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Emergence of a fluctuation relation for heat in typical open quantum processes

Philip Taranto, Felix A. Pollock, Kavan Kishore Modi

Published Oct 28, 2015• 1 citation
OpenAlex

Abstract

In a generalisation of the Landauer erasure protocol, we study bounds on the heat generated in typical open quantum processes. We derive a fluctuation relation for this heat, implying that it is almost always positive when either the system or reservoir is large, as well as in the high temperature limit. This emergence of a fluctuation relation occurs at least exponentially quickly in the dimension of the larger subsystem, and linearly in the inverse temperature $\beta$. Finally, we propose efficiently computable bounds for the remaining cases.

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In a generalisation of the Landauer erasure protocol, we study bounds on the heat generated in typical open quantum processes. We derive a fluctuation relation for this heat, implying that it is almost always positive when either the system or reservoir is large, as well as in the high temperature limit. This emergence of a fluctuation relation occurs at least exponentially quickly in the dimension of the larger subsystem, and linearly in the inverse temperature β\betaβ. Finally, we propose efficiently computable bounds for the remaining cases.
Using the operational framework of completely positive, trace preserving operations and thermodynamic fluctuation relations, we derive a lower bound for the heat exchange in a Landauer erasure process on a quantum system. Our bound comes from a nonphenomenological derivation of the Landauer principle which holds for generic nonequilibrium dynamics. Furthermore, the bound depends on the nonunitality of dynamics, giving it a physical significance that differs from other derivations. We apply our framework to the model of a spin-1/2 system coupled to an interacting spin chain at finite temperature.
A fluctuation relation for the heat exchange of an open quantum system under a thermalizing Markovian dynamics is derived. We show that the probability that the system absorbs an amount of heat from its bath, at a given time interval, divided by the probability of the reverse process (releasing the same amount of heat to the bath) is given by an exponential factor which depends on the amount of heat and the difference between the temperatures of the system and the bath. Interestingly, this relation is akin to the standard form of the fluctuation relation (for forward-backward dynamics). We also argue that the probability of the violation of the second law of thermodynamics in the form of the Clausius statement (i.e., net heat transfer from a cold system to its hot bath) drops exponentially with both the amount of heat and the temperature differences of the baths.
We study quantum thermodynamics in nonequilibrium reservoirs (NERs) that are prepared from initially thermal ones via unitary driving processes. Based on the formulation of entropy production in NERs, we establish a Landauer-like bound and show that the bound can be violated under a definite condition depending on the states of NERs. It is found that the breakdown of this Landauer-like bound implies occurrences of several anomalous phenomena in NERs, namely, the efficiency enhancement of quantum thermal machines beyond the Carnot limit, the extraction of work from a single reservoir, and the spontaneous heat flow from a cold reservoir to a hot one. The results are illustrated through a physical model taking the preparation processes of NERs into account. Our work sheds light on the reason and condition of some unusual phenomena occurring in NERs and is helpful for one to prepare effective NERs that can be used as quantum resources to realize certain thermodynamic tasks.
Landauer's principle provides a perspective on the physical meaning of information as well as on the minimum working cost of information processing. Whereas most studies have related the decrease in entropy during a computationally irreversible process to a lower bound of dissipated heat, recent efforts have also provided another lower bound associated with the thermodynamic fluctuation of heat. The coexistence of the two conceptually independent bounds has stimulated comparative studies of their close relationship or tightness; however, these studies were concerned with finite quantum systems that allowed the revival of erased information because of a finite recurrence time. We broaden these comparative studies further to open quantum systems with infinite recurrence times. By examining their dependence on the initial state, we find the independence of the thermodynamic bound from the initial coherence, whereas the entropic bound depends on both the initial coherence and the population. A crucial role is indicated by the purity of the initial state: The entropic bound is tighter when the initial condition is sufficiently mixed, whereas the thermodynamic bound is tighter when the initial state is close to a pure state. These trends are consistent with previous results obtained for finite systems.
Quantum state engineering and quantum computation rely on information erasure procedures that, up to some fidelity, prepare a quantum object in a pure state. Such processes occur within Landauer's framework if they rely on an interaction between the object and a thermal reservoir. Landauer's principle dictates that this must dissipate a minimum quantity of heat, proportional to the entropy reduction that is incurred by the object, to the thermal reservoir. However, this lower bound is only reachable for some specific physical situations, and it is not necessarily achievable for any given reservoir. The main task of our work can be stated as the minimisation of heat dissipation given probabilistic information erasure, i.e., minimising the amount of energy transferred to the thermal reservoir as heat if we require that the probability of preparing the object in a specific pure state | φ 1 ⟩ be no smaller than p φ 1 max − δ . Here p φ 1 max is the maximum probability of information erasure that is permissible by the physical context, and δ ⩾ 0 the error. To determine the achievable minimal heat dissipation of quantum information erasure within a given physical context, we explicitly optimise over all possible unitary operators that act on the composite system of object and reservoir. Specifically, we characterise the equivalence class of such optimal unitary operators, using tools from majorisation theory, when we are restricted to finite-dimensional Hilbert spaces. Furthermore, we discuss how pure state preparation processes could be achieved with a smaller heat cost than Landauer's limit, by operating outside of Landauer's framework.
The Landauer principle states that any logically irreversible information processing must be accompanied by dissipation into the environment. In this Letter, we investigate the heat dissipation associated with finite-time information erasure and the effect of quantum coherence in such processes. By considering a scenario wherein information is encoded in an open quantum system whose dynamics are described by the Markovian Lindblad equation, we show that the dissipated heat is lower bounded by the conventional Landauer cost, as well as a correction term inversely proportional to the operational time. To clarify the relation between quantum coherence and dissipation, we derive a lower bound for heat dissipation in terms of quantum coherence. This bound quantitatively implies that the creation of quantum coherence in the energy eigenbasis during the erasure process inevitably leads to additional heat costs. The obtained bounds hold for arbitrary operational time and control protocol. By following an optimal control theory, we numerically present an optimal protocol and illustrate our findings by using a single-qubit system.