Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory

Title Quantum Operation Time Reversal

Permalink https://escholarship.org/uc/item/4024j4b7

Author Crooks, Gavin E.

Publication Date 2009-05-27

eScholarship.org Powered by the California Digital Library University of California arXiv:0706.3749v1 [quant-ph] 26 Jun 2007 xoiino h okflcuto hoe o dissipative for dynamics. theorem quantum fluctuation work the an to of leads exposition the naturally to This related change. is entropy breaking environmental and symmetry environment, this the of to magnitude system the the system coupling isolated by an broken of is invariance environ- time-reversal and The of system time-reversal ment. the between the interactions consider of we chain the the Instead, In consider states. meaningfully of cannot operations. chain we quantum system, to quantum reversal a time chain Markov operator [6]. positive Neu- (POVM) von general measurement a more valued to a due or subspace measurement, a mann into sys- the projection of either environment, measurements tem, by the quantum-classical induced with disturbance mixed interacting the the and system a system, of isolated dynam- dynamics quantum an pure wide of the a linear, ics describe including can dynamics, a superoperators of These range 6]. operation, opera- 5, of 4, quantum map [3, (TCP) tors positive a complete preserving, is trace matrix transition a with equipped are operation. time-reversal dynamics natural Markov Classical chain. ∗ iiyo h poietransition opposite the of bility itiuin oevr h rbblt ftetransition probability the of equilibrium homoge- probability i same Markovian, the the Moreover, also has is distribution. and states, time left, of in sequence to neous reversed right The from right. reading to left from read a,hmgnos taysae akvcan[,2.For 2]. 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Here ρ and ρ$ are the initial and final density matrices for the quantum dynamics (at least, not without measur- of the system, ρE is the initial density matrix of the en- ing, and therefore disturbing the system) we instead focus vironment, USE is the unitary operator representing the on the sequence of transitions. Each operator of a Kraus time evolution of the combined system over some time operator-sum represents a particular interaction with the interval, and trE is a partial trace over the environment environment that an external observer could, in principle, Hilbert space. The superoperator S is a quantum opera- measure and record. We can therefore define the dynam- tion, a linear, trace preserving, complete positive (TCP) ical history by the observed sequence of Kraus operators. map of operators [3, 4, 5, 6]. For each Kraus operator of the forward dynamics, Aα, Any complete map of positive operators has an there should be a corresponding operator, Aα of the re- operator-sum (or Kraus) representation, versed dynamics such that, starting from equilibrium, the probability of observing any sequence of Kraus operators $ † & ρ = Sρ = AαρAα (4) in the forward dynamics is the same as the probability of α " observing the reversed sequence of reversed operators in Conversely, any operator-sum represents a complete, pos- the reversed dynamics. Specifically, for consecutive pairs itive superoperator. The collection {Aα} are known as of events, starting from π (the invariant, equilibrium den- Kraus operators. sity matrix of the dynamics Sπ = π) The requirement that the quantum operation con- p(α1,α2|π)=p(α2,α1|π) served the density matrix trace can be compactly written as or equivalently [by Eq. (7)] & × † S I = A Aα = I. (5) † † † † α tr Aα2 Aα1 πAα1 Aα2 = tr Aα1 Aα2 πAα2 Aα1 α " ' ( ' ( Since the invariant density matrix π, is positive definite Here, S× is the superoperator adjoint of S, the unique su- & & & & × it has a unique inverse and a positive definite square root. peroperator such that #SA, B$ = #A, S B$, where #A, B$ 1 1 We may therefore insert the identity I = π− 2 π 2 between is the Hilbert-Schmit inner product tr A†B. In the op- pairs of Kraus operators. By taking advantage of the erator sum representation the superoperator adjoint is cyclic property of the trace we find that performed by taking the adjoints of the corresponding 1 1 1 1 1 1 1 1 Kraus operators [5]. 2 † − 2 2 † − 2 − 2 − 2 2 2 tr [π Aα1 π ][π Aα2 π ] π [π Aα2 π ][π Aα1 π ] † × † Sρ = AαρAα, S ρ = AαρAα (6) ' † † ( = tr Aα1 Aα2 π Aα2 Aα1 . α α " " 1 1 ' ( 2 † − 2 Each Kraus operator of a TCP map represents a par- Therefore, Aα = π Aαπ and the superoperator& & & &S, the ticular interaction with the environment that an external reversal or π-dual of S, is observer could, in principle, measure and record without & 1 1 1 1 & † − 2 † 2 2 − 2 further disturbing the dynamics of the system. The prob- Sρ = Aα ρ Aα = [π Aαπ ] ρ [π Aαπ ] (10) α α ability of observing the αth Kraus interaction is " " & & & 1 1 † If we write D ρ for the superoperator π 2 ρπ 2 then this pα = tr AαρAα (7) π reversal may be expressed independently of any particu- and the state of the system after this interaction is lar decomposition of S into Kraus operators. † $ AαρAα × −1 S = Dπ S D (11) ρα = † . (8) π tr AαρAα By similar reasoning,& the time reversal of the Lindbla- The overall effect of the dynamics, averaging over differ- dian continuous time dynamics [Eq. (9)] takes the same ent interactions, is the full quantum operation, Eq. (4). form, L = D L×D−1, analogous to the time reversal of In the limit of small time interval we obtain a contin- π π a continuous time Markov chain [2] uous time quantum Markovian dynamic, We can& readily confirm that the quantum operator re- t versal [Eq. (11)] is an involution (a duality) on TCP maps ρ(t) = exp L(τ)dτ ρ(s) (9) with fixed point π. From Eq. 10 it is clear that the re- #$s % versed superoperator has an operator-sum representation where L is the Lindbladian superoperator [6]. and is, therefore, a complete, positive map. Note that × Dπ = Dπ is a Hermitian superoperator (Since positive † operators are Hermitian π = π), that DπI = π, and −1 Quantum operation time reversal that Dπ π = I. Therefore, the reversal is idempotent, S = S, has the correct invariant density matrix, We will now consider the time reversal of a quantum × −1 × operation. Since we cannot observe a sequence of states && Sπ = Dπ S Dπ π = Dπ S I = Dπ I = π, & 3 and is trace preserving. respect to the probability vector on the density matrix diagonal. × −1 −1 −1 S I = D SDπI = D Sπ = D π = I. π π 1 1 1 1 2 × − 2 − 2 2 Mac = Saacc = #ea|π S π |ec$#ec|π π |ea$ If R and& S are two TCP maps with the same fixed point πaa ×' ( then RS = SR. ! & = #ea|S (|ec$#ec|) |ea$ By analog with classical Markov chain terminology, we πcc π p◦ may say) that& & a quantum operation is balanced with re- aa a = Sccaa = ◦ Mca spect to a density matrix, π, if Sπ = π, and detailed πcc pc balanced if S = S. Conversely, if S is detailed balanced with respect to some density matrix π, then π is a fixed Thermostated quantum system point of S: &

