PHYSICAL REVIEW A 77, 034101 ͑2008͒
Quantum operation time reversal
Gavin E. Crooks* Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA ͑Received 25 June 2007; published 25 March 2008͒ The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.
DOI: 10.1103/PhysRevA.77.034101 PACS number͑s͒: 05.30.Ch, 05.70.Ln
I. INTRODUCTION II. BACKGROUND: MARKOV CHAIN TIME REVERSAL
Consider a sequence of states sampled from a classical, Let M be the classical Markov transition matrix of a for- ward chain of states i.e., M is the probability of moving homogeneous, steady-state, Markov chain ͓1,2͔. For ex- ͑ ji ample, from state i to state j͒, M˜ the transition matrix of the reversed chain, and p° the equilibrium probability distribution of both chains ͑i.e., Mp°=M˜ p°=p°͒. Since the probability of the A, B, C, A, A, B, C, A, B, C, A, transition i j in the forward chain will be the same as the probability→ of the opposite transition j i in the time re- versed chain, it follows that the forward→ and reversed chain where the three states are labeled ͑A, B, C͒ and time is transition matrices are related by read from left to right. The reversed sequence of states, read- ing from right to left, is also Markovian, homogeneous in ˜ ° ° Mijpj = M jipi for all i, j. ͑1͒ time, and has the same equilibrium probability distribution. Moreover, the probability of the transition i j in the for- Note that in much of the Markov chain literature ͑e.g., Ref. ward chain will be the same as the probability→ of the opposite ͓2͔͒ the transition matrix is the transpose of the matrix de- transition j i in the time reversed chain. Classical Markov fined here. Also, we have avoided the conventional notation dynamics are→ equipped with a natural time-reversal opera- for the time reversed matrix Mˆ since this may cause unfor- tion. tunate confusion in the context of quantum dynamics. The quantum mechanical generalization of a Markov tran- In matrix notation this time reversal operation can be con- sition matrix is a quantum operation, a linear, trace preserv- veniently expressed as ing, complete positive ͑TCP͒ map of operators ͓3–6͔. These superoperators can describe a wide range of dynamics, in- M˜ = diag͑p°͒MT diag͑p°͒−1. ͑2͒ cluding the pure quantum dynamics of an isolated system, Here, diag p° indicates a matrix whose diagonal elements the mixed quantum-classical dynamics of a system interact- ͑ ͒ ˜ ing with the environment, and the disturbance induced by are given by the vector p°. M is referred to as the reversal of measurements of the system, either projection into a sub- M ͓2͔ or as the dual of M ͓1͔. A transition matrix M is space due to a von Neumann measurement, or a more gen- balanced with respect to a probability distribution p° if eral positive operator valued measurement ͑POVM͓͒5͔. Mp°=p° and detailed balanced if the matrix is time-reversal In this Brief Report, we consider the generalization of invariant, M˜ =M. classical Markov chain time reversal to quantum operations. In a quantum system, we cannot meaningfully consider the III. BACKGROUND: QUANTUM OPERATIONS chain of states. Instead, we consider the time reversal of the chain of interactions between the system and environment. The dynamics of an open quantum system can be repre- The time-reversal invariance of an isolated system is broken sented as the closed system unitary dynamics of the system by coupling the system to the environment, and the magni- coupled to an extended environment, followed by a measure- tude of this symmetry breaking is related to the environmen- ment of the environment: tal entropy change. This naturally leads to an exposition of Ј = = tr U ͓ ͔U† . ͑3͒ the work fluctuation theorem for dissipative quantum dynam- S E SE E SE ics. Here and Ј are the initial and final density matrices of the system, E is the initial density matrix of the environment, USE is the unitary operator representing the time evolution of *[email protected] the combined system over some time interval, and trE is a
