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.
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.
Ojha, Sardana, and Ghosh [Phys. Rev. A 113, 042221 (2026)] propose that tracing environmental records of particle permutations produces an entropy kB ln N!, thereby explaining the Gibbs factor, and use the same construction to multiply the Saha equilibrium relation by 1/(Ne!Np!). We show that these conclusions do not follow. The reduced density matrix printed in their Eq. (32) is not a partial trace, and even under the natural corrected interpretation the entropy equals kB ln N! only when the system branch states are mutually orthogonal. That condition is not generally satisfied and, when imposed on labeled permutation branches, does not by itself restrict the particle state to one bosonic or fermionic symmetry sector. A two-particle calculation makes the contradiction explicit. Their Gibbs-paradox calculation also starts from a distinguishable-particle entropy while calling it the Sackur-Tetrode entropy. In the Saha section, the state in Eq. (52), as written, factorizes and has zero system-environment entanglement. The proposed factorial does not approach unity in the claimed dilute-gas limit, does not yield a finite nonzero intensive thermodynamic limit, and counts indistinguishability twice. We give the corrected canonical and fugacity-based formulations and identify which standard results of the paper remain unaffected.
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.
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.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
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.
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.