Geometric complexity in thermodynamics

• URL: http://arxiv.org/abs/2604.27858v1
• Date: Thu, 30 Apr 2026 13:41:37 GMT
• Title: Geometric complexity in thermodynamics
• Abstract summary: The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps.<n>We show that achieving zero error in a state-reset operation requires a divergent geometric complexity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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