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arXiv:2212.13027

Quantum ensembles and the statistical operator: a tutorial

Yinxiù Zhan, Matteo G. A. Paris

Published Dec 26, 2022• 0 citations
arXivHTMLPDF

Abstract

The main purpose of this tutorial is to elucidate in details what should be meant by ensemble of states in quantum mechanics, and to properly address the problem of discriminating, exactly or approximately, two different ensembles. To this aim, we review the notion and the definition of quantum ensemble as well as its relationships with the concept of statistical operator in quantum mechanics. We point out the implicit assumptions contained in introducing a correspondence between quantum ensembles and the corresponding single-particle statistical operator, and discuss some issues arising when these assumptions are not satisfied. We review some subtleties leading to apparent paradoxes, and illustrate the role of approximate quantum cloning. In particular, we review some examples of practical interest where different (but equivalent) preparations of a quantum system, i.e. different ensembles corresponding to the same single-particle statistical operator, may be successfully discriminated exploiting multiparticle correlations, or some a priori knowledge about the number of particles in the ensemble.

DOI
10.48550/arxiv.2212.13027

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The main purpose of this tutorial is to elucidate in details what should be meant by ensemble of states in quantum mechanics, and to properly address the problem of discriminating, exactly or approximately, two different ensembles. To this aim, we review the notion and the definition of quantum ensemble as well as its relationships with the concept of statistical operator in quantum mechanics. We point out the implicit assumptions contained in introducing a correspondence between quantum ensembles and the corresponding single-particle statistical operator, and discuss some issues arising when these assumptions are not satisfied. We review some subtleties leading to apparent paradoxes, and illustrate the role of approximate quantum cloning. In particular, we review some examples of practical interest where different (but equivalent) preparations of a quantum system, i.e. different ensembles corresponding to the same single-particle statistical operator, may be successfully discriminated exploiting multiparticle correlations, or some a priori knowledge about the number of particles in the ensemble.
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.
The conceptual basis of statistical operator is analysed by comparing two different ways of introducing it in quantum mechanics–subsystem method and ensemble method. The one-to-one correspondence of the statistical operator to the physical state properly characterized is contrasted with the deficient degrees of freedom of the wave function. The paradoxical features of the Einstein-Podolsky-Rosen type thought experiment and of the reduction of wave-packet at observation can be dissolved in the reformulation of quantum mechanics in which the statistical operator, instead of the wave function, is primarily introduced as the fundamental quantity to describe the state.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
Statistical physics examines the collective properties of large ensembles of particles, and is a powerful theoretical tool with important applications across many different scientific disciplines. This book provides a detailed introduction to classical and quantum statistical physics, including links to topics at the frontiers of current research. The first part of the book introduces classical ensembles, provides an extensive review of quantum mechanics, and explains how their combination leads directly to the theory of Bose and Fermi gases. This allows a detailed analysis of the quantum properties of matter, and introduces the exotic features of vacuum fluctuations. The second part discusses more advanced topics such as the two-dimensional Ising model and quantum spin chains. This modern text is ideal for advanced undergraduate and graduate students interested in the role of statistical physics in current research. 140 homework problems reinforce key concepts and further develop readers' understanding of the subject.
Statistical Physics examines the collective properties of large ensembles of particles, and is a powerful tool with important applications across many different disciplines. This book provides a detailed introduction to classical and quantum statistical physics, including links to topics at the frontiers of current research. The first part of the the book introduces classical ensemble, provides an extensive review of Quantum Mechanics, and explains how their combination leads to the theory of Bose and Fermi gases. This allows a detailed analysis of the quantum properties of matter and introduces the exotic features of vacuum fluctuations. The second part discusses more advanced topics such as the two-dimensional Ising model and quantum spin chains. This modern text is ideal for advanced undergraduate and graduate students interested in the role of statistical physics in current research. One hundred and forty homework problems reinforce key concepts and further develop reader's understanding of the subject.
Identical particles underpin most quantum technologies, but the role of indistinguishability in these processes extends far beyond the symmetrization of many-particle states. This Colloquium presents a unified operational perspective on how particle identity, indistinguishability, exchange statistics, and many-particle interference can instead be harnessed as resources for quantum information processing. We first clarify the notions of entanglement and indistinguishability for identical particles, emphasizing their dependence on the accessible measurements and information. We then review how spatial overlap, multiparticle interference, and path identity enable the generation, extraction, protection, and distillation of coherence and entanglement. Their applications to quantum metrology and imaging, state discrimination, teleportation, entanglement swapping, and quantum cloning are examined. Finally, we discuss the thermodynamic role of indistinguishability, from the Gibbs paradox to quantum thermal machines and statistics-driven engines. Across these settings, indistinguishability emerges as a controllable, task-dependent resource whose benefits are determined by state preparation, exchange statistics, interference, and measurement constraints.