QUANTUM MARKOVIAN DYNAMICS AND BIPARTITE ENTANGLEMENT

Transcription

1 UNIVERSITÀ DEGLI STUDI DI TRIESTE Sede ammnstratva del Dottorato d Rcerca DIPARTIMENTO DI FISICA TEORICA XXII CICLO DEL DOTTORATO DI RICERCA IN FISICA QUANTUM MARKOVIAN DYNAMICS AND BIPARTITE ENTANGLEMENT Settore scentfco-dscplnare FIS/02 DOTTORANDA: Alexandra M. Lguor COORDINATORE DEL COLLEGIO DEI DOCENTI: Char.mo prof. Gaetano Senatore Unv. Treste FIRMA: TUTORE: Dott. Fabo Benatt Unv. Treste FIRMA: RELATORE: Dott. Fabo Benatt Unv. Treste FIRMA: ANNO ACCADEMICO 2008/2009

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3 Contents Introducton 4 1 Entanglement Compound systems Bpartte systems Entanglement Postvty and complete postvty Defntons Postve maps and separablty crtera Entanglement measures Defntons and propertes Relatve entropy of entanglement Concurrence Open Quantum Systems Reversble and rreversble dynamcs Reduced dynamcs Abstract form of the generators of dynamcal semgroups Master equaton and Markovan approxmatons One- and two-qubt systems Asymptotc states One-qubt case Two-qubt case Determnaton of the nose parameters n a one-dmensonal open quantum system Introducton Determnaton of the Kossakowsk matrx elements Complete postvty Conclusons

4 4 CONTENTS 4 Open quantum systems and entanglement Entanglement generaton n a two-qubt system Asymptotc entanglement Envronment-Induced Bpartte Entanglement Introducton Entanglement generaton between two qubts Entanglement generaton n hgher dmensonal bpartte systems Conclusons Entanglement and entropy rates n open quantum systems Introducton The Reduced Dynamcs Entropy and Entanglement Rates Results Conclusons Summary and Outlook 87 Acknowledgements 89 A Calculaton of transmsson and reflecton coeffcents 91 B Integraton of the master equaton 93 C Soluton of the maxmzaton problem 95 Bblography 98

5 Introducton The focus of ths PhD work s two-fold: on one hand, open quantum systems,.e. systems whose nteractons wth the external envronment cannot be neglected, are studed, and a way of characterzng the bath s found n terms of physcal quanttes of the subsystem mmersed wthn t; on the other hand bpartte entanglement n open quantum systems s analyzed n detal, from the pont of vew both of envronment-nduced entanglement generaton and of the tme-evoluton of entanglement under partcular dsspatve dynamcs together wth the possblty of ts asymptotc persstence. The mportance of the purely quantum phenomenon of entanglement as a physcal resource for performng nformatonal tasks whch would be classcally mpossble [1] has spurred the study of ts dynamcal behavor n many dfferent systems. The tmeevoluton of most of these s reversble and generated by a Hamltonan. Many realstc systems, however, nteract wth the external envronment n a non-neglgble way and thus undergo dsspatve rreversble dynamcs. Therefore, snce the am s to be able to use entanglement as an effcent physcal resource n realstc systems that can be expermentally mplemented, t s mportant that the temporal behavor of ths purely quantum phenomenon should be studed also n systems that are drven by nosy envronments. In standard quantum mechancs the focus s manly upon closed physcal systems,.e. systems whch can be consdered solated from the external envronment and whose reversble tme-evoluton s descrbed by one-parameter groups of untary operators. On the other hand, when a system S nteracts wth an envronment E, t must be consdered as an open quantum system whose tme-evoluton s rreversble and exhbts dsspatve and nosy effects. A standard way of obtanng a manageable dsspatve tme-evoluton of the densty matrx ϱ t descrbng the state of S at tme t s to construct t as the soluton of a Louvlle-type master equaton t ϱ t = L[ϱ t ]. Ths can be done by tracng away the envronment degrees of freedom [2, 3] and by performng a Markovan approxmaton [4, 5],.e. by studyng the evoluton on a slow tme-scale and neglectng fast decayng memory effects. Then the rreversble reduced dynamcs of S s descrbed by one-parameter semgroups of lnear maps obtaned by exponentatng the generator L of Lndblad type [6, 7]: γ t = e tl, t 0, such that γ t+s = γ t γ s = γ s γ t s, t 0 and ϱ t γ t [ϱ].

6 6 Introducton In order to ensure that the reduced dynamcs thus obtaned s physcally consstent the semgroup γ t must be composed of completely postve maps[8]. Ths requrement comes from the possblty of the system of nterest, S, of beng coupled to some so-called "anclla" system, A. Naturally, every lnear map L descrbng a physcal transformaton must preserve the postvty of every state ϱ: f ths were not so, then the system densty matrx could develop some negatve egenvalues, whch would contradct the statstcal nterpretaton of the egenvalues as probabltes [9]. So, n order for L to preserve the postvty of the spectrum of every ϱ, L must be a postve map. Ths, however, s not suffcent, as the system S descrbed by ϱ mght be coupled to a so-called "anclla" system A. If a physcal transformaton, represented by the postve map L, s performed on the system S that s statstcally coupled to the system A whch does not undergo the same transformaton, then t s necessary to consder the tensor product of maps d A L on the composte system A + S, where d A s the dentty on the state space S A of system A. Therefore, n order for L to correctly represent a physcal transformaton, t s not suffcent for L to be postve: the whole tensor product d A L must be postve for any anclla A,.e. the map L must be completely postve [8]. The typcal effect of nose and dsspaton on a system S mmersed n a large envronment E s decoherence; n certan specfc stuatons, however, the envronment E may even create quantum correlatons between the subsystems whch compose S. Ths possblty depends on the form of the Kossakowsk matrx that characterzes the dsspatve part of the generator L. In [10] an nequalty was found, nvolvng the entres of such a matrx whch, f fulflled, s suffcent to ensure that a specfc ntal separable pure state of two qubts gets entangled. Part of ths PhD work s dedcated to provng that the above-mentoned condton s also necessary to create entanglement n an ntally separable state of two qubts only va the nosy evoluton due to the common envronment n whch they are mmersed. On the other hand, another mportant ssue s to understand the behavor of entanglement n an open quantum system evolvng under a dsspatve dynamcs, and t s of partcular nterest to see whether t can persst asymptotcally. Ths queston s also consdered n ths work. The topc of the frst chapter s bpartte entanglement. Frstly, composte systems are descrbed, focusng n partcular on bpartte ones,.e. systems composed of two subsystems, snce these are the systems consdered n ths PhD work. The defnton of entangled and non-entangled separable state s gven both for pure states and for statstcal mxtures. Then some defntons and theorems concernng postvty and complete postvty, whch wll be useful n the followng, are gven. Postve and completely postve maps are thus ntroduced and the use of postve maps as a tool to dstngush entangled states from separable ones s explaned. Fnally, the ssue of quantfyng entanglement

