Geometry of restricted information: the case of quantum thermodynamics

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.

💡 Research Summary

The paper develops a geometric framework in which physical laws emerge from the limited microscopic information available to an observer. The authors model measurement constraints as a gauge symmetry acting on the space of density operators. This gauge group, denoted (G_T), consists of unitary transformations that act non‑trivially only within degenerate subspaces of the instantaneous Hamiltonian. When an observer can measure only a restricted set of observables (for example, energy), many distinct density operators become indistinguishable; they belong to the same gauge orbit. By quotienting the full Hilbert space by (G_T), one obtains a reduced “gauge‑invariant” state space that contains only the physically accessible information.

Two geometric structures are introduced. First, a trivial principal (U(d))‑bundle over the time axis is equipped with a connection given by the Maurer–Cartan form (A_t=\dot u_t u_t^\dagger), where (u_t) diagonalises the Hamiltonian (H_t). The associated covariant derivative (\nabla_t=\partial_t+