Schaefer,and P. Utreras-SM, Phys. Rev. A , 062316 (2007); H.T. Cui, K. Li, and X. X. Yi, arXiv:quant-ph/0612145;F. F. Fanchini and R. d. J. Napolitano, arXiv:quant-ph/0707.4092; A. R. P. Rau, M. Ali, and G. Alber,arXiv:quant-ph/0711.0317.[11] L. Derkacz and L. Jak´obczyk, Phys. Rev. A , 032313(2006).[12] Z. Sun, X. Wang, and C. P. Sun, Phys. Rev. A , 062312(2007).[13] F. Lastra, G. Romero, C. E. L´opez, M. Fran¸ca Santos,and J. C. Retamal, Phys. Rev. A , 062324 (2007).[14] M. P. Almeida, et al ., Sience , 579 (2007).[15] J. Laurat, K.S Choi, H. Deng, C.W. Chou, H.J. Kimble,Phys. Rev. Lett. , 180504 (2007).[16] Note that this approach is different from the case of fourqubits in the purely unitary case. See M. Y¨ona¸c, Ting Yuand J. H. Eberly, J. Phys. B , S45 (2007)[17] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998).[18] A.R.R Carvalho, F. Mintert, and A. Buchleitner, Phys.Rev. Lett. , 230501 (2004).[19] P. Rungta, V. Buzek, C.M. Caves, M. Hillery, and G.J.Milburn, Phys. Rev. A , 042315 (2003).[20] K. Chen, S. Albeverio, S.M. Fei, Phys. Rev. Lett. ,210501 (2005).[21] A. Peres, Phys. Rev. Lett. , 1413 (1996).[22] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett.A223