Introducing one-shot work into fluctuation relations

For closed quantum systems driven away from equilibrium, work is often defined in terms of projective measurements of initial and final energies. This definition leads to statistical distributions of work that satisfy nonequilibrium work and fluctuation relations. While this two-point measurement definition of quantum work can be justified heuristically by appeal to the first law of thermodynamics, its relationship to the classical definition of work has not been carefully examined. In this paper, we employ semiclassical methods, combined with numerical simulations of a driven quartic oscillator, to study the correspondence between classical and quantal definitions of work in systems with 1 degree of freedom. We find that a semiclassical work distribution, built from classical trajectories that connect the initial and final energies, provides an excellent approximation to the quantum work distribution when the trajectories are assigned suitable phases and are allowed to interfere. Neglecting the interferences between trajectories reduces the distribution to that of the corresponding classical process. Hence, in the semiclassical limit, the quantum work distribution converges to the classical distribution, decorated by a quantum interference pattern. We also derive the form of the quantum work distribution at the boundary between classically allowed and forbidden regions, where this distribution tunnels into the forbidden region. Our results clarify how the correspondence principle applies in the context of quantum and classical work distributions and contribute to the understanding of work and nonequilibrium work relations in the quantum regime.

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“…(16)] as a sum over trajectories that begin and end with energies E A m and E B n , respectively [Eq. (39)]. In Sec.…”

Section: Semiclassical Theorymentioning

confidence: 99%

Quantum-Classical Correspondence Principle for Work Distributions

2015

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“…We shall employ a subset of the trajectory formalism (see e.g. [21][22][23][24][25][26][27][28] among many), but stress that this framework is just one arbitrary choice from many thermodynamic models. The bounds derived in this article, being information-theoretic in origin, hold for any model of thermal interaction which defines heat and work in a way that is consistent with Landauer's principle.…”

Section: Corollary 11 (All Non-zero) If For Somementioning

confidence: 99%

“…[7,8]), single-shot statistical mechanics (e.g. [27][28][29]), among others.In these frameworks, Landauer's principle is not an additional imposition on top of the physics, but rather manifests from microdynamical behaviour -it is an emergent law, much like the second law of thermodynamics. It is not necessary to know the particulars of the microdynamics to derive or apply the results in this article -so long as the framework obeys Landauer's principle, any model in that framework that implements the pattern manipulations presented in the following sections will be bound by our results.…”

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confidence: 99%

Thermodynamics of complexity and pattern manipulation

et al. 2017

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Many organisms capitalize on their ability to predict the environment to maximize available free energy, and reinvest this energy to create new complex structures. This functionality relies on the manipulation of patterns-temporally ordered sequences of data. Here, we propose a framework to describe pattern manipulators -devices that convert thermodynamic work to patterns or vice versa -and use them to build a 'pattern engine that facilitates a thermodynamic cycle of pattern creation and consumption. We show that the least heat dissipation is achieved by the provably simplest devices; the ones that exhibit desired operational behaviour while maintaining the least internal memory. We derive the ultimate limits of this heat dissipation, and show that it is generally non-zero and connected with the patterns intrinsic crypticity -a complexity theoretic quantity that captures the puzzling difference between the amount of information the pattern's past behaviour reveals about its future, and the amount one needs to communicate about this past to optimally predict the future.The manipulation of patterns is as important to living organisms as it is for computation. Living things capitalize on structure in their environment for available energy, and use this energy to generate new complex structures. Similarly, a crucial task in the modern era of big data is to identify patterns in large data sets in order to make predictions about future events -often at great energetic cost. Here, we consider the thermodynamic costs intrinsic to this sort of pattern manipulation and ask: is there a preferred method by which this manipulation should be done? Our intuition is that simpler is better, a longstanding tenant of natural philosophy known as Occam's razor. To formalize this, we first qualify what is meant both by simpler and by better.In complexity science, computational mechanics formalizes what is simpler in the context of pattern manipulation [1][2][3]. The premise is that everything we observe in the environment can be considered to be a patterna temporal sequence of data exhibiting certain statistical structure. Much of science then deals with building models that can explain such statistics -machines that take information from past observations, and use it to generate statistically coinciding conditional future predictions. Given two machines that exhibit same pattern of behaviour, the one that stores less information from the past is considered simpler, the motivation being that it better isolates indicators of future behaviour. The simplest such machine then defines exactly how much memory is required to produce a given pattern, and thus * ajpgarner@nus.edu.sg † cqtmileg@nus.edu.sg quantifies the pattern's intrinsic structure. Known as statistical complexity, this measure has been applied to quantify structure in diverse contexts [4][5][6].Meanwhile in thermodynamics, better originally described heat engines that produce more work with less wasted heat. This carries through to modern thermodynamics: the best approach fo...

