Holonomy and Complementarity in Open Quantum Systems

TL;DR

This paper explores the geometric interpretation of complementarity relations in open quantum systems, revealing how dissipation and coherence influence the system's thermodynamic and geometric properties.

Contribution

It introduces a geometric framework for understanding complementarity, dissipation, and work in open quantum systems, linking quantum geometry with thermodynamic response.

Findings

Complementarity variables form cylindrical coordinates on the Bloch sphere.

Openness appears as a radial deficit in the geometric representation.

Curvature of the work connection determines cyclic response behavior.

Abstract

Complementarity relations constrain the distribution of coherence, predictability, and openness in quantum systems. Here we show that, in open quantum systems, these local constraints acquire a geometric interpretation through quasistatic transport. For a driven dissipative qubit, the complementarity variables define cylindrical coordinates on the Bloch sphere, while openness appears geometrically as a radial deficit associated with reduction from a larger Hilbert space. Quasistatic driving induces a work connection on the resulting steady-state manifold whose curvature determines the cyclic response. Hamiltonian-aligned dissipation produces an exact work connection and vanishing cyclic work, whereas fixed pointer-basis dissipation generates non-integrable transport, finite curvature, and holonomic response. The resulting curvature admits a phase-resolved representation on the triality manifold and develops perturbatively with pointer--Hamiltonian mismatch. In the weak-mismatch limit, the curvature is governed by a competition between coherence-preserving and pure-dephasing channels, producing symmetry-related positive- and negative-curvature sectors. These results establish a direct connection between complementarity, dissipation, and geometric thermodynamic response, and show that cyclic quasistatic work provides an operational probe of nonequilibrium quantum geometry.