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arXiv:2009.03110

Thermal Operations in general are not memoryless

Hanna Wojewódka–Ściążko, Stankiewicz, Maciej, Edgar A. Aguilar, Christopher Perry, Piotr Ćwikliński, Andrzej Grudka, Jarzynski, Christopher, Karol Horodecki, Michał Horodecki

Published Sep 7, 2020• 3 citations
arXivHTMLPDF

Abstract

So-called Thermal Operations seem to describe the most fundamental and reasonable set of operations allowable for state transformations at an ambient inverse temperature $β$. However, a priori, they require experimentalists to manipulate very complex environments and have control over their internal degrees of freedom. For this reason, the community has been working on creating more experimentally-friendly operations. In [Perry, et al., Phys. Rev. X 8, 041049], it was shown that for states diagonal in the energy basis, Thermal Operations can be performed by so-called Coarse Operations, which need just one auxiliary qubit but are otherwise Markovian and classical in spirit. In this work, by providing an explicit counterexample, we show that this one qubit of memory is necessary. We also provide almost full characterization (i.e., full, but with one intriguing exception that remains open) of possible transitions that do not require memory in the case where the system is a qubit. We do this by analyzing arbitrary control sequences comprising level energy changes and partial thermalizations in each step.

DOI
10.48550/arxiv.2009.03110

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So-called Thermal Operations seem to describe the most fundamental and reasonable set of operations allowable for state transformations at an ambient inverse temperature βββ. However, a priori, they require experimentalists to manipulate very complex environments and have control over their internal degrees of freedom. For this reason, the community has been working on creating more experimentally-friendly operations. In [Perry, et al., Phys. Rev. X 8, 041049], it was shown that for states diagonal in the energy basis, Thermal Operations can be performed by so-called Coarse Operations, which need just one auxiliary qubit but are otherwise Markovian and classical in spirit. In this work, by providing an explicit counterexample, we show that this one qubit of memory is necessary. We also provide almost full characterization (i.e., full, but with one intriguing exception that remains open) of possible transitions that do not require memory in the case where the system is a qubit. We do this by analyzing arbitrary control sequences comprising level energy changes and partial thermalizations in each step.
We develop a resource-theoretic framework that allows one to bridge the gap between two approaches to quantum thermodynamics based on Markovian thermal processes (which model memoryless dynamics) and thermal operations (which model arbitrarily non-Markovian dynamics).Our approach is built on the notion of memory-assisted Markovian thermal processes, where memoryless thermodynamic processes are promoted to non-Markovianity by explicitly modeling ancillary memory systems initialized in thermal equilibrium states.Within this setting, we propose a family of protocols composed of sequences of elementary two-level thermalizations that approximate all transitions between energy-incoherent states accessible via thermal operations.We prove that, as the size of the memory increases, these approximations become arbitrarily good for all transitions in the infinite temperature limit, and for a subset of transitions in the finite temperature regime.Furthermore, we present solid numerical evidence for the convergence of our protocol to any transition at finite temperatures.We also explain how our framework can be used to quantify the role played by memory effects in thermodynamic protocols such as work extraction.Finally, our results show that elementary control over two energy levels at a given time is sufficient to generate all energy-incoherent transitions accessible via thermal operations if one allows for ancillary thermal systems.
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The study of thermal operations allows one to investigate the ultimate possibilities of quantum states and of nanoscale thermal machines. Whilst fairly general, these results typically do not apply to continuous variable systems and do not take into account that, in many practically relevant settings, system-environment interactions are effectively bilinear. Here we tackle these issues by focusing on Gaussian quantum states and channels. We provide a complete characterization of the most general Gaussian thermal operation acting on an arbitrary number of bosonic modes, which turn out to be all embeddable in a Markovian dynamics, and derive necessary and sufficient conditions for state transformations under such operations in the single-mode case, encompassing states with nonzero coherence in the energy eigenbasis (i.e., squeezed states). Our analysis leads to a no-go result for the technologically relevant task of algorithmic cooling: We show that it is impossible to reduce the entropy of a system coupled to a Gaussian environment below its own or the environmental temperature, by means of a sequence of Gaussian thermal operations interspersed by arbitrary (even non-Gaussian) unitaries. These findings establish fundamental constraints on the usefulness of Gaussian resources for quantum thermodynamic processes.
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Providing the microscopic behavior of a thermalization process has always been an intriguing issue. There are several models of thermalization, which often require interaction of the system under consideration with the microscopic constituents of the macroscopic heat bath. With the aim to simulate such a thermalization process, here we look at the thermalization of a two-level quantum system under the action of a Markovian master equation corresponding to memory-less action of a heat bath, kept at a certain temperature, using a single-qubit ancilla. A two-qubit interaction Hamiltonian ( H th , say) is then designed —with a single-qubit thermal state as the initial state of the ancilla— which gives rise to thermalization of the system qubit in the infinite time limit. Further, we study the general form of Hamiltonian, of which ours is a special case, and look for the conditions for thermalization to occur. We also derive a Lindblad-type non-Markovian master equation for the system dynamics under the general form of system-ancilla Hamiltonian.
Thermal operations are a generic description for allowed state transitions under thermodynamic restrictions. However, the quest for simpler methods to encompass all these processes remains unfulfilled. We resolve this challenge through the catalytic use of thermal baths, which are assumed to be easily accessible. We select two sets of simplified operations: elementary thermal operations and Markovian thermal operations. They are known for their experimental feasibility, but fail to capture the full extent of thermal operations due to their innate Markovianity. We nevertheless demonstrate that this limitation can be overcome when the operations are enhanced by ambient-temperature Gibbs state catalysts. In essence, our result indicates that free states within thermal operations can act as catalysts that provide the necessary non-Markovianity for simpler operations. Furthermore, we prove that when any catalyst can be employed, different thermal processes (thermal operations, elementary thermal operations, and Markovian thermal operations) converge. Notably, our results extend to scenarios involving initial states with coherence in the energy eigenbasis, a notoriously difficult process to characterise.