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arXiv:1504.02797 · New Journal of Physics

Foundations of statistical mechanics from symmetries of entanglement

Sebastian Deffner, Wojciech H Zurek

Published Jun 9, 2016• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/18/6/063013

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

Robert Alicki

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.

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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.
We build on the foundational work of Deffner and Zurek [S. Deffner and W. H. Zurek, New J. Phys. 18, 063013 (2016)] to show how central equilibrium structures of statistical mechanics can be understood within standard quantum mechanics using the concept of envariance (environment-assisted invariance). In particular, we show how the Binomial, Poisson, and Gaussian distributions naturally emerge from entangled system-environment states. We revisit the Gibbs paradox from a quantum information perspective, demonstrating that the standard Sackur-Tetrode entropy and its 1/N! factor arise from indistinguishability enforced through entanglement with an environment, without introducing additional thermodynamic corrections. Within the same framework, we analyze ionization equilibrium and show how the classical Saha equation is recovered, while clarifying how indistinguishability enters through an entanglement-induced reduction of permutation redundancy. Assuming the standard exchange symmetries of identical quantum particles, we further show how the Bose-Einstein and Fermi-Dirac distributions follow as the equilibrium weighting of symmetry-allowed occupation configurations. Overall, our results support the view that equilibrium statistical mechanics can be consistently interpreted as an emergent consequence of quantum information-theoretic structure and symmetry, rather than as a collection of independent phenomenological postulates.
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.
I show how probabilities arise in quantum physics by exploring the implications of environment-assisted invariance or envariance, a recently discovered symmetry exhibited by entangled quantum systems. Envariance of perfectly entangled ``Bell-like'' states can be used to rigorously justify complete ignorance of the observer about the outcome of any measurement on either of the members of the entangled pair. For more general states, envariance leads to Born's rule ${p}_{k}\ensuremath{\propto}{\ensuremath{\mid}{\ensuremath{\psi}}_{k}\ensuremath{\mid}}^{2}$ for the outcomes associated with Schmidt states. The probabilities derived in this manner are an objective reflection of the underlying state of the system---they represent experimentally verifiable symmetries, and not just a subjective ``state of knowledge'' of the observer. This envariance-based approach is compared with and found to be superior to prequantum definitions of probability including the standard definition based on the ``principle of indifference'' due to Laplace and the relative frequency approach advocated by von Mises. Implications of envariance for the interpretation of quantum theory go beyond the derivation of Born's rule: Envariance is enough to establish the dynamical independence of preferred branches of the evolving state vector of the composite system and, thus, to arrive at the environment-induced superselection (einselection) of pointer states, which was usually derived by an appeal to decoherence. The envariant origin of Born's rule for probabilities sheds light on the relation between ignorance (and, hence, information) and the nature of quantum states.
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrodinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.
We demonstrate that the one-time measurement scheme is more consistent than the standard two-time measurement scheme with respect to the fundamental symmetry of entanglement, namely entanglement-assisted invariance (envariance). In particular, the work statistics obtained from the one-time measurement scheme arise naturally as a direct consequence of envariance for any driving unitary, without involving an explicit measurement postulate, whereas the analogous identification for the two-time measurement scheme is possible only for a Haar-measure-zero set of unitaries. This envariance-based formulation eliminates the ambiguity in work statistics related to the energetic cost of measurement.