A Qubit as a Bridge Between Statistical Mechanics 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.

Figures (3)

• Figure 1: A schematic diagram of a qubit's energy levels with an energy gap $\Delta E$ of magnitude $J (>0)$ between the two levels. On the right, the orthonormality conditions of the basis states are shown.
• Figure 2: The blue line represents the real Boltzmann factor from statistical mechanics, which never intersects the isolated zero at $y = -1$ (the green point). The red unit circle corresponds to quantum time evolution. It traces the complex Boltzmann factor, periodically crossing the zero. The brown point at $y = 1$ marks both the infinite-temperature limit $\beta \to 0$ and the initial time of the quantum system $t \to 0$.
• Figure 3: Periodic logarithmic divergences are observed in the real dynamical free energy $f_L(y)$ as a function of scaled time in (i) a qubit (pink) and (ii) TFIM with open boundary conditions (green).