7 Introducton 7 n a gven state s consdered and therefore entanglement measures are defned and descrbed, focusng n partcular on two measures whch wll be used n ths thess, namely on the so-called relatve entropy of entanglement and concurrence. The second chapter s dedcated to open quantum systems. Frstly, the mathematcal character of ther rreversble dynamcs s descrbed. Then the reduced dynamcs of the system S mmersed n an external bath E s consdered: frst the constrants on the form of the generator L of Lndblad type n order to ensure physcal consstency of the dynamcal maps are llustrated; then the master equaton for S s derved. Further, the two most commonly used Markovan approxmatons [4, 5], namely the weak couplng and the sngular couplng lmts are descrbed n some detal. Then, snce one- and two-qubt systems wll be manly dealt wth n ths thess, the form of the reduced dynamcs and of ts dsspatve term s gven explctly for one- and two-qubt systems mmersed n an external bath. Fnally, the asymptotc states of dynamcal semgroups are derved n Secton 2.4 for a partcular dsspatve dynamcs of nterest for ths thess, agan focusng on one- and two-qubt states. The thrd chapter concerns the study of the propertes of open quantum systems and a way of characterzng the acton of the bath n terms of physcal quanttes of the subsystem mmersed wthn t. The system consdered here conssts of an electron that can propagate n a one-dmensonal wre n whch a spn-1/2 mpurty s embedded at a fxed poston. The electron and the mpurty magnetcally nteract, and the whole system s mmersed n an external bath whose nosy effects act only on the spn-1/2 mpurty degrees of freedom. Then, the elements of the Kossakowsk matrx descrbng ths dsspatve dynamcs,.e. the nose parameters, are wrtten n terms of the electron s transmsson and reflecton probabltes, whch can be measured. Moreover, a partcular example of the necessty of complete postvty for physcal consstency s gven, showng that f the dsspatve evoluton s descrbed by a postve but not completely postve map, negatve transmsson probabltes can arse. In the fourth chapter, entanglement n open quantum systems s consdered, n the partcular case of two-qubt systems, whch are manly of nterest n ths thess. As mentoned above, there are some partcular cases when the envronment can create quantum correlatons between the subsystems composng the system S mmersed wthn t. In ths chapter, frstly a suffcent condton for the bath to generate entanglement n an ntally separable state of the two qubts s gven [10]. Then the possblty of asymptotc persstence of entanglement n a two-qubt state subject to the dsspatve dynamcs ntroduced n Secton 2.4 s studed. Interestngly t s found that n some cases the envronmentnduced entanglement can even persst for long tmes n the asymptotc state. The ffth chapter deals wth envronment-nduced bpartte entanglement. Here, the suffcent condton presented n Secton 4.1 s proved to be also necessary to guarantee

8 8 Introducton that entanglement s generated only va the bath n an ntally separable state of two qubts. Further, the above-mentoned suffcent condton s generalzed to bpartte systems of arbtrary dmenson, and explct examples are gven both for the two-qubt and for the arbtrary-dmensonal bpartte case. The sxth chapter concerns the tme-behavor of entanglement under dsspatve dynamcs and the possblty of ts asymptotc persstence. In partcular, the behavor of the varaton n tme of entanglement the so-called entanglement rate s compared to the varaton n tme of entropy the so-called entropy rate. The dea of ths comparson stems from the fact that, on one hand, the formalsm of open quantum systems has been used to descrbe the tendency to thermal equlbrum of a small system n weak nteracton wth a large heat bath at a certan temperature and the man tool n ths thermodynamcal pcture s the quantum relatve entropy [4], whle on the other, the entanglement measure called relatve entropy of entanglement provdes a pseudo-dstance between a state and the closed convex set of separable states [11]. In a prevous paper [12] a conjecture was put forward, and proved n a partcular case, namely that for open quantum systems where the nteracton s due only to the common bath, the entanglement rate s always bounded by the entropy rate. In ths work a more general dsspatve dynamcs ntroduced n Sectons 2.4 and 4.2 s consdered, the master equaton s analytcally solved for a partcular class of ntal states and the explct form of the asymptotc states of the dynamcs s found. Thus the tme-behavor of entanglement for varous ntal states s analyzed and the entanglement and entropy rates are compared. Ths leads to a new conjecture, namely that the one put forward n [12] holds when the asymptotc state s separable but does not when the latter s entangled. Then there s a chapter that brefly summarzes the results of ths PhD work, drawng some conclusons and provdng the outlook for future work. Fnally, there are three Appendces n whch some detals of the more cumbersome calculatons are gven: Appendx A pertans to the work presented n Chapter 3, whle Appendces B and C pertan to that llustrated n Chapter 6.

9 Chapter 1 Entanglement In ths chapter, the quantum property of entanglement wll be analyzed. Snce entanglement represents quantum correlatons wthn a system, compound systems wll be descrbed n the frst secton, focusng manly on bpartte systems,.e. systems composed of two subsystems. Then the defnton of entangled and non-entangled separable state wll be gven both for pure and for mxed states. In the second secton, the focus wll be on ways to dstngush entangled states from separable ones, usng so-called postve maps as a tool for ths. Fnally, n the thrd secton, the queston of how to quantfy the amount of entanglement n a gven state wll be dealt wth, and to ths end "entanglement measures" wll be defned and analyzed. 1.1 Compound systems In general physcal systems of many partcles or wth many degrees of freedom must be dealt wth: these represent compound systems,.e. systems composed of two or more ndependent subsystems. A general quantum system composed of N subsystems S = S 1 +S 2 + +S N s descrbed by the Hlbert space H whch s the tensor product of the Hlbert spaces of the subsystems: H = H 1 H 2 H N [9]. The states of such systems can ether be pure state projectors ψ ψ on vector states ψ H or mxed states,.e. densty matrces arsng from convex combnatons of projectors ϱ := λ ψ ψ, wth 0 λ 1, λ = 1, ψ H. The space of all states of a system S wll be denoted by S d, wth d the dmenson of the Hlbert space H, and t s a closed, convex set [9].

