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Matrix Thermodynamic Uncertainty Relation for Non-Abelian Charge Transport

Domingos S. P. Salazar

TL;DR

The paper addresses the challenge of formulating a thermodynamic uncertainty relation (TUR) for transport of non-Abelian (noncommuting) charges, where no single classical frame records all currents. It derives a fully nonlinear, saturable matrix TUR for the bath-relative entropy $D_{\mathrm{bath}}$ in terms of the transported-charge vector $\Delta q$ and the pre/post covariances $V,V'$, via a χ^2λ decomposition of Petz f-divergences and an optimization over witness directions. In the symmetric-covariance limit $V'=V$, the bound reduces to a closed-form scalar function $F(s)$ with $s=\Delta q^{\mathsf T}V^{-1}\Delta q$ and a corresponding semidefinite matrix form; the general case yields a tighter, parameter-free bound that remains valid beyond linear response and strong coupling. Numerical strong-coupling qubit collision simulations show near-saturation of the bound across broad parameter ranges, with measurements limited to local bath probes, highlighting the role of non-Abelian incompatibility (no joint measurement frame) encoded in the bound and the witness $C$. The framework provides a practical, time-reversal-free dissipation bound for genuinely quantum transport and opens paths to optimized non-Abelian transport protocols and experimental tests in spin, cold-atom, and superconducting platforms.

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 $Σ= D(ρ'_{SE}\|ρ'_S\!\otimes\!ρ_E)$. Isolating the experimentally accessible bath divergence $D_{\mathrm{bath}}=D(ρ'_E\|ρ_E)$, we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors $Δq$: $D_{\mathrm{bath}} \ge B(Δq,V,V')$, where the bound depends only on the transported-charge signal $Δq$ and the pre/post collision covariance matrices $V$ and $V'$. In the small-fluctuation regime $D_{\mathrm{bath}}\geq\frac12\,Δq^{\mathsf T}V^{-1}Δq+O(\|Δq\|^4)$, 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.

Matrix Thermodynamic Uncertainty Relation for Non-Abelian Charge Transport

TL;DR

The paper addresses the challenge of formulating a thermodynamic uncertainty relation (TUR) for transport of non-Abelian (noncommuting) charges, where no single classical frame records all currents. It derives a fully nonlinear, saturable matrix TUR for the bath-relative entropy in terms of the transported-charge vector and the pre/post covariances , via a χ^2λ decomposition of Petz f-divergences and an optimization over witness directions. In the symmetric-covariance limit , the bound reduces to a closed-form scalar function with and a corresponding semidefinite matrix form; the general case yields a tighter, parameter-free bound that remains valid beyond linear response and strong coupling. Numerical strong-coupling qubit collision simulations show near-saturation of the bound across broad parameter ranges, with measurements limited to local bath probes, highlighting the role of non-Abelian incompatibility (no joint measurement frame) encoded in the bound and the witness . The framework provides a practical, time-reversal-free dissipation bound for genuinely quantum transport and opens paths to optimized non-Abelian transport protocols and experimental tests in spin, cold-atom, and superconducting platforms.

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 . Isolating the experimentally accessible bath divergence , we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors : , where the bound depends only on the transported-charge signal and the pre/post collision covariance matrices and . In the small-fluctuation regime , 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.
Paper Structure (27 sections, 4 theorems, 70 equations, 2 figures)

This paper contains 27 sections, 4 theorems, 70 equations, 2 figures.

Table of Contents

  1. Symmetric-covariance regime $V'=V$.
  2. Matrix TUR in the symmetric-covariance regime ($V'=V$).
  3. SUPPLEMENTARY MATERIAL
  4. Petz f-divergences in left/right multiplication form
  5. One collision step, entropy production, mutual information split, and bath relative entropy
  6. Quadratic contrasts: chi2-lambda and integral representations
  7. Currents, covariance matrices, and a one-dimensional witness
  8. TUR from Chapman--Robbins at fixed lambda: general f and KL corollary
  9. Theorem: scalar TUR for a witness observable
  10. Proof.
  11. Theorem: matrix TUR via optimizing over u
  12. Proof.
  13. Proof.
  14. Small-current limit: quadratic form and relation to Onsager geometry
  15. Comparison with Kubo--Mori.
  16. ...and 12 more sections

Key Result

Theorem 1

For any operator-convex $f$ with integral weight $w_f(\lambda)\ge 0$ as in SM:eq:f_integral_rep, and any $u\in\mathbb{R}^m$,

Figures (2)

  • Figure 1: Bath relative entropy bound from microscopic simulations. For each random instance ($n=10^3$) of a system--bath interaction (cartoon inset), we simulate the reduced bath state $\rho_E\!\to\!\rho_E'$ and compute the bath relative entropy$D_{\mathrm{bath}}:=D(\rho_E'\| \rho_E)$ (vertical axis). The horizontal axis shows our bound $B(\Delta q,V,V')$, built from three empirically accessible ingredients: the vector of current variations $\Delta q$ and the associated covariance matrices $V$ and $V'$ (before/after the interaction) color-coded by the relative covariance drift $\|V'-V\|/\|V\|$ after the unitary. The dashed diagonal indicates saturation, $D_{\mathrm{bath}}=B(\Delta q,V,V')$. Inset: symmetric-covariance approximation $B\simeq F(s)$ as a function of $s=\delta q^{\mathsf T}V^{-1}\delta q$ (black), where $\delta q\equiv\Delta q$ denotes the small-current regime. The small-$s$ expansion $F(s)=s/2-s^2/12+O(s^3)$ (red dashed) captures the initial curvature, while deviations appear for larger currents ($s\gg 1$), as expected beyond linear response.
  • Figure 2: Near-saturation of the matrix TUR bound. Scatter of the relative slack $1-B/D_{\mathrm{bath}}$ versus the current magnitude $\|\Delta q\|$ for the qubit collision model. Here $D_{\mathrm{bath}}$ is the bath relative entropy (entropy production contribution associated with the bath state change), and $B\equiv B(\Delta q,V,V')$ is the predicted lower bound from the full (non-Abelian) TUR. The dashed line at $0$ marks exact saturation ($B=D_{\mathrm{bath}}$). Point colors encode the noncommutativity indicator $C\in[0,1]$ used in the main text. The solid black curve is a binned running average (guide to the eye), and the light-gray band indicates the corresponding spread within bins.

Theorems & Definitions (4)

  • Theorem 1: Witness TUR for $D_f(\rho'_E\|\rho_E)$
  • Corollary 1: KL witness bound
  • Lemma 1: Optimization identity at fixed $\lambda$
  • Theorem 2: Matrix TUR for the bath relative entropy