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arXiv:2505.23360

A hierarchy of thermodynamically consistent quantum operations

Fereshte Shahbeigi, Majid Mohammady

Published May 29, 2025• 0 citations
arXivHTMLPDF

Abstract

In order to determine what quantum operations and measurements are consistent with the laws of thermodynamics, one must start by allowing all processes allowed by the framework of quantum theory, and then impose the laws of thermodynamics as a set of constraints. Here, we consider a hierarchy of quantum operations and measurements that are consistent with ($I$) the weak third law, ($II$) the strong third law, and ($III$) both the second and the third laws of thermodynamics, i.e., operations and measurements that are fully consistent with thermodynamics. Such characterisation allows us to identify which particular thermodynamic principle is responsible for the (un)attainability of a given quantum operation or measurement. In the case of channels, i.e., trace-preserving operations, we show that a channel belongs to ($I$) and ($II$) if and only if it is strictly positive and rank non-decreasing, respectively, whereas a channel belongs to ($III$) only if it is rank non-decreasing and does not perturb a strictly positive state. On the other hand, while thermodynamics does not preclude the measurability of any POVM, the realisable state-update rules for measurements are increasingly restricted as we go from ($I$) to ($III$).

DOI
10.48550/arxiv.2505.23360

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In order to determine what quantum operations and measurements are consistent with the laws of thermodynamics, one must start by allowing all processes allowed by the framework of quantum theory, and then impose the laws of thermodynamics as a set of constraints. Here, we consider a hierarchy of quantum operations and measurements that are consistent with (III) the weak third law, (IIIIII) the strong third law, and (IIIIIIIII) both the second and the third laws of thermodynamics, i.e., operations and measurements that are fully consistent with thermodynamics. Such characterisation allows us to identify which particular thermodynamic principle is responsible for the (un)attainability of a given quantum operation or measurement. In the case of channels, i.e., trace-preserving operations, we show that a channel belongs to (III) and (IIIIII) if and only if it is strictly positive and rank non-decreasing, respectively, whereas a channel belongs to (IIIIIIIII) only if it is rank non-decreasing and does not perturb a strictly positive state. On the other hand, while thermodynamics does not preclude the measurability of any POVM, the realisable state-update rules for measurements are increasingly restricted as we go from (III) to (IIIIIIIII).
In the quantum regime, the third law of thermodynamics implies the unattainability of pure states. As shown recently, such unattainability implies that a unitary interaction between the measured system and a measuring apparatus can never implement an ideal projective measurement. In this paper, we introduce an operational formulation of the third law for the most general class of physical transformations, the violation of which is both necessary and sufficient for the preparation of pure states. Subsequently, we investigate how such a law constrains measurements of general observables, or positive operator-valued measures. We identify several desirable properties of measurements which are simultaneously enjoyed by ideal projective measurements---and are hence all ruled out by the third law in such a case---and determine if the third law allows for these properties to obtain for general measurements of general observables and, if so, under what conditions. It is shown that while the third law rules out some of these properties for all observables, others may be enjoyed by observables that are sufficiently ``unsharp.''
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Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum systems undergoing a general quantum evolution. We derive the first law of ther- modynamics for a process described by a completely-positive and trace-preserving map and show consistency with the Hatano-Sasa statement of the second law. We show that heat, from the first law, is positive when the input state of the map majorises the output state. Moreover, the change in entropy is also positive for the same majorisation condition. This makes a strong connection between the two operational laws of thermodynamics. Introduction.— The laws of thermodynamics were forged in the furnaces of the industrial revolution, as engineers and scientists refined their picture of energy, studying heat and its interconversion to mechanical work with a view to powering the mines and factories of this new era of human endeavor. Followed by the development of statistical mechanics at the change of the centuries (1), far from its pragmatic inception, thermodynamics is now a theory with a remarkable range of applicability, successfully describing the properties of macro- scopic systems ranging from refrigerators to black holes (2). Moving on to the 21st century with both the industrial and electronic revolutions behind us, we are currently pushing technology towards and beyond the microscopic scale. With a view to devices operating at a scale where quantum mech- anical laws become important we may ask whether the solid and some combination of the two (8). Finally, central to the work presented here is a work extraction formalism for non-passivity of quantum states (9). Despite the range of ap- proaches a more general picture for the thermodynamics of general quantum evolutions is far from clear. In this Letter, we take an operational approach to character- izing the energy change of an open quantum process described by a completely-positive trace-preserving (CPTP) map. In analogy to the first law of thermodynamics we discuss work done, extractable work, and heat. The concepts of ergotropy and adiabatic work allow us to state our main result: An op- erational first law for general quantum processes. We show that our operational first law is in agreement with widely used Hatano-Sasa version of the second law for CPTP maps (10, 11) by explicitly stating the Clausius inequality for unital and thermal maps. We then show that both operational heat and the change in von Neumann entropy are positive when the input state of the map majorises the output state. Thermodynamics of quantum systems.— The first law of thermodynamics states that the internal energy change in a thermodynamic process can be split up into two contributions - work and heat: dE = Q + W. For a generic quantum system, the internal energy at time t is E(t) = tr( (t)H(t)), implying that the change in the internal energy dE depends only on the end points. Heat and work on the other hand are path-dependent—thus the different notation for the 'differen- tials'. As an illustration we may consider the heat expended when pushing a piston into a cylinder filled with gas: It not only depends on the initial and final positions of the piston but also on how fast it is pushed. Using the derivative of the internal energy with respect to time the following two expres- sions are motivated (10):
Quantum theory combines density matrices, Born probabilities, tensor-product composites, positive-operator-valued measures (POVMs), and quantum channels. In a finite-dimensional causal operational theory, we prove that two postulates suffice: local input-output statistics identify channels, and every state admits an equivalent-system purification, unique up to reversible dynamics. The full complex quantum formalism follows; every consistent probability rule is realized as a POVM, so measurement no-restriction is derived rather than assumed.
The second law of thermodynamics places constraints on state transformations. It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are approximately cyclic, the second law for microscopic systems takes on a different form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of constraints. We find a family of free energies which generalize the traditional one, and show that they can never increase. The ordinary second law relates to one of these, with the remainder imposing additional constraints on thermodynamic transitions. We find three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are relevant for small systems, and also apply to individual macroscopic systems interacting via long-range interactions. By making precise the definition of thermal operations, the laws of thermodynamics are unified in this framework, with the first law defining the class of operations, the zeroth law emerging as an equivalence relation between thermal states, and the remaining laws being monotonicity of our generalized free energies.
Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum system undergoing quantum evolution under a general quantum process by which we mean a completely positive and trace-preserving map. We derive an operational first law of thermodynamics for such processes and show consistency with the second law. We show that heat, from the first law, is positive when the input state of the map majorizes the output state. Moreover, the change in entropy is also positive for the same majorization condition. This makes a strong connection between the two operational laws of thermodynamics.