10 10 Entanglement Bpartte systems In the followng only bpartte systems S = S A + S B wth dscrete varables wll be consdered: these are descrbed by the Hlbert space H = H A H B, wth H A C d A, H B C d B, the Hlbert space of the frst, respectvely second, subsystem. Therefore the total Hlbert space of the bpartte system wll be H C d A C d B wth dmenson d = d A d B, snce d A and d B are the dmensons of H A and H B respectvely. An arbtrary vector ψ C d A C d B can be wrtten [9] as follows: ψ =,j c j φ A φ B j, where { φ A } wth = 1,..., d A and { φ B j } wth j = 1,..., d B are the orthonormal bases of C d A and C d B respectvely, and c j C. Therefore the total Hlbert space H wll be generated by the bass { φ A } { φb j }; moreover, gven two operators O A, O B actng on S A, respectvely S B, ther tensor product on S s defned by the way t acts on the total bass: O A O B φ A φ B j = O A φ A O B φ B j. Formally, a local operator actng only on the subsystem S A s wrtten O A I B, wth I B the dentty operator on S B, and analogously for a local operator O B actng only on S B. The space of states,.e. the convex set of densty matrces, of the bpartte system S A +S B wll be ndcated as S da d B, and ϱ A, ϱ B wll be the statstcal operators pertanng to S A, respectvely S B. The densty matrx of only one of the two subsystems,.e. the reduced densty matrx, s obtaned as the partal trace wth respect to the Hlbert space of the other subsystem, namely ϱ A = Tr B [ϱ AB ] and ϱ B = Tr A [ϱ AB ], wth ϱ AB S da d B ; t follows that that the mean values of local operators read Tr AB [ϱ AB O A I B ] = Tr A [ϱ A O A ], Tr AB [ϱ AB I A O B ] = Tr B [ϱ B O B ]. In order to study the correlatons that can be present n a bpartte system, t s useful to consder the Schmdt decomposton [9, 13] of the bpartte state. Proposton 1 Gven a state vector ψ AB C d A C d B, ts Schmdt decomposton s ψ AB = mnd A,d B =1 λ A ra r B, wth { r A } and { rb }, = 1,..., mnd A, d B, orthonormal sets for C d A, C d B and mnd A, d B the mnmum between d A and d B. respectvely, It s evdent from ths decomposton that the two partal traces ϱ A = λa ra ra and ϱ B = λa rb rb have the same egenvalues λa ; moreover, the non-zero egenvalues have the same multplcty.

11 Postvty and complete postvty Entanglement Frstly, let the bpartte system S = S A + S B be n a pure state descrbed by the state vector ψ C d A C d B. Defnton 1 The state ψ C d A C d B s separable f and only f there exst two vectors ψ A C d A and ψ B C d B such that ψ = ψ A ψ B [14]. Otherwse the state s sad to be entangled. For a pure separable state the partal traces are projectors. Indeed, gven ψ sep = ψ A ψ B, t follows: ϱ A = Tr B ψ sep ψ sep = Tr B ψ A ψ A ψ B ψ B = ψ A ψ A and analogously for ϱ B = ψ B ψ B. From the Schmdt decomposton t can be easly seen that ψ AB s separable f and only f there exsts only one coeffcent r j = 1, r = 0 j. If nstead there are more coeffcents whch are non-zero, then the state s entangled. In partcular, f the coeffcents are all the same, then the state s sad to be maxmally entangled. Further, the dstncton between separable and entangled states can be extended to statstcal mxtures as follows. Defnton 2 The densty matrx ϱ AB S da d B s sad to be separable entangled f and only f t can cannot be wrtten as follows [15]: ϱ AB = n p ϱ A ϱ B 1.1 =1 for some n N, wth weghts p 0 and such that p = 1, where ϱ A, ϱb densty matrces n S da, S db respectvely. are sets of 1.2 Postvty and complete postvty Defntons In ths secton some general defntons wll be gven, whch wll be useful n the followng. Defnton 3 Let M d C := M d be the algebra of d d complex matrces. A Hermtan operator X = X M d s sad to be postve X 0 f and only f Ψ X Ψ 0 Ψ C d [9, 13]. Defnton 4 A lnear map L : M m M n s sad to be postve f and only f LX 0 X 0 [16].

12 12 Entanglement Defnton 5 A lnear map L : M m M n s sad to be k-postve f and only f d k L : M k M m M k M n s postve, where d k s the dentty on M k [17]. Defnton 6 A lnear map L : M m M n s sad to be completely postve f and only f L s k-postve k N [17]. The tensor product of a postve map L : M m M n wth the dentty d k can be performed both as d k L : M k M m M k M n and as L d k : M m M k M n M k, and the same Defnton 6 holds for complete postvty. The followng Theorem follows from [18]: Theorem 1 The map L : M m M n s completely postve f and only f the tensor product d n L s postve for any n N. Defnton 7 A lnear map L : M m M n s sad to be k-copostve f and only f T k L : M k M m M k M n s postve, where T k ndcates the transposton on M k [17]. Defnton 8 A lnear map L : M m M n s sad to be completely copostve f and only f L s k-copostve k N [17]. Completely postve CP maps are fully characterzed by the followng Theorem 2 Kraus-Stnesprng Representaton [19, 20] The map L : M m M n s completely postve f and only L = V V, wth V M n m C := M n m. Moreover, L s such that: 1. the trace s preserved f and only f V V = I; 2. the trace s non-ncreasng f and only f V V I; 3. the dentty s preserved f and only f V V = I. Note that the Kraus-Stnesprng representaton for a gven completely postve map s not unque [19, 20]. Whle completely postve maps can be fully determned by Theorem 2, there s stll no complete characterzaton for generc postve maps. The followng Defnton, however, helps n the descrpton of some postve maps.

