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arXiv:quant-ph/0505229

On distinguishability, orthogonality, and violations of the second law: contradictory assumptions, contrasting pieces of knowledge

PierGianLuca Porta Mana, A. Maansson, Gunnar G. Bjork

Published May 31, 2005• 1 citation

Abstract

Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.

DOI: 10.48550/arxiv.quant-ph/0505229

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Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from two different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.

The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.

For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.

Differences of opinion concerning fundamental issues in statistical mechanics directly related to the thermodynamic entropy have persisted through more than a century of debate. One reason is the lack of consensus on the definitions of key terms, especially the terms “distinguishable,” “indistinguishable,” and “identical.” Several definitions occur in the literature, but are not always made explicit. The multiplicity of definitions has created confusion about the basic conditions under which entropy is to be defined. In this paper, I present an overview of definitions in current use for terms associated with distinguishability and relate them to various definitions that have been suggested for entropy. My hope is that consensus will be achievable if the definitions are clarified and agreed upon.