We present a simple and general factorization law for quantum systems shared
by two parties, which describes the time evolution of entanglement upon passage
of either component through an arbitrary noisy channel. The robustness of
entanglement-based quantum information processing protocols is thus easily and
fully characterized by a single quantity.Comment: 4 pages, 5 figure

A Jamiołkowski

Adriano Aragão

Andreas Buchleitner

ARR Carvalho

CF Roos

CH Bennett

Christian Kasztelan

E Joos

F Mintert

Fernando de Melo

J-W Pan

K Życzkowski

M Horodecki

M Schlosshauer

M Żukowski

MA Nielsen

Markus Tiersch

MF Santos

MP Almeida

PJ Dodd

T Yu

Thomas Konrad

W Dür

WH Zurek

WK Wootters

English

arXiv

Crossref

Evolution equation for quantum entanglement

Konrad, T.

de Melo, F.

Tiersch, M.

Kasztelan, C.

Aragão, A.

Buchleitner, A.

MPG.PuRe

A Factorization Law for Entanglement DecayThomas Konrad,1, ∗ Fernando de Melo,2, † Markus Tiersch,2, ‡ ChristianKasztelan,3, § Adriano Aragão,4, 2, ¶ and Andreas Buchleitner2, ∗∗1Quantum Research Group, School of Physics, University of KwaZulu-Natal,Private Bag X54001, Durban 4000, South Africa2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str.38, D-01187 Dresden, Germany3Institut für Theoretische Physik C, RWTH Aachen, D-52056 Aachen, Germany4Instituto de F́ısica, Universidade Federal do Rio de Janeiro,Caixa Postal 68.528, CEP 21945-970, Rio de Janeiro, RJ, BrazilWe present a simple and general factorization law for quantum systems shared by two parties,which describes the time evolution of entanglement upon passage of either component through anarbitrary noisy channel. The robustness of entanglement-based quantum information processingprotocols is thus easily and fully characterized by a single quantity.Whenever we contemplate the potential technologicalapplications of quantum information theory [1], from se-cure quantum communication over quantum teleporta-tion [2] to quantum computation [3], we need to worryabout the unavoidable and detrimental coupling of anysuch quantum device to uncontrolled degrees of freedom– typically lumped together under the label “environ-ment”. Environment coupling induces decoherence [4–6],i.e., it gradually destroys the phase relationship betweenquantum states, and thus their ability to interfere. Incomposite quantum systems, these phase relationships(or “coherences”) are at the origin of strong quantumcorrelations between measurements on distinct systemconstituents – which then are entangled. The promises ofquantum information technology rely on exploring pre-cisely these non-classical correlations.Yet, entanglement is not equivalent to many-particlecoherences: it is an even stronger property, and hardto quantify – all commonly accepted entanglement mea-sures [7, 8] are nonlinear functions of the density matrixwhich describes the state of the composite quantum sys-tem, and in particular the coherences. While an elab-orate theory on the time evolution of quantum statesunder environment coupling is at our hands, virtuallyno general results on entanglement dynamics have beenstated. Hitherto, the time evolution of entanglement al-ways needed to be deduced from the time evolution of thestate [9–18]. In the present Letter, a direct relationshipbetween the initial and final entanglement of an arbitrarybipartite state of two qubits (basic units of quantum in-formation) subject to incoherent dynamics in one sys-tem component is derived, which, as illustrated in