13 Postvty and complete postvty 13 Defnton 9 From [17, 21] a postve map L s sad to be decomposable not decomposable f and only f t can cannot be wrtten as the sum of a completely postve map and a completely copostve map,.e. L = Λ 1 CP + Λ 2 CP T where Λ 1 CP and Λ2 CP are completely postve maps and T s the transposton. The followng characterzaton of postve maps n low dmensons holds [17, 21]: Theorem 3 All postve maps L : M 2 M 2, M 2 M 3, M 3 M 2 are decomposable. If, nstead, one of the subspaces of the bpartte system has hgher dmenson, then there exst postve maps whch are not decomposable [17, 21] Postve maps and separablty crtera In the prevous secton general lnear maps L actng on postve operators 0 X M d, wth M d C := M d the algebra of d d complex matrces, were consdered. In the followng, lnear maps actng on the states 0 ϱ S d, wth S d the state space, wll be dealt wth. Therefore the followng Defnton s useful: Defnton 10 Gven a lnear map L : M m M n, ts dual map L : M n M m s defned as Tr n ϱl[x] = Tr m L [ϱ]x X M m, ϱ S n [22]. Every lnear map L descrbng a physcal transformaton must preserve the postvty of every state ϱ: f ths were not so, then the system state could develop some negatve egenvalues, whch would contradct the statstcal nterpretaton of ts egenvalues as probabltes [9]. In order for L to preserve the postvty of the spectrum of every ϱ, L must be a postve map. Ths, however, s not suffcent, as the system S d descrbed by ϱ mght be coupled to a so-called "anclla" system S n. If a physcal transformaton, represented by the postve map L, s performed on the system S d whch s statstcally coupled to the system S n, then t s necessary to consder the tensor product of maps d n L on the composte system S n + S d, where d n s the dentty on the state space S n of system S n. Therefore, n order for L to correctly represent a physcal transformaton, t s not suffcent for L to be postve: the whole tensor product d n L must be postve for any n,.e. the map L must be completely postve [8]. The necessty of complete postvty s due to the exstence of entangled states of the composte system S n +S d [16]. If all of the physcal states of a bpartte system were separable

14 14 Entanglement as n Defnton 2, then the postvty of map L would suffce. Indeed, from Defnton 2, we know that f ϱ 0 s separable, then ϱ ϱ nd = p ϱ n ϱd, and thus t follows that d n L[ϱ] = p d n [ϱ n ] L[ϱ d ] = p ϱ n L[ϱ d ] 0. If, nstead, the state of the bpartte system s entangled, ϱ ϱ ent, then t cannot be wrtten as n 1.1 and therefore, n order to have d n L[ϱ ent ] 0 for all n, the tensor product d n L must be postve,.e. the map L must be completely postve. However, although postve maps do not descrbe consstent physcal transformatons, they represent an mportant tool n the dentfcaton of entangled states [23], as wll be shown n the Theorems enuncated n ths secton. In the followng, to smplfy the notaton, bpartte states ϱ S d d wll be consdered, but all the results hold also n the general case of bpartte states ϱ AB S da d B. Theorem 4 A state ϱ S d d s entangled f and only f there exsts a postve map L on S d such that d d L [ϱ] s non-postve. The smplest example of postve but not completely postve map s the transposton. The followng one-way separablty crteron comes from [24]: Theorem 5 Peres crteron If a state ϱ S d d s separable, then ϱ T B := d d T B [ϱ] 0, where T B represents the transposton on the second subsystem. Therefore, f ϱ T B := d d T B [ϱ] s not postve, then the state ϱ s entangled: nonpostvty under partal transposton mples entanglement; the opposte, however, s not true n general. A general two-way separablty crteron through postve but not completely postve maps comes from [16]: Theorem 6 Horodeck crteron A state ϱ S d d s separable f and only f d d L[ϱ] 0 for all maps L : M d M d whch are postve but not completely postve. From a result obtaned n [16] t follows: Corollary f and only f A state ϱ AB S da d B wth d A d B 6.e. 2 2, 2 3, 3 2 s separable ϱ T B AB := d A T B [ϱ AB ] 0. The above Corollary follows from Theorems 3 and 6,.e. from the fact that for d A d B 6 all postve maps are decomposable: therefore, n ths case, the partal transposton detects all entangled bpartte states and the above Corollary offers a necessary and suffcent crteron for separablty. Instead, for d A d B > 6 not all postve maps are decomposable [17, 21] and ths mples that n ths case postvty under partal transposton s only a necessary, but not suffcent, condton for the separablty of a bpartte state ϱ AB S da d B [16].

15 Entanglement measures Entanglement measures Defntons and propertes So far the focus has been on the queston of how to qualfy an entangled state versus a separable state. Another mportant, though dffcult to answer, queston s how to quantfy the amount of entanglement wthn a gven state. There are varous ways n whch an entanglement measure can be constructed or defned, and these lead to dfferent "famles" of entanglement measures, namely [25]: axomatc measures convex-roof measures operatonal measures. In the followng, the dfferent approaches to the constructon of entanglement measures wll be brefly analyzed, focusng on bpartte systems; then two entanglement measures, whch wll be of partcular nterest n the rest of ths work, wll be consdered n more detal. Axomatc approach In the axomatc approach entanglement measures Eϱ are constructed by allowng any functon of state to be a measure, provded t satsfes the followng postulates [11, 26 28]: 1. For any separable state σ the entanglement measure should be zero,.e. Eσ = 0; 2. For any state ϱ and any local untary transformaton,.e. for any untary transformaton of the form U A U B, the amount of entanglement remans unchanged: Eϱ = EU A U B ϱu A U B ; 3. Monotoncty under LOCC: entanglement cannot ncrease under local operatons and classcal communcaton. Namely, for any LOCC operaton,.e. for any operaton of the form Λ Λ A Λ B aded only wth classcal communcaton [9], t must hold true that EΛ[ϱ] Eϱ; 4. Contnuty: n the lmt of vanshng dstance between two densty matrces the dfference between ther entanglement should tend to zero,.e. Eϱ 1 Eϱ 2 0 for ϱ 1 ϱ 2 0, wth ϱ 1 ϱ 2 := Tr ϱ 1 ϱ 2 2 ; 5. Addtvty: a number n of dentcal copes of the state ϱ should contan n tmes the entanglement of one copy,.e. Eϱ n = neϱ;

