Quantifying Symmetry Breaking

This paper addresses the fundamental challenge of quantifying symmetry breaking in quantum systems. While pure-state symmetry breaking is well-understood through the quantum geometric tensor, a complete characterization for general mixed states has remained elusive. The authors investigate the resource theory of asymmetry, specifically focusing on the i.i.d. asymptotic regime where many identical copies of a quantum state are manipulated. They employ quantum local asymptotic normality (QLAN) to reduce the complex problem of state conversion to a more tractable convertibility problem between quantum Gaussian shift models.

The central contribution is a single-letter formula for the optimal conversion rate between arbitrary finite-dimensional states. This rate is governed by a continuous one-parameter family of quantum Fisher information (QFI) matrices that interpolates between the symmetric logarithmic derivative (SLD) and right logarithmic derivative (RLD) QFIs.

Key implications of this result include:

• Completeness: Unlike pure-state conversion, which relies on a single measure, general mixed-state conversion requires the entire QFI family; no finite subset of these measures is sufficient.
• Distillation and Reversibility: The formula provides an exact distillation rate for pure-state extraction and a necessary and sufficient condition for asymptotic reversibility, which requires the proportionality of the full QFI family.
• Asymmetry Activation: The authors identify a novel activation mechanism where two states, neither of which can be converted to a target state individually, can achieve a non-zero conversion rate when processed jointly. This phenomenon arises from the complementarity of different QFI constraints.

This work provides a rigorous operational framework for understanding symmetry breaking in realistic, mixed-state quantum systems. By bridging the gap between abstract resource theory and experimentally accessible quantities—via the link between QFIs and linear-response theory—the findings offer a powerful tool for analyzing quantum clocks, error-correcting codes, and thermodynamic processes. The identification of the QFI family as a complete set of measures resolves a long-standing open problem in quantum information theory.