Geometry of restricted information: the case of quantum thermodynamics

Abstract

We formulate a geometric framework in which physical laws emerge from restricted access to microscopic information. Measurement constraints are modeled as a gauge symmetry acting on density operators, inducing a gauge-reduced space of physically distinguishable states. In the case of quantum thermodynamics, this construction leads to a gauge-invariant formulation in which the invariant entropy admits a stochastic description and satisfies a general detailed fluctuation theorem. From this result, we derive an integrated fluctuation theorem and a Clausius-like inequality that unifies the first and second laws in terms of invariant work and coherent heat. Entropy production is identified with the relative entropy between forward and backward probability measures on the gauge-reduced space of thermodynamic trajectories, revealing irreversibility as a geometric consequence of limited observability. The third law emerges as a singular zero-temperature limit in which thermodynamic orbits collapse and entropy production vanishes. Since the framework applies to arbitrary information constraints, it encompasses energy-based thermodynamics as a particular case of more general measurement scenarios.

Figures (2)

• Figure 1: The thermodynamic work $W_u = W_{\mathrm{inv}} - Q_c$ is compared to the bound of eq. \ref{['geometric_second_law']}. Particularly, we see that quantum coherence plays a very significant role in tightening the Clausius Inequality for this model.
• Figure 2: The thermodynamic work $W_u = W_{\mathrm{inv}} - Q_c$ is plotted against the bound of eq. \ref{['geometric_second_law']}. At the phase transition, an energy cost $T S_\Gamma$ associated to the generation of degeneracies arises. Due to the dynamics depending only on the $z$-magnetization, the system presents no coherence.