16 16 Entanglement 6. Subaddtvty: the entanglement of the tensor product of two states ϱ 1 and ϱ 2, wth ϱ 1 ϱ 2, should not be larger than the sum of the entanglement of each of the states,.e. Eϱ 1 ϱ 2 Eϱ 1 + Eϱ 2 ; 7. Convexty: the entanglement measure should be a convex functon,.e. Eλϱ λϱ 2 λeϱ λeϱ 2 for 0 < λ < 1. Regardng the frst condton, n general t s not possble to mpose that Eϱ = 0 for a generc state ϱ mply that the latter s separable: ndeed, there exst some entanglement measures whch vansh on partcular classes of entangled states see, e.g., the negatvty defned n the followng. The thrd condton s the one that really restrcts the class of possble entanglement measures and was proposed as the most mportant postulate for the latter [26], although usually t s also the most dffcult one to prove [11]. Most of the known entanglement measures, however, satsfy the monotoncty condton on average [25], namely p Eσ Eϱ, where {p, σ } s the ensemble obtaned from the state ϱ by means of LOCC. Fnally, t should be noted that the monotoncty condton can also be gven n more coarse graned" terms as monotoncty under SLOCC stochastc LOCC [25, 29, 30],.e. n terms of operatons that transform a state ϱ nto a new state σ = Λ[ϱ] TrΛ[ϱ] wth some nonzero probablty TrΛ[ϱ] < 1 1. For pure bpartte states t s rather smple to fnd entanglement measures: ndeed, snce there are no classcal probablstc correlatons contaned n pure states, any untarly nvarant, concave functon of the reduced densty matrx defnes one. The most mportant entanglement measure for pure states whch satsfes the above condtons s the entropy of entanglement, and ths was proved to be the unque entanglement measure for pure states [11]. The defnton of entropy of entanglement s based on that of von Neumann entropy, whch for a generc state ϱ reads S vn ϱ := Tr[ϱ ln ϱ] = d r ln r, where r are the egenvalues of ϱ. The von Neumann entropy measures the amount of uncertanty about the state ϱ and t s zero f and only f the state s pure. The entropy of entanglement s defned as the von Neumann entropy of the reduced operator ϱ A := Tr B [ϱ AB ] ϱ B := Tr A [ϱ AB ] of the bpartte state ϱ AB [11, 27],.e. E vn ϱ AB = S vn ϱ A = Tr[ϱ A ln ϱ A ] = S vn ϱ B = Tr[ϱ B ln ϱ B ] The case of LOCC corresponds to TrΛ[ϱ] = 1. =1

17 Entanglement measures 17 Therefore the property of a pure bpartte state ϱ AB of beng entangled or not s related to the mxedness of ts reduced operator ϱ A ϱ B : from 1.2 t turns out that the pure state ϱ AB s separable f and only f the reduced operator ϱ A ϱ B s also pure, snce n ths case the von Neumann entropy of the latter s zero 2. In the case of mxed states, however, the entropy of entanglement cannot dstngush between classcal and quantum correlatons [27] and therefore fals to be a good entanglement measure. For mxed states, a class of axomatc entanglement measures s defned, based on the natural ntuton that the closer the state s to a separable state, the less entangled t s. Therefore, these entanglement measures are bult as the mnmum dstance D between the gven state ϱ and the set of separable states S sep,.e. E D,Ssep ϱ = nf σ S sep Dϱ, σ. 1.3 In [32] t was shown that f one takes as dstance D n 1.3 the Bures metrc 3 or the quantum relatve entropy Sϱ σ := Tr[σ log σ σ log ϱ], 1.4 whch s well defned f Suppσ Suppϱ, then 1.3 ndeed satsfes the condtons of beng zero for separable states, untarly nvarant, monotone under LOCC and convex, and s thus a good entanglement measure 4. The measure based on the quantum relatve entropy 1.4 s the so-called relatve entropy of entanglement, and t wll be consdered n more detal n Secton Moreover, the quantum relatve entropy s very mportant also n systems whch are n contact wth large heat baths snce n ths case, as shown n Secton 6.3, ths quantty s related to the free energy of the system. Ths concept, however, wll be dscussed more precsely n Chapter 6, after havng descrbed the dynamcs of systems n contact wth external baths n Chapters 2 and 4. Convex roof approach For mxed states the stuaton s much more nvolved, because there are both classcal and quantum correlatons that must be dstngushed between each other through an entanglement measure. Therefore, the generalzaton of a pure state entanglement measure to a mxed state entanglement measure s by no means straghtforward. A proper 2 In the case of mxed states, only some partal results exst for the relaton between the propertes of entanglement of a bpartte state and the mxedness of ts reduced operators see, e.g., [31]. 3 The Bures metrc s defned as B 2 := 2 2 p F ϱ, σ, wth F ϱ, σ = [Tr ϱσ ϱ 1/2 ] 2 the fdelty [33, 34]. 4 There exst some nterestng results concernng the possblty of ntroducng geometrc quantfcatons of entanglement n terms of relatve entropes also for partcular pure bpartte states see, e.g., [35] for dscrete systems and [36] for contnuous varables systems.

