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arXiv:1807.02942 · Physical Review A

Thermal operations involving a single-mode bosonic bath

Xueyuan Hu, Feng Ding

Published Jan 8, 2019• 8 citations
arXivHTMLPDF

Abstract

We study the limitations on coherence evolutions under the constraints of thermodynamic laws, and focus on the optimal thermal operations (TO) reaching the bounds. For the qubit case, we prove that the TO that preserves maximal coherence can be reproduced by thermal operations involving only a single-mode reservoir, which we call single-mode thermal operations (STO). For higher dimensional systems, we derive tight limitations on coherence merging which are allowed by TO, and find that these bounds can also be reached by STO. By applying the coherence merging bounds to a two-qubit system, we observe that for some states, it is impossible to erase correlations while preserving the marginal states by TO. Our results reveal the strong ability of STO in coherence processing. Because interactions between quantum systems and single-mode reservoirs are realizable under recent techniques, our theoretical results are helpful to experimentally demonstrate the limitations of thermodynamic laws.

DOI
10.1103/physreva.99.012104

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We study the limitations on coherence evolutions under the constraints of thermodynamic laws, and focus on the optimal thermal operations (TO) reaching the bounds. For the qubit case, we prove that the TO that preserves maximal coherence can be reproduced by thermal operations involving only a single-mode reservoir, which we call single-mode thermal operations (STO). For higher dimensional systems, we derive tight limitations on coherence merging which are allowed by TO, and find that these bounds can also be reached by STO. By applying the coherence merging bounds to a two-qubit system, we observe that for some states, it is impossible to erase correlations while preserving the marginal states by TO. Our results reveal the strong ability of STO in coherence processing. Because interactions between quantum systems and single-mode reservoirs are realizable under recent techniques, our theoretical results are helpful to experimentally demonstrate the limitations of thermodynamic laws.
The second law of thermodynamics places a limitation into which states a system can evolve into. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and it says that a system can only evolve into another if the free energy goes down. Recently, it's been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are, in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of a single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture that more severe constraints exist. We also introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to a class of operations known as thermal operations.
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.
Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask whether several coherence transfers that can be optimized separately can also be optimized simultaneously by the same thermodynamic process. Using an energy-resolved Stinespring representation of thermal operations, we express elementary transfer amplitudes as inner products of bath-weighted vectors. Joint saturation imposes collinearity and phase-closure conditions on these vectors, which must all arise from one energy-preserving system--bath unitary. This gives necessary compatibility conditions for arbitrary finite-dimensional Hamiltonians. We show that joint incompatibility persists in a system with both degenerate and nondegenerate energy levels: rigorous computer-assisted semidefinite-program bounds, combined with explicit thermal constructions, prove that three individual thermal optima cannot be approached simultaneously. We then determine how well different coherences can be protected when their optima are incompatible. For a four-level family at infinite temperature, we obtain the complete closed retention region and the cost of balancing the coherences relative to their individual optima. At positive temperature, covariance and Gibbs preservation permit joint retention that no thermal operation can approach. Finite-bath constructions and certified semidefinite relaxations bound the balanced thermal performance. When two coherences carry independent phases, these bounds describe simultaneous preservation of interference visibility and phase Fisher information.
Heat-Bath Algorithmic Cooling is a technique for producing pure quantum systems by utilizing a surrounding heat-bath. Here we connect the study of these cooling techniques to the resource theory of thermal operations, enabling us to derive provably optimal cooling protocols under a variety of experimental restrictions on the available control. For qubit systems, we find that a surprisingly simple, optimal protocol consisting of repeated application of a Pauli XXX unitary and a thermal operation can achieve purity converging exponentially quickly to one. What is more, this thermal operation can be well approximated using a Jaynes Cummings interaction between the system and a single thermal bosonic mode and we consider experimental implementations of this. In addition, we investigate the role of quantum coherence and non-Markovianity in cooling protocols and extend our results to higher dimensional systems. Finally, by considering the role of correlations with auxiliary systems in cooling, we show that purity arbitrary close to one can be achieved in a fixed number of operations. Our results serve to find practical applications for the resource theoretic approach to quantum thermodynamics and suggest relevant experimental implementations.
The first law of thermodynamics imposes not just a constraint on the energy content of systems in extreme quantum regimes but also symmetry constraints related to the thermodynamic processing of quantum coherence.We show that this thermodynamic symmetry decomposes any quantum state into mode operators that quantify the coherence present in the state.We then establish general upper and lower bounds for the evolution of quantum coherence under arbitrary thermal operations, valid for any temperature.We identify primitive coherence manipulations and show that the transfer of coherence between energy levels manifests irreversibility not captured by free energy.Moreover, the recently developed thermomajorization relations on block-diagonal quantum states are observed to be special cases of this symmetry analysis.
To what extent is the resource theory approach to thermodynamics relevant for realistic experimental scenarios? We deconstruct this framework, showing that each transition among incoherent states allowed in the theory can be obtained by sequentially applying elementary thermal operations. These only couple the bath to single transitions in the system and satisfy detailed balance. We show that most of them can be realised within one of the simplest thermodynamic models, a resonant Jaynes-Cummings interaction with a single mode bosonic bath, and discuss the link to thermalisation models. No external control of the Hamiltonian or ancillae are needed. We also present an extension of Birkhoff's theorem for bistochastic matrices to thermodynamics, and discuss its consequences for the problem of simulating arbitrary thermal processes. Finally, we show that the constraints found within thermal operations for transitions among incoherent states are to a large extent \emph{universal}, i.e. they apply to a broad class consisting of very different thermodynamic frameworks.