Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States

TL;DR

This paper introduces a new form of spontaneous symmetry breaking in mixed quantum states, called strong-to-weak SSB, which is characterized by the fidelity correlator and is robust across quantum phases and scenarios.

Contribution

It demonstrates the universality and robustness of strong-to-weak SSB in mixed states, and connects it to phase transitions in classical and quantum models, including the decohered Ising model.

Findings

SW-SSB is a universal property of mixed quantum phases.

SW-SSB transition in decohered Ising model aligns with channel recoverability.

Thermal states at nonzero temperature exhibit spontaneous strong symmetry breaking.

Abstract

Symmetry in mixed quantum states can manifest in two distinct forms: strong symmetry, where each individual pure state in the quantum ensemble is symmetric with the same charge, and weak symmetry, which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SW-SSB) is measured by the fidelity correlator. We prove that SW-SSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SW-SSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is "spontaneous" in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study non-thermal scenarios where decoherence induces SW-SSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SW-SSB transition of a decohered Ising model can be viewed as the "ungauged" version of the celebrated toric code decodability transition. We confirm that, in the decohered Ising model, the SW-SSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SW-SSB, through correlation functions with higher Renyi indices.