18 18 Entanglement method of obtanng entanglement measures for mxed states s the so-called convex roof approach [37], n whch one starts by mposng a measure E on pure states and then extends t to mxed states n the followng way. Any mxed state ϱ can be expressed as a convex sum of pure states ψ : ϱ = p ψ ψ, wth p 0, p = 1. Ths decomposton, however, s not unque and dfferent decompostons n general lead to dfferent values for a gven entanglement measure. Therefore, a proper, unambguous generalzaton of a pure state entanglement measure conssts n takng the nfmum over all decompostons nto pure states,.e. the so-called convex roof : Eϱ = nf p Eψ, p 0, p = 1. The frst entanglement measure to be constructed n ths way was the so-called entanglement of formaton [26], whch s defned as the averaged von Neumann entropy of the reduced densty matrces ϱ,red of the pure states, mnmzed over all possble decompostons,.e. E F ϱ = nf p Sϱ,red. 1.5 It can be easly checked that ths entanglement measure s convex and t has been proved to be monotone under LOCC [26], but t has recently been shown [38] not to be addtve 5, so t cannot be strctly consdered an entanglement measure n the axomatc sense. For generc two-qubt states an explct expresson was found whch leads to an easly computable formula of the entanglement of formaton n ths partcular case [39, 40]: the entanglement measure thus obtaned s the so-called concurrence, whch wll be descrbed n detal n Secton Operatonal approach Ths method of constructng entanglement measures s connected to the dea of quantfyng entanglement wth respect to ts "usefulness" n terms of communcaton [23, 26]. The two man entanglement measures constructed n ths way are the followng [25, 28]: Dstllable entanglement E D, whch quantfes the amount of maxmal entanglement that can be extracted from a gven entangled state,.e. the rato of maxmally entangled output states Φ + 2 over the needed nput states ϱ. Namely, startng from n copes of the gven state ϱ and havng appled an LOCC operaton Λ, one obtans the fnal state σ n, whch s requred to approach the desred maxmally entangled state Φ + 2 m n = Φ + 2 m n for large n. If ths s mpossble, then E D = 0; otherwse the m rate of dstllaton s gven by R D := lm n n n and the dstllable entanglement s E D ϱ = sup{r : lm nf n Λ Λϱ n Φ + 2 rn = 0}, 5 It has been recently proved [38] that the entanglement of formaton E F does not satsfy the addtvty request,.e. E F ϱ ϱ < 2E F ϱ.

19 Entanglement measures 19 where s the trace norm. Entanglement cost E C, whch s dual to E D,.e. t measures how many maxmally entangled states are needed n order to create an entangled state. In other terms, t quantfes the rato of the number of maxmally entangled nput states Φ + 2 over the produced output states ϱ, mnmzed over all LOCC operatons,.e. Some other entanglement measures E C ϱ = nf{r : lm nf n Λ Λϱ n Φ + 2 rn = 0}. There exst many more entanglement measures and n ths subsecton a few of them, whch for dfferent reasons present an nterest, wll be brefly descrbed. Negatvty: ths smple computable measure was ntroduced n [41] and s defned as the sum of the negatve egenvalues of the partal transpose of the gven state ϱ,.e. N ϱ = λ<0 λ. Further, the logarthmc negatvty E Nϱ := log N ϱ+1 2 was shown to be an upper bound for dstllable entanglement [42]. It must be noted, however, that ths entanglement measure vanshes for a partcular class of entangled states [41], namely for PPT entangled states see, e.g., [43]: ndeed, these are entangled states whch reman postve under the partal transposton operaton, and therefore ther partal transpose has no negatve egenvalues, leadng to a vanshng negatvty. Nevertheless, ths measure and ts logarthmc verson are very commonly used quanttes see, e.g., [44 46] snce they have the major advantage that they can be computed straghtforwardly, whle many other entanglement measures cannot. Geometrc measure of entanglement: ths entanglement measure s based on the dea of quantfyng the degree to whch a pure quantum state s entangled n terms of ts dstance or angle to the nearest unentangled state. The geometrc measure of entanglement was frst ntroduced n the settng of bpartte pure states [47] and then generalzed to the multpartte settng [48]. Note that the geometrc measure of entanglement dffers from the measures proposed n [27] based on the mnmal dstance between the entangled mxed state and the set of separable mxed states, as the former s frst defned as the mnmal dstance between the entangled pure state and the set of separable pure states, and then t s extended to mxed states by convex roof constructon. For an N-partte pure state ψ the geometrc measure of entanglement as a quantfer of global mult-partte entanglement 6 s defned as [49, 50]: E G ψ := 1 max φ φ ψ 2, 6 A more detaled and nterestng classfcaton of the geometrc measure of entanglement as relatve.e. partton-dependent and absolute.e. partton-ndependent n the multpartte settng, whch s a bt beyond ths thess, s gven n [49].

20 20 Entanglement where the maxmum s taken wth respect to all pure states that are fully factorzed,.e. φ = φ 1 φ N, wth φ j, j = 1,..., N, the sngle-qubt pure states. Then the geometrc measure of entanglement s extended to an N-partte mxed state ϱ by convex roof as follows [50]: Eϱ := mn p,ψ p E G ψ, wth the decomposton nto pure states ϱ = p ψ ψ. Rans bound: Rans [51] combned the two concepts of relatve entropy of entanglement and negatvty to defne ths measure as E R ϱ = nf σ Sϱ σ + σ Γ, where the nfmum s taken over the set of all states, Sϱ σ s the relatve entropy, σ Γ s the partal transpose of state σ and s the trace norm. Apart from beng explctly computable when the relatve entropy and the negatvty are, ths entanglement measure s also the best known upper bound on dstllable entanglement Relatve entropy of entanglement The defnton of relatve entropy of entanglement s based on dstngushablty and geometrcal dstance [11]: the man dea s to compare a gven quantum state ϱ of a bpartte system wth separable states σ S sep, and then fnd the separable state that s closest to ϱ. Havng taken the relatve entropy Sϱ σ := Tr[σ log σ σ log ϱ] as the dstance n 1.3, the defnton of the relatve entropy of entanglement s E D,Ssep ϱ = where S sep represents the set of separable states. nf Tr[σ ln σ σ ln ϱ], 1.6 σ S sep Although the relatve entropy Sϱ σ s not really a dstance n the mathematcal sense because t s not symmetrc, nevertheless 1.6 defnes a good entanglement measure that satsfes all the condtons lsted n the prevous secton 7. Moreover, for pure states the relatve entropy of entanglement reduces to the entropy of entanglement [27], whch s a satsfyng property Concurrence Ths entanglement measure was defned frstly for all mxed states of two qubts havng no more than two non-zero egenvalues [39] and then generalzed to arbtrary states 7 The addtvty has only been confrmed numercally but all other propertes have been proved analytcally [11, 27]

