A Qubit as a Bridge Between Statistical Mechanics and Quantum Dynamics

TL;DR

This paper explores how a single qubit can serve as a bridge between statistical mechanics and quantum dynamics by analyzing partition functions, Loschmidt amplitudes, and their zeros, revealing deep connections between equilibrium and dynamical properties.

Contribution

It introduces a unified framework linking thermal equilibrium and quantum dynamics using the analytic properties of a qubit, extending to spin chains.

Findings

Zeros of Loschmidt amplitude encode dynamical features.

High-temperature specific heat relates to early-time evolution.

Unified perspective connects partition functions and quantum dynamics.

Abstract

This work presents a unified perspective on thermal equilibrium and quantum dynamics by examining the simplest quantum system, a qubit, as a minimal model. We show that both the thermal partition function and the Loschmidt amplitude can be understood as extensions of a single analytic function along different paths in the complex plane. The zeros of Loschmidt amplitude encode dynamical features such as orthogonality, rate function singularities, and quantum speed limits, in analogy with the role of partition function zeros in equilibrium statistical mechanics. We further establish, through the Cauchy-Riemann equations, that the high-temperature specific heat corresponds to early-time evolution. The discussion follows a pedagogical progression from a single qubit to an interacting spin chain, all with finite dimensional Hilbert spaces.