1050-2947/2008/77͑3͒/034101͑4͒ 034101-1 ©2008 The American Physical Society BRIEF REPORTS PHYSICAL REVIEW A 77, 034101 ͑2008͒ partial trace over the environment Hilbert space. The super- starting from equilibrium, the probability of observing any operator is a quantum operation, a linear, trace preserving, sequence of Kraus operators in the forward dynamics is the completeS positive ͑TCP͒ map of operators ͓3–6͔. same as the probability of observing the reversed sequence Any complete map of positive operators has an operator- of reversed operators in the reversed dynamics. Specifically, sum ͑or Kraus͒ representation, for consecutive pairs of events, starting from ͑the invari- ant, equilibrium density matrix of the dynamics =͒ † S Ј = = ͚ A␣A␣. ͑4͒ S ␣ p͉͑␣1,␣2͉͒ = ˜p͉͑␣2,␣1͉͒, Conversely, any operator-sum represents a complete, positive or equivalently ͓by Eq. ͑7͔͒ superoperator. The collection ͕A␣͖ are known as Kraus op- tr͑A A A† A† ͒ = tr͑˜A ˜A ˜A† ˜A† ͒. erators. ␣2 ␣1 ␣1 ␣2 ␣1 ␣2 ␣2 ␣1 The requirement that the quantum operation conserved Since the invariant density matrix is positive definite it has the density matrix trace can be compactly written as a unique inverse and a positive definite square root. We may ϫ † therefore insert the identity I=−1/21/2 between pairs of I = ͚ A␣A␣ = I. ͑5͒ S ␣ Kraus operators. By taking advantage of the cyclic property of the trace we find that Here, ϫ is the superoperator adjoint of , the unique super- S ϫ S tr͓͑1/2A† −1/2͔͓1/2A† −1/2͔͓−1/2A −1/2͔ operator such that ͗ A,B͘=͗A, B͘, where ͗A,B͘ is the ␣1 ␣2 ␣2 Hilbert-Schmit innerS product tr A†SB. In the operator sum rep- ϫ͓1/2A 1/2͔͒ = tr͑˜A ˜A ˜A† ˜A† ͒. resentation the superoperator adjoint is performed by taking ␣1 ␣1 ␣2 ␣2 ␣1 the adjoints of the corresponding Kraus operators ͓4͔ ˜ 1/2 † −1/2 ˜ Therefore, A␣ = A␣ and the superoperator , the re- † ϫ † S = ͚ A␣A␣, = ͚ A␣A␣. ͑6͒ versal or dual of , is S ␣ S ␣ S ˜ ˜ ˜ † −1/2 † 1/2 1/2 −1/2 = ͚ A␣A␣ = ͚ ͓ A␣ ͔͓ A␣ ͔. Each Kraus operator of a TCP map represents a particular S ␣ ␣ interaction with the environment that an external observer could, in principle, measure and record without further dis- ͑10͒ 1/2 1/2 turbing the dynamics of the system. The probability of ob- If we write for the superoperator then this serving the ␣th Kraus interaction is reversal mayD be expressed independently of any particular † decomposition of into Kraus operators p␣ = tr A␣A␣, ͑7͒ S ˜ = ϫ −1. ͑11͒ and the state of the system after this interaction is S DS D A A† By similar reasoning, the time reversal of the Lindbladian ␣ ␣ ˜ ␣Ј = † . ͑8͒ continuous time dynamics ͓Eq. ͑9͔͒ takes the same form, tr A␣A␣ ϫ −1 L = , analogous to the time reversal of a continuous The overall effect of the dynamics, averaging over different timeD L MarkovD chain ͓2͔. interactions, is the full quantum operation, Eq. ͑4͒. We can readily confirm that the quantum operator reversal In the limit of small time interval we obtain a continuous ͓Eq. ͑11͔͒ is an involution ͑a duality͒ on TCP maps with time quantum Markovian dynamic fixed point . From Eq. ͑10͒ it is clear that the reversed t superoperator has an operator-sum representation and is, therefore, a complete, positive map. Note that = ϫ is a ͑t͒ = exp ͵ ͑͒d ͑s͒, ͑9͒ D D ͩ s L ͪ Hermitian superoperator ͑since positive operators are Her- † −1 mitian =͒, that I=, and that =I. Therefore, the where is the Lindbladian superoperator ͓5͔. D ˜ D L reversal is idempotent, ˜ = , has the correct invariant den- S S IV. QUANTUM OPERATION TIME REVERSAL sity matrix, ˜ = ϫ −1 = ϫI = I = , We will now consider the time reversal of a quantum S DS D DS D operation. Since we cannot observe a sequence of states for and is trace preserving, the quantum dynamics ͑at least not without measuring and therefore disturbing the system , we instead focus on the ˜ ϫ −1 −1 −1 ͒ I = I = = = I. sequence of transitions. Each operator of a Kraus operator S D SD D S D sum represents a particular interaction with the environment If and are two TCP maps with the same fixed point then R S that an external observer could, in principle, measure and =˜ ˜ . record. We can therefore define the dynamical history by the RSByS analogyR with classical Markov chain terminology, we observed sequence of Kraus operators. For each Kraus op- may! say that a quantum operation is balanced with respect to erator of the forward dynamics, A␣, there should be a corre- a density matrix, , if =, and detailed balanced if ˜ ˜ S sponding operator, A␣, of the reversed dynamics such that, = . Conversely, if