21 Entanglement measures 21 of two qubts [40]. As mentoned n Secton 1.3.1, the advantage of ths entanglement measure les n the fact that t s an explct formulaton of the entanglement of formaton for two-qubt states and leads to an easly computable formula of the latter, as wll be shown here. Ths formula for entanglement makes use of the so-called spn-flp transformaton, whch for a pure state of a sngle qubt ψ leads to ψ := σ 2 ψ, wth ψ the complex conjugate of ψ and σ 2 σ y the Paul matrx, whle for a mxed state of two qubts ϱ t leads to wth ϱ the complex conjugate of ϱ. ϱ := σ 2 σ 2 ϱ σ 2 σ 2, In [39, 40] t was shown that the entanglement of a pure two-qubt state can be wrtten as where the concurrence C s defned as and the functon E s gven by Eψ = ECψ, Cψ = ψ ψ, 1.7 EC = H C 2, where Hx := x log x 1 x log1 x s the bnary entropy. For pure two-qubt states the concurrence can also be wrtten explctly as Cψ = 21 Tr[ϱ 2 red ], 1.9 wth ϱ red the reduced state. For two qubts ths leads to Cψ = 2a 1 a 2, where a 1, a 2 are the Schmdt coeffcents of state ψ. Another smple explct expresson for the concurrence of a pure two-qubt state s n terms of the coeffcents of the state wrtten n the standard computatonal bass { 0, 1 }: for ψ = a a a a Cψ = 2 a 00 a 11 a 01 a 10, 1.10 Further, n [40] t was shown that the entanglement of formaton for an arbtrary mxed state of two qubts can be wrtten as Eϱ = ECϱ,

22 22 Entanglement where E s the functon n 1.8 and the concurrence s defned as Cϱ = max{0, λ 1 λ 2 λ 3 λ 4 }, 1.11 wth λ, = 1,..., 4, the egenvalues of the Hermtan matrx R ϱ ϱ ϱ, taken n decreasng order. Alternatvely, the real non-negatve numbers λ can be seen as the square roots of the egenvalues of the non-hermtan matrx ϱ ϱ, taken n decreasng order. The mportance of ths entanglement measure les not only n the fact that t leads to an explct and easly computable formula for the entanglement of formaton of arbtrary two-qubt states, but also n the fact that expresson 1.9 can be generalzed to defne concurrence n hgher dmensons.

23 Chapter 2 Open Quantum Systems In ths Chapter, open quantum systems,.e. systems whose nteracton wth an external envronment cannot be neglected, wll be consdered, and ther dynamcs wll be descrbed n some detal, followng especally [5]. In partcular, snce n Chapters 3, 5 and 6 one- and two-qubt systems are manly studed, n Secton 2.2.3, the master equaton and the dsspatve term for one- and two-qubt systems mmersed n an external bath wll be gven explctly. Then, n Secton 2.3, the dervaton of the asymptotc states of a dsspatve dynamcs wll be gven, agan focusng manly on partcular one- and two-qubt cases of nterest n ths thess. 2.1 Reversble and rreversble dynamcs In standard quantum mechancs the focus s manly upon closed physcal systems,.e. systems whch can be consdered solated from the external envronment. Consderng a closed system of fnte dmenson n n a pure state ψ t, ts dynamcs s determned by a Hamltonan operator H M n C through the Schrödnger equaton settng = 1: ψ t t = H ψ t. 2.1 For mxtures ϱ t, ths leads to the so-called Louvlle-von Neumann equaton on the state space S whose soluton, wth ntal condton ϱ t=0 = ϱ, s ϱ t t = [H, ϱ t], 2.2 ϱ t = U t ϱu t, U t = e Ht. 2.3 Havng denoted the dynamcal map 2.3 by ϱ U t [ϱ] := ϱ t, and the lnear acton of

24 24 Open Quantum Systems the generator on the left hand sde of 2.2 by ϱ L H [ϱ] := [H, ϱ], 2.4 the Schrödnger untary dynamcs amounts to exponentaton of L H : ϱ t = U t [ϱ] = e tl H [ϱ] = k t k k! L H L H L H [ϱ], where ndcates the composton of maps. Therefore, the dynamcal maps U t form a one-parameter group of lnear maps on the state space S: U t U s = U t+s for all t, s R. Ths fact mathematcally descrbes the reversble character of the Schrödnger dynamcs,.e. the fact that the dynamcal maps U t can be nverted. Moreover, these maps preserve the spectrum of all states ϱ, leave the von Neumann entropy unchanged and transform pure states nto pure states. On the other hand, n ths work open quantum systems wll be consdered,.e. systems S whose nteractons wth the external envronment E, n whch they are mmersed, cannot be neglected. Snce, n prncple, the envronment conssts of nfntely many degrees of freedom, the proper approach would be that of statstcal mechancs [52, 53]; here, however, for sake of clarty, the envronment wll be descrbed by densty matrces ϱ E n an nfnte dmensonal Hlbert space H E. The compound system of the subsystem together wth the envronment, S + E, s a closed system whose Hlbert space s the tensor product C n H E, where C n s the Hlbert space of the n-dmensonal subsystem. Therefore, a state ϱ S+E belongng to the state space S S+E of the compound system wll evolve reversbly under the acton of a group of dynamcal maps U S+E t e tl S+E. Formally, ths group s generated by the exponentaton of the total generator L S+E [ϱ S+E ] := [H S+E, ϱ S+E ], where the total Hamltonan s H S+E = H S I E + I S H E + λh, 2.5 wth H S I S, H E I E the Hamltonans dentty operators pertanng to the subsystem, respectvely, envronment, H the Hamltonan descrbng the nteracton between the subsystem and the envronment, and λ an admensonal couplng constant. The total generator L S+E can thus be decomposed n the sum: L L S+E = L S + L E + λl. 2.6 Often, when consderng a system S mmersed n an external envronment, t s mportant to study the statstcal propertes of S alone, whch are descrbed by the state ϱ S S S. As seen n Secton 1.1.1, when dealng wth compound systems, ths can be

