Temperature Beyond Equilibrium in Isolated Quantum Many-Body Systems and Their Subsystems

Temperature is one of the central concepts of thermodynamics, yet its meaning far from equilibrium remains unclear. The problem is especially challenging in isolated quantum many-body systems, whose states evolve unitarily, may remain far from equilibrium, and retain energy coherence, a genuinely quantum feature with no direct classical counterpart. Specifically, energy fluctuations in a nonstationary quantum state have two distinct origins. Part of these fluctuations comes from uncertainty in the energy populations and has the usual thermodynamic meaning. The rest comes from quantum coherence between energy sectors and is responsible for the state's time dependence. We propose that, even away from equilibrium, temperature identifies the state within the family of regular states sharing the same energy-coherence structure. The resulting inverse temperature is obtained from a principle of minimum discrimination information, which selects the least distinguishable state compatible with the prescribed energy and coherence structure. This replaces the usual maximum-entropy principle, which, applied naively, can yield a biased estimator of the inverse temperature. We also lay the groundwork for an extension of the construction to subsystems and show that, although the inverse temperature of a subsystem is not determined by its reduced state alone, under an alignment assumption, its instantaneous rate of change is a local quantity for sufficiently large subsystems.