Matrix Thermodynamic Uncertainty Relation for Non-Abelian Charge Transport

TL;DR

This paper derives a matrix thermodynamic uncertainty relation for non-Abelian charge transport, providing a nonlinear, saturable lower bound on bath divergence that accounts for quantum noncommuting charges and is validated through numerical simulations.

Contribution

It introduces a process-level matrix TUR for non-Abelian charges, extending classical bounds to quantum regimes with nonlinear, experimentally accessible bounds.

Findings

Derived a nonlinear, saturable lower bound for non-Abelian charge transport.

Validated the bound through numerical simulations of qubit collisions.

Bound remains accurate beyond linear response regime.

Abstract

Thermodynamic uncertainty relations (TURs) bound the precision of currents by entropy production, but quantum transport of noncommuting (non-Abelian) charges challenges standard formulations because different charge components cannot be monitored within a single classical frame. We derive a process-level matrix TUR starting from the operational entropy production $\Sigma = D(\rho'_{SE}\|\rho'_S\!\otimes\!\rho_E)$. Isolating the experimentally accessible bath divergence $D_{\mathrm{bath}}=D(\rho'_E\|\rho_E)$, we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors $\Delta q$: $D_{\mathrm{bath}} \ge B(\Delta q,V,V')$, where the bound depends only on the transported-charge signal $\Delta q$ and the pre/post collision covariance matrices $V$ and $V'$. In the small-fluctuation regime $D_{\mathrm{bath}}\geq\frac12\,\Delta q^{\mathsf T}V^{-1}\Delta q+O(\|\Delta q\|^4)$, while beyond linear response it remains accurate. Numerical strong-coupling qubit collisions illustrate the bound and demonstrate near-saturation across broad parameter ranges using only local measurements on the bath probe.