Modern quantum platforms naturally realize many-body states shaped by measurement, decoherence, and dissipation. The appropriate object is often not a wave function or ground state, but a density matrix. This thesis asks how phases and phase transitions can be defined for such mixed states, especially when universal structure appears only through nonlinear, replicated, or doubled diagnostics. The thesis develops this perspective using nonlinear diagnostics, doubled-Hilbert- space representations, statistical-mechanics mappings, and generalized symmetry. It shows that weak measurement can suppress higher-form symmetry breaking and drive mixed-state transitions; maps fidelity diagnostics of decohered Chern insulators to effective conformal-field-theory partition functions; and maps Rényi diagnostics of noisy stabilizer and CSS codes to dual statistical-mechanics models connected to decodability. It also shows that measurement and Lindbladian evolution can prepare density matrices whose doubled-space representations realize spin-liquid-like states and superconducting order. Finally, it identifies strong-to-weak symmetry breaking as a route to universal mixed-state behavior connected to hydrodynamic emergence. Together, these results show that openness is not only a perturbation that degrades pure-state physics. Measurement, decoherence, and dissipation can hide, preserve, trans- form, or generate structure in density matrices. The resulting phases and transitions are tied to information processing: what can be decoded, distinguished, recovered, or lost to local probes.
Modern quantum platforms naturally realize many-body states shaped by measurement, decoherence, and dissipation. The appropriate object is often not a wave function or ground state, but a density matrix. This thesis asks how phases and phase transitions can be defined for such mixed states, especially when universal structure appears only through nonlinear, replicated, or doubled diagnostics. The thesis develops this perspective using nonlinear diagnostics, doubled-Hilbert- space representations, statistical-mechanics mappings, and generalized symmetry. It shows that weak measurement can suppress higher-form symmetry breaking and drive mixed-state transitions; maps fidelity diagnostics of decohered Chern insulators to effective conformal-field-theory partition functions; and maps Rényi diagnostics of noisy stabilizer and CSS codes to dual statistical-mechanics models connected to decodability. It also shows that measurement and Lindbladian evolution can prepare density matrices whose doubled-space representations realize spin-liquid-like states and superconducting order. Finally, it identifies strong-to-weak symmetry breaking as a route to universal mixed-state behavior connected to hydrodynamic emergence. Together, these results show that openness is not only a perturbation that degrades pure-state physics. Measurement, decoherence, and dissipation can hide, preserve, trans- form, or generate structure in density matrices. The resulting phases and transitions are tied to information processing: what can be decoded, distinguished, recovered, or lost to local probes.
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.
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.
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.
Symmetry plays a fundamental role in quantum many-body physics, and a central concept is spontaneous symmetry breaking, which imposes crucial constraints on the possible quantum phases and their transitions. Recent developments have extended the discussion of symmetry and its breaking to mixed states, enhancing our understanding of novel quantum phases that have no counterpart in pure states. Specific attention has been paid to scenarios where a strongly symmetric density matrix exhibits spontaneous symmetry breaking to weak symmetry, characterized by the fidelity correlator. In this work, we propose the Wightman correlator as an alternative diagnostic tool. This construction relies on the introduction of the thermofield double state for a generic density matrix, which maps the strong symmetry of the density matrix to the doubled symmetry of the pure state, allowing the Wightman correlator to emerge naturally as a standard probe of symmetry breaking. We prove the equivalence between the Wightman function and the fidelity correlator in defining strong-to-weak symmetry breaking, and examine explicit examples involving spin glasses, thermal density matrices, and the decohered Ising model. Additionally, we discuss a susceptibility interpretation of the Wightman correlator.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl, SU(2), SU(1,1), and SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi Q-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan P-function to the Q-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In SU(2) and SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale, t∗, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.
The interplay of unitary evolution and local measurements in many-body systems gives rise to a stochastic state evolution and to measurement-induced phase transitions in the pure state entanglement. In realistic settings, however, this dynamics may be spoiled by decoherence, e.g., dephasing, due to coupling to an environment or measurement imperfections. We investigate the impact of dephasing and the inevitable evolution into a non-Gaussian, mixed state, on the dynamics of monitored fermions. We approach it from three complementary perspectives: (i) the exact solution of the conditional master equation for small systems, (ii) quantum trajectory simulations of Gaussian states for large systems, and (iii) a renormalization group analysis of a bosonic replica field theory. For weak dephasing, constant monitoring preserves a weakly mixed state, which displays a robust measurement-induced phase transition between a critical and a pinned phase, as in the decoherence-free case. At strong dephasing, we observe the emergence of a new scale describing an effective temperature, which is accompanied with an increased mixedness of the fermion density matrix. Remarkably, observables such as density-density correlation functions or the subsystem parity still display scale invariant behavior even in this strongly mixed phase. We interpret this as a signature of gapless, classical diffusion, which is stabilized by the balanced interplay of Hamiltonian dynamics, measurements, and decoherence.