Purity of thermal mixed quantum states

TL;DR

This paper introduces a formula to evaluate the purity of thermal mixed quantum states using a new statistical quantity called NFPF, enabling efficient numerical estimation without prior knowledge of the quantum state.

Contribution

It develops a novel formula for purity based on NFPF and analyzes the efficiency of thermal mixed quantum state construction, validated through numerical experiments.

Findings

NFPF quantifies the efficiency of TMQ state mixtures.

Gibbs state has the largest NFPF and requires many samples.

Thermal pure quantum state has the smallest NFPF with a single sample.

Abstract

We develop a formula to evaluate the purity of a series of thermal equilibrium states that can be calculated in numerical experiments without knowing the exact form of the quantum state \textit{a priori}. Canonical typicality guarantees that there are numerous microscopically different expressions of such states, which we call thermal mixed quantum (TMQ) states. Suppose that we construct a TMQ state by a mixture of $N_\mathrm{samp}$ independent pure states. The weight of each pure state is given by its norm, and the partition function is given by the average of the norms. To qualify how efficiently the mixture is done, we introduce a quantum statistical quantity called "normalized fluctuation of partition function (NFPF)". For smaller NFPF, the TMQ state is closer to the equally weighted mixture of pure states, which means higher efficiency, requiring a smaller $N_\mathrm{samp}$. The largest NFPF is realized in the Gibbs state with purity-0 and exponentially large $N_\mathrm{samp}$, while the smallest NFPF is given for thermal pure quantum state with purity-1 and $N_\mathrm{samp}=1$. The purity is formulated using solely the NFPF and roughly gives $N_\mathrm{samp}^{-1}$. Our analytical results are numerically tested and confirmed by the two random sampling methods built on matrix-product-state-based wave functions.