× −1 × Sπ = Sπ = DπS Dπ π = DπS I = DπI = π The reduced dynamics of a quantum system interacting with an external environment or bath can be de- In passing,& it is interesting to note that the time rever- rived by considering the deterministic dynamics of the × −1 sal operation, S = Dπ S Dπ is an anti-linear operator joint system, and then tracing over the bath degrees of of a superoperator, an anti-super-duper operator. freedom, leaving a quantum operation description of the & system dynamics alone. In particular, this approach pro- vides a concise description of a quantum system interact- Isolated quantum system ing with a thermal environment of constant temperature. Let the total Hamiltonian of the combined system be The operator sum representation of a closed system HSB = HS ⊕ IB + IS ⊕ HB + &Hint, (14) dynamic contains a single, unitary Kraus operator, U = − i Ht e ! , where H is the system Hamiltonian. Any density where IS and IB are system and bath identity operators, matrix that is diagonal in the energy eigenbasis will be HS is the Hamiltonian of the system, HB is the bath a ﬁxed point of this dynamics, and any such diagonal Hamiltonian, Hint is the bath-system interaction Hamil- operator will commute with the unitary Kraus operator. tonian and & is a coupling constant. Therefore, the quantum operator reversal corresponds to We assume that initially the system and bath are un- the time-reversal of the unitary dynamics. correlated, and therefore the initial combined state is ρS ⊕ πB, where πB is the thermal equilibrium density † − i Ht + i Ht S(t)ρ = UρU = e ! ρe ! (12) matrix of the bath. i i † + ! Ht − ! Ht S(t)ρ = U ρU = e ρe βEB e− i πB = |b $#b | Z i i & i B Classical Markov chain " B Here {Ei } are the energy eigenvalues, {|bi$} are the orthonormal energy eigenvectors of the bath, and Z is the Given an orthonormal basis set {|ei$} we can extract B the “matrix elements” of a superoperator S. bath partition function. We follow the dynamics of the combined system for some time, then measure the state

