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.
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36 pages, 4 figures in the main text, 21 figures in the Supplemental Material
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-based classification of quantum phases of matter is one of the most foundational organizing principles in physics; however, an analogous framework for mixed, decohered quantum states has only begun to emerge. A central new concept is strong-to-weak spontaneous symmetry breaking (SW-SSB), a sharp transition in mixed quantum states that is invisible to any observable linear in the density matrix and that has since been predicted across a broad class of open and monitored quantum systems. It also provides a unifying language for phenomena as disparate as the decodability of topological quantum memories and the emergence of classical hydrodynamics from decohered quantum dynamics. Here we report the first experimental observation of SW-SSB, in dephased single-component fermionic matter imaged by a quantum gas microscope. A quantum-classical estimator built on a machine-learned Gaussian reference state gives direct access to the nonlinear Rényi-1 and Rényi-2 correlators that diagnose SW-SSB, and reveals long-range Rényi order in the dephased Fermi liquid. Adding a commensurate superlattice drives the underlying fermions through a metal-to-insulator transition that, after full dephasing, manifests as a sharp SW-SSB phase transition. Our results uncover the symmetry principle behind information-theoretic transitions in open quantum systems, and extend Landau's symmetry paradigm into the regime of real, decohering quantum devices.
Open quantum systems can exhibit inherently mixed quantum orders, such as strong-to-weak spontaneous symmetry breaking (SW-SSB), which have no analogue in pure states. It is generally expected that a finite-depth local quantum channel cannot induce a phase transition to SW-SSB in one spatial dimension. We circumvent this obstruction by introducing a long-range quantum channel, which implements two-body dephasing on pairs of sites with probability falling off according to a power-law in the site separation. Applying this channel to a one-parameter family of matrix product states interpolating between trivial and symmetry-protected topological (SPT) phases, we construct a phase diagram that hosts trivial, mixed SPT, and SW-SSB phases with respect to Rényi-2 diagnostics. Notably, the mixed SPT phase is distinct from previously studied decohered SPT phases: it hosts two logical qubits and exhibits nonvanishing Rényi-2 string order, despite vanishing conventional string order. Through a series of mappings to a classical statistical mechanics model, we establish a direct connection between the loss of logical information and the onset of SW-SSB in this model. Finally, we consider the feasibility of experimental realization of the phenomena we have studied. We emphasize that our family of mixed states can be tuned to regimes of high purity, substantially reducing measurement overhead and enabling their experimental realization and detection on near-term quantum hardware.
We experimentally investigate how symmetry-protected topological order in a one-dimensional cluster state is transformed by measurement and decoherence in a five-qubit superconducting array. We first characterize the state's nonlocal string order and show that controlled dephasing selectively suppresses one symmetry sector while leaving the other robust, consistent with average symmetry-protected topological order. We then measure one sublattice in a tunable basis and show that the remaining qubits are driven between a long-range-entangled GHZ state and a paramagnetic state. When the measurement record is discarded, the conventional long-range correlator vanishes while a nonlinear fidelity correlator remains finite, providing a finite-size signature of strong-to-weak spontaneous symmetry breaking. These experiments demonstrate how conditioning, averaging, and decoherence reveal distinct manifestations of order encoded in the same underlying cluster state, and establish a superconducting-circuit setting for probing mixed-state symmetry and topology.
Symmetry-based classification of quantum phases of matter is one of the most foundational organizing principles in physics; however, an analogous framework for mixed, decohered quantum states has only begun to emerge.A central new concept is strong-to-weak spontaneous symmetry breaking (SW-SSB), a sharp transition in mixed quantum states that is invisible to any observable linear in the density matrix and that has since been predicted across a broad class of open and monitored quantum systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14].It also provides a unifying language for phenomena as disparate as the decodability of topological quantum memories [1,15,16] and the emergence of classical hydrodynamics from decohered quantum dynamics [17,18].Here we report the first experimental observation of SW-SSB, in dephased single-component fermionic matter imaged by a quantum gas microscope.A quantum-classical estimator built on a machine-learned Gaussian reference state gives direct access to the nonlinear Rényi-1 and Rényi-2 correlators that diagnose SW-SSB, and reveals long-range Rényi order in the dephased Fermi liquid.Adding a commensurate superlattice drives the underlying fermions through a metal-to-insulator transition that, after full dephasing, manifests as a sharp SW-SSB phase transition.Our results uncover the symmetry principle behind information-theoretic transitions in open quantum systems, and extend Landau's symmetry paradigm into the regime of real, decohering quantum devices.
We explore the onset of spontaneous strong-to-weak symmetry breaking (SW-SSB) under U(1)-symmetric (i.e., charge-conserving) open-system dynamics. We define this phenomenon for quantum states and classical probability distributions, and explore it in three complementary models, one of which exhibits nontrivial quantum coherence at short times. Our main conclusions are as follows. In one dimension, the strong symmetry is not spontaneously broken at any finite time; however, correlators probing strong-to-weak symmetry breaking develop order on length scales that grow linearly in time, parametrically faster than charge diffusion. We provide numerical evidence for this scaling in multiple distinct probes of SW-SSB, and derive it from a field-theory analysis. Moreover, we relate this scaling to the problem of inferring the charge inside a subregion by measuring its surroundings, and construct explicit decoding protocols that illustrate its origin. In two dimensions, field theory and numerical simulations support a finite-time Berezinskii-Kosterlitz-Thouless-like SW-SSB transition. Within continuum hydrodynamics, by contrast, SW-SSB happens at infinitesimal time in two or more dimensions. The SW-SSB transition time can thus be interpreted as marking the emergence of a continuum hydrodynamic description, or (more precisely) the timescale beyond which non-hydrodynamic information such as discrete particle worldlines can no longer be inferred. We support this picture by analyzing a model in which we exploit SW-SSB to derive a classical stochastic hydrodynamic description from the underlying quantum dynamics.
We propose a local notion of strong-to-weak spontaneous symmetry breaking (SW-SSB), through a local one-point fidelity correlator. Compared with the previous definition in terms of a two-point fidelity correlator, our local formulation offers two key advantages: (1) it is easier to detect in large systems: for a system of size N and with poly(N) amount of resources, one can detect the local fidelity order up to volume scale O(log(N)); and (2) the local SW-SSB order remains well defined in the thermodynamic limit, where the density matrix itself is not well defined. We show that key features of SW-SSB, including stability under finite-depth symmetric channels and long-range conditional mutual information, persist within this local framework. Our definition is conceptually analogous to local thermalization, as exemplified by pure states obeying the eigenstate thermalization hypothesis (ETH). For critical states, the local one-point fidelity correlator defines an interesting class of defect problems. We demonstrate the applicability of the local formulation through several concrete examples, and derive the universal scaling behavior of the local fidelity correlator in a range of critical systems, including ground states of conformal field theories as well as ballistic and diffusive free-fermion metals.