Fig. 1,renders the solution of the corresponding state evolutionequation obsolete. Our result can be directly applied toinput/output processes, such as gates used in sequentialquantum computing. Moreover, it allows to infer theevolution of entanglement under certain time-continuousinfluences of the environment, e.g. phase- and amplitudedamping.Let us consider entangled states of qubit pairs, withC(|χ〉) C(ρ)C(1⊗ $|φ+〉〈φ+|)|χ〉 ρ1⊗ $FIG. 1: From state evolution to entanglement dynamics.Hitherto, the time evolution of entanglement C under opensystem dynamics had to be deduced from the time evolutionof the state |χ〉. However, it turns out that knowlegde of theentanglement evolution of the maximally entangled state |φ+〉under the channel 1 ⊗$ suffices to establish a direct mappingof C(|χ〉) onto C(ρ), without detour over the state.one qubit being subject to an arbitrary channel $ – whichmay represent the influence of an environment, of a mea-surement, or of both. In order to illustrate the situation,we consider a source which emits a particle to the left andanother one to the right. Each particle on its own carriesone qubit of quantum information (in general a super-position or mixture of two basis states |0〉 and |1〉). Wetherefore also refer to the particles as “left” and “right”qubit. Let the particles leaving the source be in a purestate |χ〉:|χ〉 = √ω|00〉+√1− ω|11〉 , (1)with 0 ≤ ω ≤ 1, i.e., for values of ω between zero andone the particles are in a coherent superposition of bothbeing in state |0〉 and both being in state |1〉. Any purestate can be written in this form, modulo local unitaryoperations. Since our results are not affected by these lo-cal unitaries, as we will see below, this choice of the pureinitial state does not impose a restriction of generality.2FIG. 2: Entanglement decay scenarios. a) Laboratory sce-nario: one of the qubits of the initial state |χ〉, the “right”one, undergoes the action of a general quantum channel $.b) Dual scenario: the same end result is obtained by inter-changing the role of states and channels: the “left” qubit ofa mixed state ρ$ undergoes the action of the quantum chan-nel $χ. Circles represent sources of entangled states, squaressymbolize channels.We quantify entanglement by concurrence C [19],which implies for the state |χ〉: C(|χ〉) = 2√ω(1− ω).For ω in (1) equal to zero or one, the state’s entangle-ment and hence its concurrence vanishes; ω = 1/2 implies|χ〉 = |φ+〉, one of the maximally entangled Bell states,with maximal concurrence one.Now the right qubit traverses an arbitrary quantumchannel $, as illustrated in Fig. 2a), and we want to de-rive the qubits’ entanglement hereafter. To do so, notethat the qubits’ final state must be the same as in a dualpicture [20], where the roles of initial state and channelare interchanged, as depicted in Fig. 2b). Thus, the two-qubit state |χ〉 is identified with a qubit channel $χ, andthe qubit channel $ with a two-qubit state ρ$; symboli-cally:(1 ⊗ $) |χ〉〈χ|p′=($χ ⊗ 1 ) ρ$p. (2)Here, p′ = Tr[(1 ⊗ $) |χ〉〈χ|] and p = Tr[($χ ⊗ 1 ) ρ$] arethe probabilities for channels $ and $χ to act on the states|χ〉〈χ| and ρ$, respectively. Thus, we also account fornon-trace-preserving channels, where the particle numberis not conserved.We now need to determine $χ and ρ$ explicitely. Forthis purpose, we first remember that Quantum Telepor-tation [2] is a means to transfer the state of one systemto another one, in principle with perfect fidelity. Conse-quently, teleporting the right qubit of the state |χ〉 as-sisted by the maximally entangled state |φ+〉 leaves the!