Sabcd = #ea|S(|ed$#ec|)|eb$ (13) of the bath. † There are several diﬀerent conventions for the ordering SρS = trB USB [ρS ⊕ πB ]USB B of the indices. Caves [5] would write Sad,bc and Terhal −βEi e † and DiVincenzo [7] S . The ordering used here is = #bj|USB ρS ⊕ |bi$#bi| U |bj$ ab,dc Z SB ac j + , i B -. convenient when transitioning to a tensor (S bd) or dia- a→ →d " B " grammatic (b← S←c ) notation. −βEi e † In any basis the matrix M = S is a Markov = #bj|USB |bi$ ρS #bi|U |bj$ (15) ac aacc Z SB stochastic transition matrix; The elements are real and ij B " positive Mac ≥ 0 and the rows sum to 1. (Since |ec$#ec| Here, tr is a partial trace over the bath degrees of free- is a positive operator and S is a positive map, S(|ec$#ec|) B SB ! must also be a positive operator, and therefore the ele- dom, USB = exp(−iH t/ ) is the unitary dynamic of ments are real and positive. The trace preserving con- the total system, and we have assumed that the coupling × constant & is small. It follows that the Kraus operators dition requires that S I = I. Since I = a |ea$#ea|, therefore Saacd = δcd and Mac = 1.) for this dynamics are a a * In the diagonal basis of the equilibrium density matrix, B − 1 βE a time-reversal* of the quantum* operation induces a time e 2 i Aij = #bj|USB |bi$ . (16) reversal of the embedded Markov transition matrix with ZB / 4 and the corresponding reversed operators are chain

1 † 1 A = π 2 A π− 2 (17) p(e ; α ,α , ··· ,α ; e ) ij ij 0 0 1 τ τ = exp {−βQ} (18) 1 B − 2 βEi p(eτ ; ατ , ··· ,α1,ατ ; e0) 1 e † − 1 & = π 2 [ #bi|U |bj$]π 2 Z SB B since for every Kraus interaction of the forward dynam- 1 1 1 & + 2 + 2 † − 2 = #bi|[π/S ⊕ πB ]USB [πS ⊕ IB ]|bj$ ics there is a corresponding interaction in the reverse dy- 1 † 1 † + 2 namics such that Aα = Aα exp{+ 2 βQα} [Eq. (17)] where ≈#bi|USB [IS ⊕ πB ]|bj$ B B 1 B Q = −(Ej − Ei ) is the heat, the ﬂow of energy from the − 2 βEj e † bath to the system& during the forward time step. = #bi|USB |bj$ . ZB p(e0; α1,α2, ··· ,ατ ; eτ ) If we compare the/ reversed operator with the correspond- (τ) (2) (1) (1)† (2)† (τ)† = tr#eτ |A ···A A |e0$#e0|A A ···A |eτ $ ing forward operator we can see that taking the time re- ατ α2 α1 α1 α2 ατ (1) (2) (τ) (τ)† (2)† (1)† versal of a quantum operation that acts on the system = tr#e0|Aα1 Aα2 ···Aατ |eτ $#eτ |Aατ ···Aα2 Aα1 |e0$ subspace is equivalent (in the small coupling limit) to × exp {−β(Qα1 + Qα2 + ···+ Qατ )} taking the time reversal of the entire system-bath dy- & & & & & & = p(e ; α , ··· ,α ,α ; e ) exp {−βQ} namics. τ τ 1 τ 0

Recall& that Q is the heat flow into the system, β is Driven quantum dynamics the inverse temperature of the environment, and therefore −βQ is the change in entropy of that environment. This property of microscopic reversibility as expressed in Consider a system with a time-dependent Hamiltonian, Eq. (18) [8], immediately implies that the work fluctu- interacting with an external, constant temperature heat ation relation [9] and Jarzynski identity [10] can be ap- bath. We split time into a series of intervals, labeled plied to a driven quantum system coupled to a thermal by the integer t. The system Hamiltonian changes from environment [11, 12, 13, 14, 15]. The crucial difference one interval to the next, but is fixed within each interval. between the classical and quantum regimes is that in the (We can recover a continuously varying system Hamilto- quantum case we must avoid explicitly measuring the nian by making the intervals short.) The dynamics are work directly, which is tantamount to continuously mon- described by dissipative quantum operations [Eq. 15], St itoring the system, and instead measure the heat flow (t) with Kraus operators [Eq. (16)] {Aα }. The dynami- from the environment [13, 15]. cal history of the system is defined by the initial system Financial support was provided by the DOE/Sloan state |e0$#e0|, a sequence of interactions between the sys- Postdoctoral Fellowship in Computational Biology; by tem and environment, described by the Kraus operators the Office of Science, Biological and Environmental Re- (1) (2) (τ) Aαt ,Aα2 , ··· ,Aατ and the final system state |eτ $#eτ |. search, U.S. Department of Energy under Contract No. The probability of observing this history is related to the DE-AC02-05CH11231; and by California Unemployment probability of the reversed history in the time reversed Insurance.

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