Maximum entropy distributions of wavefunctions at thermal equilibrium

Abstract

Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wavefunctions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wavefunction ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wavefunctions because they lack a definite energy. In this manuscript, we present a maximum entropy principle for the quantum wavefunction ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the Rényi divergence of the ensemble with respect to the Gibbs state, indicating that the Rényi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.

Figures (8)

• Figure 1: Maximum entropy distribution derived from different physical constraints. (a) An ensemble of wavefunctions $P(\Gamma)$ can be compactly represented by a density operator $\rho$. Maximizing the entropy of $P(\Gamma)$ is not equivalent to maximizing the entropy of $\rho$, subject to the same constraint. (b) The energy constrained ensemble (ECE) is the quantum analogue of the classical canonical ensemble, maximizing entropy under an average energy constraint. (c) The Gibbs-state constrained ensemble (GCE) maximizes entropy among distributions that average to the Gibbs state. (d) The Scrooge ensemble maximizes entropy under a constraint on the average Rényi divergence $D(\Gamma\|\rho)$ of $\Gamma$ from $\rho$.
• Figure 2: The populations of the ECE (red stars) do not agree with the Gibbs state (black stars).(a) The thermal populations at high temperature are similar. (b) At low temperature, the ground state and higher excited state populations are increased in the ECE. Inset shows populations on a logarithmic scale. Results were calculated for the Hamiltonian spectrum $\sigma\{H\}=\{k\}_{k=0}^{9}$.
• Figure 3: The hereditary property is violated by the GCE and satisfied by the Scrooge ensemble. The Kolmogorov-Smirnov (KS) statistic of the projected ensemble and thermal distribution of the system as a function of inverse temperature $\beta$. The GCE satisfies the KS test at high temperatures for the significance levels of $\alpha = 5\%$ and $0.5\%$, but fails at low temperatures. The Scrooge ensemble satisfies the hereditary property for all temperatures.Goldstein2006 Results are calculated by Monte Carlo sampling, with the system and bath spectra given by $\sigma\{H\} = \{0,1\}$. Error bars are standard error in the mean over $10$ trials with $10^7$ samples.
• Figure S1: The statistical temperature $\beta'$ of the ECE is not the thermodynamic temperature $\beta$. Results are calculated for Hamiltonians with equally spaced eigenvalues $\sigma\{H\}=\{k\}_{k=0}^{N-1}$, where $N$ is the dimension. Dashed lines are the high temperature limit $\beta'=(N+1)\beta$.
• Figure S2: The Lagrange multipliers $\mu_i$ of the GCE can be well-approximated by an ansatz. (a) The numerical solution to \ref{['eqn:mu_equations']} (stars) is well approximated by the ansatz \ref{['eqn:mu_ansatz']} (lines). (b) The difference between the numerical solution and ansatz vanishes at high and low temperatures. At intermediate temperatures, the difference is small compared to the absolute value of $\mu_i$. The results are calculated for a Hamiltonian spectrum $\sigma\{H\} = \{0,1,2,3\}$.