+=1M!+FIG. 3: Quantum Teleportation identity. The teleportationprotocol transfers the incoming state from the left to the out-going state on the right [23]. The procedure consists of acomposite Bell measurement on the incoming qubit and onone qubit of an auxiliary, maximally entangled state |φ+〉.We restrict to the case where the measurement results in aprojection Mφ+ on the state |φ+〉.state |χ〉 invariant. This invariance is depicted in Fig. 3.We therefore obtain the same final state as in the situ-ation we considered so far (Fig. 2a)) – if we replace thesource preparing state |χ〉 by a source which prepares |χ〉followed by a teleportation of the right qubit as shown inFig. 4. Let us now consider the source of the qubit pairin state |φ+〉, which we inserted with the teleportation.The succession of processes influencing the left qubit andthose acting on the right qubit of the pair can be alteredwithout consequences for the final state. For this reasonwe can replace the source producing the state |φ+〉 to-gether with the channel $ acting on the right qubit byyet another source which immediately prepares the stateρ$ := (1 ⊗ $)|φ+〉〈φ+|/p′′ , (3)where p′′ = Tr[(1 ⊗ $)|φ+〉〈φ+|], see Fig. 4. The result-ing scheme in Fig. 5 transfers entanglement between thequbit pairs prepared in states |χ〉 and ρ$ to entanglementbetween the left qubit of the first pair and the right qubitof the second pair. This scheme is called entanglementswapping [2, 21, 22].Finally, we define $χ to be the channel correspondingto the change of the left qubit of ρ$ in Fig. 5, whichincludes a projection Mφ+ of the left qubit of state ρ$and the right qubit of |χ〉 on |φ+〉. Channel $χ can beinterpreted as imperfect teleportation assisted by state|χ〉, leaves the resulting state in general non-normalized,and can be expressed in the particularly simple form:($χ ⊗ 1 ) ρ$ = (M ⊗ 1 ) ρ$(M† ⊗ 1 ) , (4)with M =(√ω|0〉〈0| + √1− ω|1〉〈1|)/√2. The nor-malized final state ($χ ⊗ 1 )ρ$/p is the same as (1 ⊗$)|χ〉〈χ|/p′, as spelled out by (2) and in Fig. 2, butthe entanglement evolution induced by the particularchannel $χ can be deduced more easily, as we will nowdemonstrate. The concurrence C of the final state3M!+"$$!+#FIG. 4: Building in the teleportation identity. The right qubitof |χ〉 undergoes the action of a quantum channel $, afterintermediate teleportation. The maximally entangled state|φ+〉 together with the action of the channel 1 ⊗ $ yield thesource of the mixed state ρ$.!$!"$M#+FIG. 5: Entanglement swapping. Transfer of entanglementbetween the qubit pairs of |χ〉 and ρ$, respectively, to en-tanglement between the outgoing pair of qubits. The BellmeasurementMφ+ together with the source of the entangledstate |χ〉 constitute the quantum channel $χ for the left qubitof state ρ$.ρ′ = (1 ⊗ $) |χ〉〈χ|/p′ is given byC(ρ′) = max{0,√ξ1 −√ξ2 −√ξ3 −√ξ4}, (5)where the ξi are the eigenvalues of the matrix ρ′ · ρ̃′, indecreasing order, with ρ̃′ = (σy⊗σy) · ρ′∗ · (σy ⊗σy), andρ′∗ the complex conjugate of ρ′, in the canonical basis.In order to evaluate this expression, we use relation (2)together with (4). We write explicitly:ρ′ · ρ̃′ = 1p2(M ⊗ 1 ) ρ$ · [MσyM ⊗ σy] · ρ∗$ · [Mσy ⊗ σy] ,(6)where we employed that M = M† = M∗. For invertableM [24], it follows that the eigenvalues of ρ′ · ρ̃′ and ρ$ · ρ̃$are proportional, sincedet[ρ′ · ρ̃′ − ξ1 ]= det[(M ⊗ 1 )−1]det [M ⊗ 1 ] det [ρ′ · ρ̃′ − ξ1 ]= det[(M ⊗ 1 )−1 ρ′ · ρ̃′ (M ⊗ 1 )− ξ1]= 14p2ω(1− ω) det [ρ$ · ρ̃$ − ξ1 ] ,(7)where we used MσyM =√ω(1− ω)σy/2 in order toobtain the last equality. Eq. (5), together with the defi-nitions of ρ$ = (1 ⊗$)|φ+〉〈φ+|/p′′, ρ′ = (1 ⊗$)|χ〉〈χ|/p′,and C(|χ〉) = 2√ω(1− ω), thus lead [25] to our centralresult:C [(1 ⊗ $) |χ〉〈χ|] = C [(1 ⊗ $) |φ+〉〈φ+|] C(|χ〉) (8)– the entanglement reduction under a one-sided noisychannel is independent of the initial state |χ〉 and com-pletely determined by the channel’s action on the maxi-mally entangled state. Thus, if we know the time evolu-tion of the Bell state’s entanglement, we know it for anypure initial state [26]. This result can also be interpretedin terms of entanglement swapping between a pure state|χ〉 and a mixed state ρ$, leading to the final state ρ′, dueto the equivalence of the processes represented in Figs. 2and 5.The factorization law (8) can be generalized for mixedinitial states ρ0, by virtue of the convexity of entan-glement monotones such as concurrence, and given anoptimal pure state decomposition ρ0 =∑j pj |ψj〉〈ψj |,in the sense that the average concurrence over thispure state decomposition is minimal [27]. It then im-mediately follows, by convexity, that C [(1 ⊗ $) ρ0] =C[∑j pj (1 ⊗ $) |ψj〉〈ψj |]≤ ∑j pjC [(1 ⊗ $) |ψj〉〈ψj |],and application of (8) leaves us withC [(1 ⊗ $) ρ0] ≤ C[(1 ⊗ $) |φ+〉〈φ+|] C(ρ0) . (9)This inequality holds for all one-sided channels $, and hasan immediate generalization for local two-sided channels$1 ⊗ $2 = ($1 ⊗ 1 )(1 ⊗ $2):C [($1 ⊗ $2) ρ0] ≤C[($1 ⊗ 1 ) |φ+〉〈φ+|](10)× C [(1 ⊗ $2) |φ+〉〈φ+|]C(ρ0) .The concurrence after passage through a two-sided chan-nel is thus bounded from above, which immediatelyimplies a sufficient criterion for finite-time disentangle-ment [9, 10, 13] of arbitrary initial states, in terms ofthe evolution of the concurrence of the maximally en-tangled state under either one of the one-sided channels(e.g., choose $1 or $2 induced by infinite temperature ordepolarizing environments).Let us finally identify relevant cases when equality in(9) holds. For that purpose, we consider mixed statesthat are obtained after the application of a one-sidedchannel to an arbitrary pure state, ρ0 = (1 ⊗ $)|ψ0〉〈ψ0|.This occurs, for instance, if the qubit originally pre-pared in a pure state suffers amplitude decay, and theresulting mixed state again is subject to decay dynamics.This is tantamount to the concatenation of channels onone side, (1 ⊗ $2) (1 ⊗ $1) |ψ0〉〈ψ0|, what can be lumpedtogether as one channel which combines both actions,(1 ⊗ $2) (1 ⊗ $1) = 1 ⊗$2,1. In a similar vein as for (10),and using the factorization relation (8) for pure states,we deduceC[(1 ⊗ $2,1) |φ+〉〈φ+|] ≤C [(1 ⊗ $2)|φ+〉〈φ+|] (11)× C [(1 ⊗ $1)|φ+〉〈φ+|] .The initial state’s concurrence rescales both sides of theequation by the same amount, and therefore it is omit-ted in above equation. It is now sufficient to investigate4the time dependence of the maximally entangled state’sconcurrence under the concatenated channels (much asfor the evaluation of (8)): if all of these are of the formC(t) = exp(−Γt) (which is the case, e.g., for $1 an am-plitude decay and $2 a dephasing channel), then equalityholds in (9), with $ = $1, ρ0 = (1 ⊗ $2)|ψ0〉〈ψ0|, andC(ρt) = exp(−Γ1t)C(ρ0) (and, equivalently, for the rolesof channels 1 and 2 interchanged).In conclusion, equations (8,9,10,11) provide us with thefirst closed expression for the time evolution of a bipartiteentangled state under general local, single- and two-sidedchannels, without recourse to the time evolution of theunderlying quantum state itself. This is a general re-sult inherited from the Jamio lkowski isomorphism [20],which is here “lifted” from state to entanglement evolu-tion (see Fig. 1), and eases the experimental characteriza-tion of entanglement dynamics under unknown channelsdramatically: instead of exploring the time-dependentaction of the channel on all initial states, it suffices toprobe the entanglement evolution of the maximally en-tangled state alone.We thank J. Audretsch, H.