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arXiv:2005.00261

Temperature of a finite-dimensional quantum system

Andrés Vallejo, A. Romanelli, R. Donangelo

Published May 1, 2020• 1 citation
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Abstract

A general expression for the temperature of a finite-dimensional quantum system is deduced from thermodynamic arguments. At equilibrium, this magnitude coincides with the standard thermodynamic temperature. Furthermore, it is well-defined even far from equilibrium. Explicit formulas for the temperature of two and three-dimensional quantum systems are presented, and some additional relevant aspects of this quantity are discussed.

DOI
10.48550/arxiv.2005.00261

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A general expression for the temperature of a finite-dimensional quantum system is deduced from thermodynamic arguments. At equilibrium, this magnitude coincides with the standard thermodynamic temperature. Furthermore, it is well-defined even far from equilibrium. Explicit formulas for the temperature of two and three-dimensional quantum systems are presented, and some additional relevant aspects of this quantity are discussed.
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A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterised by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order (i.e. continuous) phase transitions at finite temperatures.
Abstract While temperature is well understood as an intensive quantity in standard thermodynamics, it is less clear whether the same holds in quantum systems displaying correlations with no classical analogue. The problem lies in the fact that, under the presence of non-classical correlations, subsystems of a system in thermal equilibrium are, in general, not described by a thermal state at the same temperature as the global system and thus one cannot simply assign a local temperature to them. However, there have been identified situations in which correlations in thermal states decay sufficiently fast so that the state of their subsystems can be very well approximated by the reduced states of equilibrium systems that are only slightly bigger than the subsystems themselves, hence allowing for a valid local definition of temperature. In this work, we address the question of whether temperature is locally well defined for a bosonic system with local interactions that undergoes a phase transition at non-zero temperature. We consider a three-dimensional bosonic model in the grand canonical state and verify that a certain form of locality of temperature holds regardless of the temperature, and despite the presence of infinite-range correlations at and below the critical temperature of the phase transition.
We present two approaches to study the thermodynamic properties of d -dimensional classical systems in equilibrium. In the first approach we reduce this problem to the computation of the ground state properties of a d -dimensional quantum model. This classical-to-quantum mapping allows us to extend the scope of standard optimization methods by unifying them under a general framework. Particularly, we extend the quantum annealing method to study classical systems at finite temperatures. We derive the rates to assure convergence to the optimal thermodynamic state using the adiabatic theorem of quantum mechanics. In the second approach we present a quantum algorithm that performs numerical integration and computes, for example, the partition function of the system under study. We show that this quantum algorithm provides a quadratic speed-up with respect to the classical algorithm that samples with the uniform distribution and computes physical quantities of interest. Other quantum strategies, as well as their potential speed-up, are also discussed.
Considering an entropy-based division of energy transferred into heat and work, we develop an alternative theoretical framework for the thermodynamic analysis of two-level systems. When comparing these results with those obtained using the standard definitions of these quantities, we observe the appearance of a different term of work, which represents the energy cost of rotating the Bloch vector in the presence of the external field that defines the local Hamiltonian. Additionally, we obtain explicit expressions for the temperature, the heat capacity, and the internal entropy production of the system in both paradigms. In order to illustrate our findings we study, from both perspectives, matter-radiation interaction processes for two different systems.
In finite-dimensional quantum systems, temperature cannot be uniquely defined. This, in turn, implies that there are several ways to define entropy production in finite-dimensional quantum systems, because the classical entropy production depends on temperature. We propose a unified definition of entropy production based on the difference in quantum relative entropy with respect to reference states characterized by effective temperatures. We demonstrate that the proposed definition naturally decomposes into a Clausius-type entropy production and an additional contribution arising from the time dependence of the effective temperature. Furthermore, we show that requiring the entropy production rate to take the conventional form as the sum of the entropy change and the heat flow constrains the effective temperature to be either constant or equal to a specific energy-matching effective temperature. For general initial states, entropy production can become negative, in which case we derive lower bounds on entropy production and establish sufficient conditions for its non-negativity using the trace distance.