is detailed balanced with respect to S S S 034101-2 BRIEF REPORTS PHYSICAL REVIEW A 77, 034101 ͑2008͒ some density matrix , then is a fixed point of , then tracing over the bath degrees of freedom, leaving a S quantum operation description of the system dynamics alone. = ˜ = ϫ −1 = ϫI = I = . In particular, this approach provides a concise description of S S DS D DS D a quantum system interacting with a thermal environment of In passing, it is interesting to note that the time reversal constant temperature. Let the total Hamiltonian of the com- ˜ ϫ −1 operation, = , is an antilinear operator of a super- bined system be operator, anS anti-super-duperD S D operator. HSB = HS IB + IS HB + ⑀Hint, ͑14͒ V. ISOLATED QUANTUM SYSTEM where IS and IB are system and bath identity operators, HS is the Hamiltonian of the system, HB is the bath Hamiltonian, The operator sum representation of a closed system dy- Hint is the bath-system interaction Hamiltonian, and ⑀ is a namic contains a single, unitary Kraus operator, U=e−i/បHt, coupling constant. where H is the system Hamiltonian. Any density matrix that We assume that initially the system and bath are uncorre- is diagonal in the energy eigenbasis will be a fixed point of lated, and therefore the initial combined state is S B, this dynamics, and any such diagonal operator will commute where B is the thermal equilibrium density matrix of the with the unitary Kraus operator. Therefore, the quantum op- bath erator reversal corresponds to the time reversal of the unitary B dynamics −Ei B e = ͉bi͗͘bi͉. ͑t͒ = UU† = e−i/បHte+i/បHt, ͚ Z S i B B Here ͕E ͖ are the energy eigenvalues, ͕͉bi͖͘ are the orthonor- ˜ † +i/បHt −i/បHt i ͑t͒ = U U = e e . ͑12͒ mal energy eigenvectors of the bath, and Z is the bath par- S B tition function. We follow the dynamics of the combined system for some time, then measure the state of the bath: VI. CLASSICAL MARKOV CHAIN † S = trB USB͓S B͔USB Given an orthonormal basis set ͕͉e ͖͘ we can extract the S i B “matrix elements” of a superoperator e−Ei S † = ͚ ͗bj͉USB S ͚ ͉bi͗͘bi͉ USB͉bj͘ j ͩ ͫ i ZB ͬͪ Sabcd = ͗ea͉ ͉͑ed͗͘ec͉͉͒eb͘. ͑13͒ S B There are several different conventions for the ordering of e−Ei = b U b b U† b . 15 the indices. Caves ͓4͔ would write S and Terhal and Di- ͚ ͗ j͉ SB͉ i͘ S͗ i͉ SB͉ j͘ ͑ ͒ ad,bc ij ZB Vincenzo ͓7͔ Sab,dc. The ordering used here is convenient ac when transitioning to a tensor ͑ bd͒ or diagrammatic Here, trB is a partial trace over the bath degrees of freedom, a d S SB ͑ b→ →c͒ notation. USB=exp͑−iH t/ប͒ is the unitary dynamic of the total sys- ←S← In any basis the matrix Mac=Saacc is a Markov stochastic tem, and we have assumed that the coupling constant ⑀ is transition matrix; The elements are real and positive small. It follows that the Kraus operators for this dynamics MacՆ0 and the rows sum to 1. ͓Since ͉ec͗͘ec͉ is a positive are operator and is a positive map, ͉͑e ͗͘e ͉͒ must also be a c c −1 2EB positive operator,S and therefore theS elements are real and e / i Aij = ͗bj͉USB͉bi͑͘16͒ positive. The trace preserving condition requires that ͱZB ϫ I=I. Since I=͚a͉ea͗͘ea͉, therefore ͚aSaacd=␦cd and S and the corresponding reversed operators are ͚aMac=1.͔ In the diagonal basis of the equilibrium density matrix a ˜ 1/2 † −1/2 time reversal of the quantum operation induces a time rever- Aij = Aij B sal of the embedded Markov transition matrix. The reversal −1/2Ei 1/2 e † −1/2 is with respect to the probability density formed by the diag- = ͗bi͉USB͉bj͘ ͫ ͱZ ͬ onal elements of the density matrix B = b +1/2 +1/2 U† −1/2 I b ˜ ˜ 1/2 ϫ −1/2 −1/2 1/2 ͗ i͉͓ S B ͔ SB͓ S B͔͉ j͘ Mac = Saacc = ͗ea͉ ͑ ͉ec͗͘ec͉ ͒ ͉ea͘ S † +1/2 ° Ϸ͗bi͉USB͓IS B ͔͉bj͘ aa ϫ aa pa = e e e e = S = M . −1 2EB ͗ a͉ ͉͑ c͗͘ c͉͉͒ a͘ ccaa ° ca e / j cc S cc pc † = ͗bi͉USB͉bj͘. ͑17͒ ͱZB
VII. THERMOSTATED QUANTUM SYSTEM If we compare the reversed operator with the corresponding forward operator we can see that taking the time reversal of The reduced dynamics of a quantum system interacting a quantum operation that acts on the system subspace is with an external environment or bath can be derived by con- equivalent ͑in the small coupling limit͒ to taking the time sidering the deterministic dynamics of the joint system, and reversal of the entire system-bath dynamics.
