Strong-to-weak spontaneous symmetry breaking (SWSSB) has recently emerged as a universal feature of quantum mixed-state phases of matter. While various information-theoretic diagnostics have been proposed to define and characterize SWSSB phases, relating these diagnostics to observables which can be efficiently and scalably probed on modern quantum devices remains challenging. Here we propose a new observable for SWSSB in mixed states, called the Rényi-1 correlator, which naturally suggests a route toward scalably detecting certain SWSSB phases in experiment. Specifically, if the canonical purification (CP) of a given mixed state can be reliably prepared, then SWSSB in the mixed state can be detected via ordinary two-point correlation functions in the CP state. We discuss several simple examples of CP states which can be efficiently prepared on quantum devices, and whose reduced density matrices exhibit SWSSB. The Rényi-1 correlator also satisfies several useful theoretical properties: it naturally inherits a stability theorem recently proven for the closely related fidelity correlator, and it directly defines SWSSB as a particular pattern of ordinary spontaneous symmetry breaking in the CP state.
Strong-to-weak spontaneous symmetry breaking (SWSSB) has recently emerged as a universal feature of quantum mixed-state phases of matter. While various information-theoretic diagnostics have been proposed to define and characterize SWSSB phases, relating these diagnostics to observables which can be efficiently and scalably probed on modern quantum devices remains challenging. Here we propose a new observable for SWSSB in mixed states, called the Rényi-1 correlator, which naturally suggests a route toward scalably detecting certain SWSSB phases in experiment. Specifically, if the canonical purification (CP) of a given mixed state can be reliably prepared, then SWSSB in the mixed state can be detected via ordinary two-point correlation functions in the CP state. We discuss several simple examples of CP states which can be efficiently prepared on quantum devices, and whose reduced density matrices exhibit SWSSB. The Rényi-1 correlator also satisfies several useful theoretical properties: it naturally inherits a stability theorem recently proven for the closely related fidelity correlator, and it directly defines SWSSB as a particular pattern of ordinary spontaneous symmetry breaking in the CP state.
Symmetry in mixed quantum states can manifest in two distinct forms: , where each individual pure state in the quantum ensemble is symmetric with the same charge, and , 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 (SWSSB) is measured by the . We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB 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 nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB 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 SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices.
Strong-to-weak spontaneous symmetry breaking (SWSSB) is a novel phase transition in mixed states associated with the loss of global charge information. In this work we study fermions with U(1) symmetry under infinite density dephasing and derive the SWSSB properties of the resulting mixed states. Our central observation is that at infinite dephasing every Renyi correlator is exactly the spin-spin correlator of a compact XY model, whose Boltzmann weight is the squared full-counting-statistics generating function of a classical ensemble built from the parent state. For the Renyi-2 correlator it is the full counting statistics of the parent state itself. We complement the infinite dephasing mapping with a novel diagrammatic expansion for the Renyi-1 correlator, applicable to any diagonal mixed state, which computes the couplings of the Renyi-1 XY model. We also study a replica field theory for even Renyi index quantities, applicable at finite dephasing. Our analysis indicates that fully dephased metals display long-range SWSSB in d≥2, as confirmed numerically and experimentally in d=2, and quasi-long-range SWSSB in d=1. For insulating parent states both an SWSSB and a trivial phase are possible at infinite dephasing depending on UV details. Long-range SWSSB in d≥3, or quasi-long-range SWSSB in d=2, survives above a nonuniversal, Renyi-index-dependent threshold and is lost when the parent state is deep in the insulating phase, with the exception of quantum Hall insulators which generically have quasi-long-range SWSSB. For even Renyi indices we also determine the stability of the SWSSB phase away from infinite dephasing and derive the universality class of the SWSSB transition.
Strong-to-weak spontaneous symmetry breaking (SW-SSB) has recently emerged as a useful framework for studying phases of matter in open systems, quantum or classical. Beginning with the simple idea of extending symmetry breaking to general mixed states, and the familiar equivalence between canonical and grand-canonical ensembles in statistical mechanics, the concept has grown into a unifying perspective connecting many different ideas in physics, including topological orders, emergent hydrodynamics, and information-theoretic characterization of phases of matter. This review provides a bird's-eye view of some of these recent developments.
Spontaneous symmetry breaking (SSB) is the cornerstone of our understanding of quantum phases of matter. Recent works have generalized this concept to the domain of mixed states in open quantum systems, where symmetries can be realized in two distinct ways dubbed strong and weak. Novel intrinsically mixed phases of quantum matter can then be defined by the spontaneous breaking of strong symmetry down to weak symmetry. However, proposed order parameters for strong-to-weak SSB (based on mixed-state fidelities or purities) seem to require exponentially many copies of the state, raising the question: Is it possible to efficiently detect strong-to-weak SSB, in general? Here, we answer this question negatively in the paradigmatic cases of Z_{2} and U(1) symmetries. We construct ensembles of pseudorandom mixed states that do not break the strong symmetry yet are computationally indistinguishable from states that do. This rules out the existence of efficient state-agnostic protocols to detect strong-to-weak SSB.
Mixed quantum states can exhibit unconventional forms of symmetry breaking without pure-state counterparts. A prominent example is strong-to-weak spontaneous symmetry breaking (SW-SSB), which cannot be diagnosed by observables linear in the density matrix. Here, we experimentally resolve finite-size critical signatures of SW-SSB transitions on a superconducting quantum processor. We develop a measurement protocol that combines quantum measurements with classical processing based on the known dephasing channel to reconstruct nonlinear mixed-state correlators. Using only two measurement settings, the protocol reconstructs fidelity, Rényi-2, and linear correlators, as well as string observables. Applying this approach to matrix-product states subjected to power-law-correlated dephasing, we observe finite-size crossings and scaling collapses in nonlinear correlators, accompanied by decaying linear correlations. We further map the estimated SW-SSB transition points across initial-state and channel parameters. Measurements of linear and Rényi-2 string observables further reveal signatures of mixed-state symmetry-protected topological (SPT) phases, with a nonlinear string-order crossover aligned with an independently estimated SW-SSB transition point. Our results establish a quantum-classical approach to probing SW-SSB critical behavior and mixed-state SPT order.
Symmetry breaking plays a central role in classifying the phases of quantum many-body systems. Recent developments have highlighted a novel symmetry-breaking pattern, in which the strong symmetry of a density matrix spontaneously breaks to the weak symmetry. This strong-to-weak symmetry breaking is typically detected using multireplica correlation functions, such as the Rényi-2 correlator. In this Letter, we propose a practical protocol for detecting strong-to-weak symmetry breaking in experiments using the randomized measurement toolbox. Our scheme involves collecting the results of random Pauli measurements for (i) the original quantum state and (ii) the quantum state after evolution with the charged operators. Based on the measurement results, with a large number of samples, we can obtain an unbiased estimate of the Rényi-2 correlator. With a small sample size, we can still provide an alternative approach to estimate the phase boundary to a decent accuracy. We perform numerical simulations of Ising chains with all-to-all decoherence as an exemplary demonstration. Our result opens the opportunity for the experimental studies of the novel quantum phases in mixed quantum states.