25 Reversble and rreversble dynamcs 25 done by performng a partal trace over the degrees of freedom of the envronment E,.e.: S S+E ϱ S+E ϱ S Tr E [ϱ S+E ] = ψj E ϱ S+E ψj E, j wth { ψ E j } an orthonormal bass n H E. Analogously, t s often nterestng to analyze only the dynamcs of the state of the subsystem mmersed n the external envronment. Agan, ths can be done, n prncple, by calculatng the acton of the total dynamcal map on the total state and then takng the partal trace over the envronment s degrees of freedom,.e., gven a state ϱ S at tme t = 0, the state of S at any tme t s ϱ S t = Tr E U S+E t [ϱ S+E ]. On the other hand, the evoluton of ϱ S can be rewrtten as the acton of a famly of maps on the ntal state of the subsystem alone as: ϱ S t G t [ϱ S ]. In general these maps depend on ϱ S, and n order for them to preserve the convex structure of the state space of the subsystem S S,.e. to be such that [ G t λ j ϱ j ] S = λ j G t [ϱ j S ], j the ntal state of the compound system must factorze [54]. Ths means that the subsystem and the envronment must be ntally uncorrelated and therefore that the ntal state of the total system must be of the form ϱ S+E = ϱ S ϱ E. Although ths s not true n general, nevertheless ths condton s fully consstent n many nterestng physcal contexts and gves rse to a famly of dynamcal maps G t whch depend on the envronment reference state ϱ E but act lnearly on the state space of the subsystem S. Snce G t [ϱ S ] Tr E U S+E t [ϱ S+E ] and the partal trace breaks tme-reversal symmetry, the famly of maps G t, t 0, descrbes an rreversble dynamcs. In general, G t G s G t+s, for t, s 0, and therefore ths famly lacks a semgroup composton law. If, however, the nteracton of the open quantum system wth the envronment s weak or the envronment tme-correlatons decay rapdly wth respect to the tme-varaton of the subsystem, then the equalty holds, G t G s = G t+s, for t, s 0, and ths ndcates the absence of cumulatve memory effects. The techncal procedures to elmnate the latter, and thus recover semgroups of dynamcal maps as reduced tme-evolutons for the subsystem alone, are known as Markovan approxmatons and wll be dscussed n the next secton. j

26 26 Open Quantum Systems As seen n Chapter 1, n order to have physcal consstency, the lnear map G t must be completely postve. From Theorem 2 t follows that ϱ S G t [ϱ S ] = V tϱ S V t, 2.7 wth V t M n C. Indeed, the evolved state of the subsystem alone can be obtaned by partal trace over the envronment degrees of freedom as ϱ S t = Tr E Ut S+E [ϱ S+E ] and, havng taken a factorzed ntal compound state ϱ S+E = ϱ S ϱ E, t follows that ϱ S t = Tr E U S+E t [ϱ S ϱ E ] = j,k rk E re j Ut S+E rk E ϱ S rk E S+E U t rj E = V tϱ S V t, 2.8 where { rj E } form a bass of egenvectors of ϱ E wth correspondng egenvalues rj E and V t := rk E re S+E j Ut rk E. Therefore the complete postvty of the map G t follows from Theorem 2 n Chapter Reduced dynamcs The completely postve maps G t gve a physcally consstent descrpton of the dynamcs of a system S nteractng wth an envronment E, wth the only necessary condton that the total ntal state be factorzed, ϱ S+E = ϱ S ϱ E : ths descrpton s closed,.e. t depends lnearly on the ntal state and can be expressed n terms of operators pertanng to S alone. From the defnton V t := rk E re S+E j Ut rk E n 2.8 t s evdent that the operators V t descrbe the dsspatve and nosy effects due to the envronment; therefore, snce the famly of maps G t can contan memory effects, n general t s not straghtforward to obtan the reduced dynamcs of the subsystem S alone. If, however, the nteracton between subsystem and envronment s suffcently weak or the envronment tmecorrelatons decay rapdly wth respect to the tme-varaton of the subsystem, then not only can the dynamcs of S be consdered dsentangled from that of the total system, but also approxmately descrbed by a one-parameter semgroup of maps γ t such that γ t γ s = γ t+s for t, s 0. The fact of the maps γ t formng a semgroup, and therefore satsfyng only a forward-n-tme composton law, reflects the rreversble character of the subsystem s dynamcs. In order to reveal the memory effects due to the envronment and contaned n the famly of maps G t, t s convenent to wrte the formal ntegro-dfferental evoluton equaton of whch G t are solutons. Ths equaton s derved va the so-called projecton technque [2, 3] and leads to master equatons of the form ϱ t t = L H[ϱ t ] + D[ϱ t ]. = [H eff, ϱ t ] + D[ϱ t ] 2.9

27 Reduced dynamcs 27 Here both operators L H and D act on the subsystem s state space S S : L H acts as n 2.4, L H [ϱ t ] = [H eff, ϱ t ] wth the effectve Hamltonan H eff = H eff M nc, whereas D s a lnear operator whch cannot be wrtten n commutator form and contans the dsspatve and nosy effects due to the envronment. Master equatons of the form 2.9 can be derved wth several so-called Markovan approxmatons, as wll be seen n Secton 2.3. Solvng 2.9, the reduced dynamcs of the subsystem S s descrbed n terms of a semgroup of lnear maps γ t, t 0, on S S, obtaned by exponentaton of the generator: γ t = e tlh+d In the next secton, the condtons to ensure the physcal consstency of the semgroup of lnear maps γ t wll be gven, focusng n partcular on the property of complete postvty. Then, n Secton 2.2.2, the detaled dervaton of the master equaton 2.9 wll be descrbed Abstract form of the generators of dynamcal semgroups In ths secton some condtons on the semgroups of maps γ t wll be gven, n order to guarantee ther full physcal consstency. Frst of all, t must be noted that the exstence of a generator and an exponental structure as n 2.10 s due to the tme-contnuty of the semgroup of maps γ t [55],.e.: wth X = Tr X X, X M n C. lm γ t[ϱ] ϱ = 0 ϱ S S, t 0 Further, the semgroup of maps γ t must satsfy three constrants n order to be physcally consstent. Frstly, snce they map states nto states, they must preserve the hermtcty of densty matrces. Secondly, they must preserve the trace: ths means that the overall probablty s constant, so phenomena wth loss of probablty, such as partcle decays, wll not be consdered. Ths constrant corresponds to the request of untalty for the dual map,.e. γ t [I d ] = I d, where I d s the d d dentty matrx and γ t s as n Defnton 11. These frst two constrants are suffcent to partally fx the form of the generator [56], as shown n the followng theorem. Theorem 7 Let γ t : M d C M d C, t 0, form a tme-contnuous semgroup of hermtctypreservng and trace-preservng lnear maps. Then the semgroup can be wrtten as γ t = e tl H+D, where the acton of the two terms of the generator on any densty matrx ϱ S S s L H [ϱ] = [H, ϱ], 2.11 D[ϱ] = d 2 1,j=1 K j F j ϱf 1 2 {F F j, ϱ}. 2.12