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VIII. DRIVEN QUANTUM DYNAMICS ϫexp −  Q + Q + + Q ͕ ͑ ␣1 ␣2 ¯ ␣͖͒ Consider a system with a time-dependent Hamiltonian, interacting with an external, constant temperature heat bath. = ˜p͑e;␣,...,␣2,␣1;e0͒exp͕− Q͖. We split time into a series of intervals, labeled by the integer t. The system Hamiltonian changes from one interval to the next, but is fixed within each interval. ͑We can recover a Note that  is the inverse temperature of the environment continuously varying system Hamiltonian by making the in- and, therefore, that −Q is the change in entropy of that tervals short.͒ The dynamics are described by dissipative environment. This property of microscopic reversibility as quantum operations ͓Eq. ͑15͔͒ t with Kraus operators ͓Eq. expressed in Eq. ͑18͓͒8͔ immediately implies that the work ͑t͒ S ͑16͔͒ ͕A␣ ͖. The dynamical history of the system is defined fluctuation relation ͓9͔ and Jarzynski identity ͓10͔ can be by the initial system state ͉e0͗͘e0͉, a sequence of interactions applied to a driven quantum system coupled to a thermal between the system and environment, described by the Kraus ͑1͒ ͑2͒ ͑͒ environment ͓11–15͔. The crucial difference between the operators A ,A ,...,A and the final system state ͉e͗͘e͉. ␣1 ␣2 ␣ classical and quantum regimes is that in the quantum case we The probability of observing this history is related to the must avoid explicitly measuring the work directly and in- probability of the reversed history in the time reversed chain stead measure the heat flow from the environment ͓13,15͔. p͑e0;␣1,␣2,...,␣;e͒ Measuring the work and the energy of the environment is = exp͕− Q͖. ͑18͒ tantamount to continuously monitoring the energy of the sys- ˜p͑e;␣,...,␣2,␣1;e0͒ tem, which would disturb the quantum dynamics of the sys- This follows because for every Kraus interaction of the for- tem in which we are interested. ward dynamics there is a corresponding interaction in the ˜ † 1 reverse dynamics A␣ =A␣ exp͕+ 2 Q␣͖͓Eq. ͑17͔͒ where B B Q=−͑Ej −Ei ͒ is the heat, the flow of energy from the bath to ACKNOWLEDGMENTS the system during the forward time step,
p͑e0;␣1,␣2,...,␣;e͒ Financial support was provided by the DOE/Sloan, ͑͒ ͑2͒ ͑1͒ ͑1͒† ͑2͒† ͑͒† by the Office of Science, Biological and Environmental = tr͗e͉A A A ͉e ͗͘e ͉A A A ͉e͘ ␣ ¯ ␣2 ␣1 0 0 ␣1 ␣2 ¯ ␣ Research, U.S. Department of Energy under Contract No. ˜ ͑1͒˜ ͑2͒ ˜ ͑͒ ˜ ͑͒† ˜ ͑2͒†˜ ͑1͒† DE-AC02-05CH11231, and by California Unemployment = tr͗e ͉A A A ͗e͉A A A ͉e ͘ 0 ␣1 ␣2 ¯ ␣ ␣ ¯ ␣2 ␣1 0 Insurance.
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