The Role of Completeness in Probing Symmetry Breaking
Abstract
Completeness is widely recognized in quantum information as an important property of a family of monotones, because it ensures that no information relevant to state conversion is lost. We show that completeness is also physically important in many-body systems: an incomplete measure of symmetry breaking can miss essential features of symmetry-restoration dynamics. We study the logarithmic characteristic function (LCF), also known as the string order parameter, whose full family is complete for exact i.i.d. pure-state conversion under symmetry-preserving operations for finite groups. We introduce a fidelity-based extension of the LCF to mixed states and identify two concrete advantages over entanglement asymmetry (EA). First, for a finite symmetry group G, EA is bounded by log |G| and can approach the same saturated value for different initial states in the thermodynamic limit, making their relaxation curves indistinguishable and preventing the identification of a discrete-symmetry Mpemba effect. By contrast, the LCF can remain extensive, with a coefficient that depends on the initial state and time, and therefore continues to distinguish their relaxation dynamics. Second, different LCF components can exhibit distinct relaxation and crossing behavior, including cases in which EA shows no crossing. We demonstrate these advantages in spin-chain quenches and a single-qubit system under depolarizing noise. We also develop a replica construction for the fidelity-based LCF in quantum field theory and derive analytical results for conformal field theory.
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BibTeX
@misc{kusuki2026role,
title = {{The Role of Completeness in Probing Symmetry Breaking}},
author = {Yuya Kusuki and Hiroyasu Tajima and Shion Yamashika},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16153},
eprint = {2609.16153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Role of Completeness in Probing Symmetry Breaking
AU - Yuya Kusuki
AU - Hiroyasu Tajima
AU - Shion Yamashika
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16153
UR - https://arxiv.org/abs/2609.16153
AB - Completeness is widely recognized in quantum information as an important property of a family of monotones, because it ensures that no information relevant to state conversion is lost. We show that completeness is also physically important in many-body systems: an incomplete measure of symmetry breaking can miss essential features of symmetry-restoration dynamics. We study the logarithmic characteristic function (LCF), also known as the string order parameter, whose full family is complete for exact i.i.d. pure-state conversion under symmetry-preserving operations for finite groups. We introduce a fidelity-based extension of the LCF to mixed states and identify two concrete advantages over entanglement asymmetry (EA). First, for a finite symmetry group G, EA is bounded by log |G| and can approach the same saturated value for different initial states in the thermodynamic limit, making their relaxation curves indistinguishable and preventing the identification of a discrete-symmetry Mpemba effect. By contrast, the LCF can remain extensive, with a coefficient that depends on the initial state and time, and therefore continues to distinguish their relaxation dynamics. Second, different LCF components can exhibit distinct relaxation and crossing behavior, including cases in which EA shows no crossing. We demonstrate these advantages in spin-chain quenches and a single-qubit system under depolarizing noise. We also develop a replica construction for the fidelity-based LCF in quantum field theory and derive analytical results for conformal field theory.
ER -