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arXiv:1504.06572

Comments on the paper "Foundation of statistical mechanics from symmetries of entanglement" by S. Deffner and W. Zurek

Robert Alicki

Published Apr 23, 2015• 0 citations
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Abstract

The authors of the recent paper [1] boldly claim to discover a new fully quantum approach to foundation of statistical mechanics: "Our conceptually novel approach is free of mathematically ambiguous notions such as probability, ensemble, randomness, etc." The aim of this note is to show that this approach is neither specific for quantum systems nor really conceptually different from the standard textbook arguments supporting microcanonical or canonical ensembles in statistical mechanics.

DOI
10.48550/arxiv.1504.06572

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Comments

Bernard d’Espagnat

This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.

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The authors of the recent paper [1] boldly claim to discover a new fully quantum approach to foundation of statistical mechanics: "Our conceptually novel approach is free of mathematically ambiguous notions such as probability, ensemble, randomness, etc." The aim of this note is to show that this approach is neither specific for quantum systems nor really conceptually different from the standard textbook arguments supporting microcanonical or canonical ensembles in statistical mechanics.
The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective quantum uncertainties due to entanglement and uncertainty relations. No additional randomness is added by hand and no assumptions about a priori probabilities are made, instead measure concentration results are used to justify the methods of Statistical Physics. The approach explains the applicability of the microcanonical and canonical ensemble and the tendency to equilibrate in a natural way. This work contains a pedagogical review of the existing literature and some new results. The most important of which are: i) A measure theoretic justification for the microcanonical ensemble. ii) Bounds on the subsystem equilibration time. iii) A proof that a generic weak interaction causes decoherence in the energy eigenbasis. iv) A proof of a quantum H-Theorem. v) New estimates of the average effective dimension for initial product states and states from the mean energy ensemble. vi) A proof that time and ensemble averages of observables are typically close to each other. vii) A bound on the fluctuations of the purity of a system coupled to a bath.
The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective quantum uncertainties due to entanglement and uncertainty relations. No additional randomness is added by hand and no assumptions about a priori probabilities are made, instead measure concentration results are used to justify the methods of Statistical Physics. The approach explains the applicability of the microcanonical and canonical ensemble and the tendency to equilibrate in a natural way. This work contains a pedagogical review of the existing literature and some new results. The most important of which are: i) A measure theoretic justification for the microcanonical ensemble. ii) Bounds on the subsystem equilibration time. iii) A proof that a generic weak interaction causes decoherence in the energy eigenbasis. iv) A proof of a quantum H-Theorem. v) New estimates of the average effective dimension for initial product states and states from the mean energy ensemble. vi) A proof that time and ensemble averages of observables are typically close to each other. vii) A bound on the fluctuations of the purity of a system coupled to a bath.
This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.
Envariance-entanglement assisted invariance-is a recently discovered symmetry of composite quantum systems.We show that thermodynamic equilibrium states are fully characterized by their envariance.In particular, the microcanonical equilibrium of a system  with Hamiltonian  H is a fully energetically degenerate quantum state envariant under every unitary transformation.The representation of the canonical equilibrium then follows from simply counting degenerate energy states.Our conceptually novel approach is free of mathematically ambiguous notions such as ensemble, randomness, etc., and, while it does not even rely on probability, it helps to understand its role in the quantum world.
Envariance—entanglement assisted invariance—is a recently discovered symmetry of composite quantum systems. We show that thermodynamic equilibrium states are fully characterized by their envariance. In particular, the microcanonical equilibrium of a system S with Hamiltonian HS is a fully energetically degenerate quantum state envariant under every unitary transformation. The representation of the canonical equilibrium then follows from simply counting degenerate energy states. Our conceptually novel approach is free of mathematically ambiguous notions such as ensemble, randomness, etc., and, while it does not even rely on probability, it helps to understand its role in the quantum world.