Geometric complexity in thermodynamics

TL;DR

This paper establishes a universal geometric limit on the physical realization of reset operations in thermodynamics and quantum computation, linking complexity to error and unattainability of perfect reset.

Contribution

It introduces a fundamental, dynamics-independent bound on implementing reset maps based on geometric complexity, applicable to both classical and quantum systems.

Findings

Geometric complexity of maps is bounded below by execution error.

Zero-error reset requires divergent geometric complexity.

The bound applies universally across classical and quantum regimes.

Abstract

The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps. Although this unattainability principle has been quantified for specific state-to-state transitions, a universal, dynamics-independent bound for implementing a state-agnostic reset map remains elusive. In this work, we unveil the fundamental limits of physical map implementation by deriving a trade-off relation based on geometric complexity. By analyzing continuous paths of maps on a geometric manifold, we prove that the geometric complexity of any classical stochastic map or quantum channel is bounded from below by its execution error. As a consequence, we show that achieving zero error in a state-reset operation requires a divergent geometric complexity -- a unified measure that naturally incorporates disparate physical resources, including infinite time, energetic cost, or control bandwidth. This unattainability principle holds universally across both classical and quantum regimes, establishing a strict geometric limit on the physical realization of reset operations in thermodynamic control and quantum computation.