-P. Breuer,L. Davidovich,L. Diosi and C. Viviescas for stimulating discussions.Partial support by DAAD/CAPES under a PROBRALgrant is gratefully acknowledged.∗ Electronic address: konradt@ukzn.ac.za† Electronic address: fmelo@pks.mpg.de‡ Electronic address: mtiersch@pks.mpg.de§ Electronic address: kasztelan@physik.rwth-aachen.de¶ Electronic address: aharagao@if.ufrj.br∗∗ Electronic address: abu@pks.mpg.de[1] C. Bennett and D. DiVincenzo, Nature 404, 247 (2000).[2] C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).[3] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,2000).[4] M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2004).[5] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).[6] E. Joos et al., Decoherence and the Appearance of a Clas-sical World in Quantum Theory (Springer, 2003).[7] F. Mintert et al., Phys. Rep. 415, 207 (2005).[8] R. Horodecki et al., (2007), quant-ph/0702225.[9] K. Życzkowski et al., Phys. Rev. A 65, 012101 (2001).[10] P. J. Dodd and J. J. Halliwell, Phys. Rev. A 69, 052105(2004).[11] C. F. Roos et al., Phys. Rev. Lett. 92, 220402 (2004).[12] A. R. R. Carvalho, F. Mintert, and A. Buchleitner, Phys.Rev. Lett. 93, 230501 (2004).[13] T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403(2006).[14] M. F. Santos et al., Phys. Rev. A 73, 040305 (2006).[15] W. Dür and H.-J. Briegel, Phys. Rev. Lett. 92, 180403(2004).[16] M. P. Almeida et al., Science 316, 579 (2007).[17] A. R. R. Carvalho et al., Phys. Rev. Lett. 98, 190501(2007).[18] M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math.Phys. 15, 629 (2003).[19] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).[20] A. Jamio lkowski, Rep. Math. Phys. 3, 275 (1972).[21] M. Żukowski et al., Phys. Rev. Lett. 71, 4287 (1993).[22] J.-W. Pan et al., Phys. Rev. Lett. 80, 3891 (1998).[23] Here only the measurement of |φ+〉 is used. The remain-ing three possible measurement outcomes would implylocal operations on the outgoing qubit, which can be in-corporated in the action of the channel $.[24] This can be assumed without loss of generality, since oth-erwise, for ω = 0 or ω = 1, and hence MσyM = 0,concurrence vanishes.[25] In this last equation we used that C(αρ) = αC(ρ) forpositive real α and p′ = 4p p′′. The latter can be derivedstraightforwardly, by careful comparison of the individualnormalization factors. In order to compute the entangle-ment of the normalized final state C [(1 ⊗ $) |χ〉〈χ|] hasto be devided by p′.[26] Note that our choice (1) of |χ〉 does not restrict thegenerality of relation (8). An initial state |χ′〉〈χ′| =(Ua ⊗ Ub) |χ〉〈χ| leads to the same result (8) since uni-tary operation Ua does not affect the concurrence ofthe final state and unitary operation Ub can be in-corporated into the channel: C [(1 ⊗ $) |χ′〉〈χ′|] =C [(1 ⊗ $Ub) |χ〉〈χ|] = Cˆ(1 ⊗ $Ub) |φ+〉〈φ+|˜C(|χ〉) =Cˆ(1 ⊗ $) |φ+〉〈φ+|˜C(|χ′〉).[27] C. H. Bennett et al., Phys. Rev. A 54, 3824 (1996).

A Factorization Law for Entanglement Decay

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