Statistical entropy of quantum systems

TL;DR

The paper investigates when the quantum von Neumann entropy $S_{VN}$ serves as the thermodynamic entropy for quantum systems, addressing foundational questions in quantum statistical mechanics and thermalization. It develops quantum analogues of Boltzmann and Gibbs entropies, $S_{qBN}$ and $S_{qGB}$, and shows that, in a basis-independent form, these entropies coincide with $S_{VN}$ under appropriate conditions, particularly for large systems in or near equilibrium. It discusses two interpretations and key objections to $S_{VN}$ (time-invariance and subadditivity), arguing that these concerns can be reconciled within equilibrium and open-system frameworks, and extends the equivalence to subsystems via reduced states and canonical typicality. Numerical results from a spin-1/2 chain illustrate how subsystem VN entropy scales with the density of states and supports the equivalence in nonintegrable (chaotic) regimes, while highlighting potential deviations in integrable settings relevant to ETH and thermalization.

Abstract

Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.

Figures (4)

• Figure 1: The average VN entropy, $\overline{S}^{sb}_{VN}$, is plotted as a function of the subsystem size $l_1$. Here total system size is $N=16$.
• Figure 2: The subsystem VN entropy, $S^{sb}_{VN}(\rho_{sb})$, for the individual eigenkets and $\ln (DOS)$ are plotted against the energy of the full system. Here total system size is 16 and subsystem size is 6.
• Figure 3: The subsystem VN entropy, $S^{sb}_{VN}(\overline{\rho}_{sb})$, when the full system is in the microcanonical density matrix $\rho_{mc}$. Here, the system (subsystem) size is 16 (6).
• Figure 4: The average VN entropy, $\overline{S}^{sb}_{VN}(\rho_{sb})$, of subsystem is plotted against $\ln (DOS)$ across the energy spectrum (separately for the left and the right halves of the spectrum). Here, the system (subsystem) size is 16 (6) and $\Delta_2=0.5$.