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Gaussian time-translation covariant operations: structure, implementation, and thermodynamics

Xueyuan Hu, Lea Lautenbacher, Giovanni Spaventa, Martin B. Plenio, Nelly H. Y. Ng, Jeongrak Son

TL;DR

Time-translation symmetry constrains Gaussian dynamics, and this work provides a complete classification of Gaussian covariant operations (GCOs). It introduces a minimal parameterization by $(A,B)$ that governs the moment updates $\vec{\alpha}\to A\vec{\alpha}$, $\mu\to A\mu A^{\dagger}+B$, and $\chi\to A^{*}\chi A^{\dagger}$ with $B\ge\frac{1}{2}(\mathbb{I}-AA^{\dagger})$, and shows frequency decoupling reduces the analysis to the single-frequency case. It proves a free-dilation criterion and establishes the equivalence of Gaussian enhanced thermal operations (GEnTO) and Gaussian thermal operations (GTO), thereby closing the gap in the Gaussian regime; it also introduces the completely non-extensive Sl_± monotones, complete for single-mode transformations, as well as a frequency-agnostic framework for thermodynamic resource theories in Gaussian optics. Collectively, these results illuminate the interplay between symmetry, Gaussianity, and thermodynamics, and provide practical tools for metrology and quantum thermodynamics in continuous-variable systems.

Abstract

Time-translation symmetry strongly constrains physical dynamics, yet systematic characterization for continuous-variable systems lags behind its discrete-variable counterpart. We close this gap by providing a rigorous classification of Gaussian quantum operations that are covariant under time translations, termed Gaussian covariant operations. We show that several key results known for discrete-variable covariant operations break down in the Gaussian optical setting: discrepancies arise in physical and thermodynamic implementation, in the extensivity of asymmetry, and in catalytic advantages. Our results provide comprehensive mathematical and operational toolkits for Gaussian covariant operations, including a peculiar pair of asymmetry measures that are completely non-extensive. Our findings also reveal surprising consequences of the interplay among symmetry, Gaussianity, and thermodynamic constraints, suggesting that real-world scenarios with multiple constraints have a rich structure not accessible from examining individual constraints separately.

Gaussian time-translation covariant operations: structure, implementation, and thermodynamics

TL;DR

Time-translation symmetry constrains Gaussian dynamics, and this work provides a complete classification of Gaussian covariant operations (GCOs). It introduces a minimal parameterization by that governs the moment updates , , and with , and shows frequency decoupling reduces the analysis to the single-frequency case. It proves a free-dilation criterion and establishes the equivalence of Gaussian enhanced thermal operations (GEnTO) and Gaussian thermal operations (GTO), thereby closing the gap in the Gaussian regime; it also introduces the completely non-extensive Sl_± monotones, complete for single-mode transformations, as well as a frequency-agnostic framework for thermodynamic resource theories in Gaussian optics. Collectively, these results illuminate the interplay between symmetry, Gaussianity, and thermodynamics, and provide practical tools for metrology and quantum thermodynamics in continuous-variable systems.

Abstract

Time-translation symmetry strongly constrains physical dynamics, yet systematic characterization for continuous-variable systems lags behind its discrete-variable counterpart. We close this gap by providing a rigorous classification of Gaussian quantum operations that are covariant under time translations, termed Gaussian covariant operations. We show that several key results known for discrete-variable covariant operations break down in the Gaussian optical setting: discrepancies arise in physical and thermodynamic implementation, in the extensivity of asymmetry, and in catalytic advantages. Our results provide comprehensive mathematical and operational toolkits for Gaussian covariant operations, including a peculiar pair of asymmetry measures that are completely non-extensive. Our findings also reveal surprising consequences of the interplay among symmetry, Gaussianity, and thermodynamic constraints, suggesting that real-world scenarios with multiple constraints have a rich structure not accessible from examining individual constraints separately.
Paper Structure (8 sections, 19 theorems, 96 equations, 1 figure)

This paper contains 8 sections, 19 theorems, 96 equations, 1 figure.

Table of Contents

  1. Full characterization of Gaussian Covariant Channels
  2. Full characterization of freely dilatable Gaussian covariant channels
  3. Closure of gap between TOs and EnTOs in the Gaussian regime
  4. The monotones of Gaussian covariant channels
  5. Proof of finiteness and faithfulness (P1 in main text)
  6. Proof of monotonicity (P2 in main text)
  7. Proof of complete non-extensiveness (P3 in main text)
  8. Monotonicity under correlated catalysis (P4 in main text)

Key Result

Lemma 1

A GCO $(A,B)$ is freely dilatable if and only if the following two conditions are satisfied. (F1) $\mathbb{I}-AA^\dagger\geq0$; (F2) $\mathrm{supp}(B)= \mathrm{supp}(\mathbb{I}-AA^\dagger)$.

Figures (1)

  • Figure 1: Single-mode state transformations for the second moment $(\mu,\chi)$. Colored parts represent the reachable states from a given initial state (red dot) via GCO (purple region), GTO (red line), and GTO with a correlated catalysis.

Theorems & Definitions (37)

  • Definition 1: Freely dilatable GCO
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Lemma 2
  • proof
  • Theorem 3: No-catalysis
  • proof
  • Proposition S1
  • ...and 27 more