Information-theoretic equilibrium and observable thermalization

Abstract

A crucial point in statistical mechanics is the definition of the notion of thermal equilibrium, which can be given as the state that maximises the von Neumann entropy, under the validity of some constraints. Arguing that such a notion can never be experimentally probed, in this paper we propose a new notion of thermal equilibrium, focused on observables rather than on the full state of the quantum system. We characterise such notion of thermal equilibrium for an arbitrary observable via the maximisation of its Shannon entropy and we bring to light the thermal properties that it heralds. The relation with Gibbs ensembles is studied and understood. We apply such a notion of equilibrium to a closed quantum system and show that there is always a class of observables which exhibits thermal equilibrium properties and we give a recipe to explicitly construct them. Eventually, an intimate connection with the Eigenstate Thermalisation Hypothesis is brought to light.

New Equilibrium Ensembles for Isolated Quantum Systems

“…This implies that, in the thermodynamic limit, the probability distribution can become so narrow that the first moment might be representative of the whole probability distribution. This argument is usually invoked in synergy with the Eigenstate Thermalization Hypothesis to argue for the emergence of microcanonical expectation values [ 35 , 36 ]. For the same reason, in the thermodynamic limit, the Gaussian-shaped energy probability distribution that we obtain from can be so narrow that, also thanks to the action of the density of states , it effectively acts as a microcanonical probability distribution.…”

“…Borrowing the terminology from Seth Lloyd’s PhD thesis [ 31 ], we put all these works under the name of “Pure States Quantum Statistical Mechanics”. The theory is not yet a coherent and well understood set of statements, but it is founded on four main approaches: The Quantum Chaos approach [ 21 , 22 , 23 , 24 , 25 ], the Eigenstate Thermalisation Hypothesis (ETH) [ 14 , 32 , 33 , 34 , 35 ], the so-called Typicality Arguments [ 36 , 37 , 38 , 39 , 40 ], and the Dynamical Equilibration Approach [ 30 ]. All these approaches have a highly non-trivial overlap and their interplay is not yet fully understood.…”

Eigenstate thermalization hypothesis, time operator, and extremely quick relaxation of fidelity

“…Several different approaches have been studied, including through a restriction on the macroscopic observables [10,11], the general evaluation of relaxation time [12][13][14][15], the Eigenstate thermalization hypothesis (ETH) [1,[16][17][18][19][20][21], and dynamical experiments in autonomous cold atomic systems [22][23][24]. Of these, we focus on the foundation of ETH in terms of the time-energy uncertainty by noting that the energy eigenstates are globally distributed in the basis of suitably defined 'time operator' as detailed below (1) and in section 2.…”

“…In particular, we show that each energy eigenstate can be expressed as a superposition of many mutually almost orthogonal pure states that are considered thermal. Note that [20] quantified the degree of superposition with the use of Shannon entropy, which is basis dependent and maximized to guarantee ETH. Subsequently, [21] addressed the issue to specify a class of observables, such as local and extensive quantities, that satisfy ETH in terms of mutually unbiased basis with respect to the Hamiltonian.…”

Logarithmic growth of local entropy and total correlations in many-body localized dynamics

“…Therefore, in the MBL phase, the Néel state polarized along the z direction is very close to an energy eigenstate and has very short dynamics before saturation. A set of initial states which can be used to avoid this issue is given by the elements of a Hamiltonian Unbiased Basis [42,43] (HUB). A basis B := {|v µ } D µ=1 is called a HUB whenwhere |E ν are the Hamiltonian eigenstates and D is the dimension of the Hilbert space.…”