Derivation and Improvements of the Quantum Canonical Ensemble from a Regularized Microcanonical Ensemble

For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptio...

Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e. the wavefunction, of an isolated system which is determined to calculate molecular properties and to consider the time evolution according to the unitary Schr\"odinger equation. On the other hand a mixed state, i.e. a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical qua...

The standard assumption for the equilibrium microcanonical state in quantum mechanics, that the system must be in one of the energy eigenstates, is weakened so as to allow superpositions of states. The weakened form of the microcanonical postulate thus asserts that all quantum states giving rise to the same energy expectation value must be realised with equal probability. The consequences that follow from this assertion are investigated. In particular, a closed-form expressio...

In this didactical note I review in depth the rationale for using generalised canonical distributions in quantum statistics. Particular attention is paid to the proper definitions of quantum entropy and quantum relative entropy, as well as to quantum state reconstruction on the basis of incomplete data. There are two appendices in which I outline how generalised canonical distributions link to the conventional formulation of statistical mechanics, and how classical probabilit...

It is shown, that the only reason for possibility of using microcanonical ensemble is that there are probabilistic processes in microworld, that are not described by quantum mechanic. On a simple example it is demonstrated, that canonical distribution is not independent and equal to microcanonical, but is a result of averaging by microcanonical ensemble.

This work explores fundamental statistical and thermodynamic properties of short-and long-range-interacting systems. The purpose of this study is twofold. Firstly, we rigorously prove that the probability distribution of arbitrary few-body observables is restricted by a Gaussian concentration bound (or Chernoff--Hoeffding inequality) above some threshold temperature. This bound is then derived for arbitrary Gibbs states of systems that include long-range interactions Secondly...

Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of mutually interacting subsystem and heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the "hypothesis of equal weights for eigenstates", we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (ins...

We investigate how the temperature calculated from the microcanonical entropy compares with the canonical temperature for finite isolated quantum systems. We concentrate on systems with sizes that make them accessible to numerical exact diagonalization. We thus characterize the deviations from ensemble equivalence at finite sizes. We describe multiple ways to compute the microcanonical entropy and present numerical results for the entropy and temperature computed in these var...

We introduce two kinds of quantum algorithms to explore microcanonical and canonical properties of many-body systems. The first one is a hybrid quantum algorithm that, given an efficiently preparable state, computes expectation values in a finite energy interval around its mean energy. This algorithm is based on a filtering operator, similar to quantum phase estimation, which projects out energies outside the desired energy interval. However, instead of performing this operat...

In the microcanonical thermal pure quantum (mTPQ) method, the canonical ensemble is derived using Taylor series expansions. We prove that the truncation error decreases exponentially with system size when the effective temperature of the mTPQ state is smaller than the target temperature, and otherwise, the error remains constant. We also show the discipline to set the mTPQ parameter by considering the trade-off between the error and the numerical cost.