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“…The equality unites two powerful tools that have diverse applications in quantum information, high-energy physics, statistical mechanics, and condensed matter. The union sheds new light on both fluctuation relations and the OTOC, similar to the light shed when fluctuation relations were introduced into "one-shot" statistical mechanics [37][38][39][40][41][42]. The union also relates the OTOC, known to signal quantum behavior in high energy and condensed matter, to a quasiprobability, known to signal quantum behavior in optics.…”

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confidence: 86%

Jarzynski-like equality for the out-of-time-ordered correlator

Halpern

2017

Phys. Rev. A

117

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The out-of-time-ordered correlator (OTOC) diagnoses quantum chaos and the scrambling of quantum information via the spread of entanglement. The OTOC encodes forward and reverse evolutions and has deep connections with the flow of time. So do fluctuation relations such as Jarzynski's equality, derived in nonequilibrium statistical mechanics. I unite these two powerful, seemingly disparate tools by deriving a Jarzynski-like equality for the OTOC. The equality's left-hand side equals the OTOC. The right-hand side suggests a protocol for measuring the OTOC indirectly. The protocol is platform-nonspecific and can be performed with weak measurement or with interference. Time evolution need not be reversed in any interference trial. The equality enables fluctuation relations to provide insights into holography, condensed matter, and quantum information and vice versa. DOI: 10.1103/PhysRevA.95.012120The out-of-time-ordered correlator (OTOC) F (t) diagnoses the scrambling of quantum information [1][2][3][4][5][6]: Entanglement can grow rapidly in a many-body quantum system, dispersing information throughout many degrees of freedom. F (t) quantifies the hopelessness of attempting to recover the information via local operations.Originally applied to superconductors [7], F (t) has undergone a revival recently. F (t) characterizes quantum chaos, holography, black holes, and condensed matter. The conjecture that black holes scramble quantum information at the greatest possible rate has been framed in terms of F (t) [6,8]. The slowest scramblers include disordered systems [9][10][11][12][13]. In the context of quantum channels, F (t) is related to the tripartite information [14]. Experiments have been proposed [15][16][17] and performed [18,19] to measure F (t) with cold atoms and ions, with cavity quantum electrodynamics, and with nuclear-magnetic-resonance quantum simulators.F (t) quantifies sensitivity to initial conditions, a signature of chaos. Consider a quantum system S governed by a Hamiltonian H . Suppose that S is initialized to a pure state |ψ and perturbed with a local unitary operator V . S then evolves forward in time under the unitary U = e −iH t for a duration t, is perturbed with a local unitary operator W, and evolves backward under U † . The state |ψ := U † WUV |ψ = W(t)V |ψ results. Suppose, instead, that S is perturbed with V not at the sequence's beginning, but at the end: |ψ evolves forward under U , is perturbed with W, evolves backward under U † , and is perturbed with V . The state |ψ := V U † WU |ψ = V W(t)|ψ results. The overlap between the two possible final states equals the correlator: push an ion-trap potential along some direction in space [26]. Let F := F (H f ) − F (H i ) denote the difference between the equilibrium free energies at the inverse temperature β: 1 F (H ) = − 1 β ln Z β, , wherein the partition function is Z β, := Tr(e −βH ) and = i or f . The free-energy difference has applications in chemistry, biology, and pharmacology [27]. One could measure F , in principle, by measuring th...

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Introducing one-shot work into fluctuation relations

Cited by 64 publications

References 46 publications

Quantum-Classical Correspondence Principle for Work Distributions

Quantum-Classical Correspondence Principle for Work Distributions

Thermodynamics of complexity and pattern manipulation

Jarzynski-like equality for the out-of-time-ordered correlator

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