Strong-to-Weak SSB in Quantum Systems

• SW-SSB is defined as the transition from strong symmetry in each pure state to weak symmetry preserved only in the ensemble, revealed through nonlinear diagnostics.
• Key methodologies involve using Rényi-2 and fidelity correlators to detect persistent hidden order in open quantum systems where standard correlations vanish.
• Measurement and postselection enable controlled phase transitions between SW-SSB and fully symmetric phases, offering a robust framework to classify quantum phases.

Strong-to-Weak Spontaneous Symmetry Breaking (SW-SSB) refers to a distinctive symmetry-breaking phenomenon in quantum many-body systems—particularly in mixed (open or decohered) states—where a "strong" symmetry, realized at the level of each pure state in an ensemble, breaks down to a "weak" symmetry that is preserved only in the ensemble-average sense. Unlike traditional spontaneous symmetry breaking (SSB) in pure states, SW-SSB reveals "hidden" order detectable only by nonlinear observables—such as fidelity or Rényi correlators—while linear (ordinary) two-point correlation functions vanish at large distances. SW-SSB has emerged as a universal organizing principle for classifying nontrivial quantum phases in open quantum systems and is robust in a variety of contexts, including steady states of noisy quantum circuits, dissipative condensates, quantum critical chains under decoherence, and topologically ordered states under disorder.

1. Fundamental Principles and Diagnostics

SW-SSB distinguishes between two symmetry realizations in quantum mixed states:

• Strong symmetry: A density matrix $\rho$ is strongly symmetric if $U_g \rho = e^{i\theta_g} \rho$ for all $g \in G$. Every pure state in the decomposition of $\rho$ shares the same symmetry charge.
• Weak symmetry: $\rho$ is weakly symmetric if $U_g \rho U_g^\dagger = \rho$; only the ensemble, not each component, is symmetric.

SW-SSB occurs when a state loses strong symmetry spontaneously—so that non-linear order diagnostics, but not conventional observables, signal robust long-range order. The canonical diagnostics include:

Diagnostic	Definition	Feature
Rényi-2 correlator	$$C^{(2)}(i,j)[\rho] = \frac{\operatorname{Tr}(O_i \rho O_i^\dagger O_j \rho O_j^\dagger)}{\operatorname{Tr} (\rho^2)}$$	Remains $\mathcal{O}(1)$ at long distances in SW-SSB
Fidelity correlator	$$F_O(i,j)[\rho] = \operatorname{Tr}\sqrt{\sqrt{\rho} O_i^\dagger O_j \rho O_j^\dagger O_i \sqrt{\rho}}$$	Equivalently detects SW-SSB; robust under symmetric channels
Rényi-1 ("Wightman") correlator	$$R_1(i,j)[\rho] = \operatorname{Tr}( \sqrt{\rho} O_i O_j^\dagger \sqrt{\rho} O_i^\dagger O_j )$$	Efficiently measurable if canonical purification is accessible

In SW-SSB states, ordinary two-point correlators $\langle O_i O_j^\dagger \rangle$ decay rapidly, but the nonlinear functions above maintain long-range value.

2. Steady States, Maximally Mixed Invariant States, and Phase Structure

A key mechanism for realizing SW-SSB arises in steady states of noisy quantum evolution, such as random quantum circuits or dissipative systems. The maximally mixed invariant state (MMIS) in a fixed symmetry sector is

$$\rho_N^\infty = \frac{1}{\dim V_\theta(N)} \int_G e^{-i\theta_g} U_g\, dg,$$

where $V_\theta(N)$ is a symmetry sector. MMIS are "infinite temperature" states within the symmetry sector but exhibit SW-SSB:

$$\lim_{|i-j|\to\infty} C^{sw}(i,j)[\rho_N^\infty] = \frac{\|O\|_2^4}{(\dim \mathcal{H}_\mathrm{loc})^2 | \mathcal{I}_O | },$$

where $| \mathcal{I}_O |$ is the irrep dimension under which $O$ transforms. This remains nonzero even as linear order parameters vanish, revealing hidden order.

When such systems are subjected to measurement and postselection, the state can be "steered" across a boundary between SW-SSB and fully strongly symmetric phases. For example, in $\mathbb{Z}_2$-invariant random circuit steady states, a postselection protocol (projectively measuring, then postselecting a paramagnetic outcome) induces a continuous phase transition from SW-SSB to a trivial symmetric phase, mapped onto a transverse-field Ising model (TFIM):

$$P_{\mathbb{Z}_2}^{\mathrm{tot}} = 2J \sum_{\langle i,j\rangle} [1 - \widetilde{X}_i \widetilde{X}_j ] - \frac{p s}{2} \sum_i \widetilde{Z}_i + \frac{p s N}{2}.$$

A critical postselection rate $ps_c$ marks the transition.

3. Mathematical Framework and Nonlinear Correlators

The defining aspect of SW-SSB diagnostics is their nonlinearity in $\rho$. The Rényi-2 correlator, fidelity correlator, or Wightman correlator all involve functions beyond the expectation value: they typically require access to multiple copies of $\rho$, or a purification in a doubled Hilbert space.

• Doubled Hilbert space/Choi state: Nonlinear correlators map to linear observables in a purified or doubled Hilbert space, making numerical and analytical computations tractable.
• Stability: The fidelity correlator satisfies data-processing inequalities under symmetric, finite-depth quantum channels, ensuring robustness of SW-SSB order under local noisy quantum evolution.
• Exact formulas: In models with compact Lie symmetry groups, explicit expressions for the correlator limits and the associated Markov lengths (from Rényi-2 conditional mutual information) can be derived; see formulas in the entries (e.g., Markov length, MMIS formula, and effective Hamiltonians).

4. Role of Measurements, Postselection, and Absence of Transitions in Lindbladian Steady States

Continuous transitions from SW-SSB to strongly symmetric phases cannot be realized via standard trace-preserving (Lindbladian) channels. The degeneracy of steady states under strong symmetry is constrained by the commutant algebra:

$\dim \mathcal{C} = \sum_\lambda (d_\lambda)^2,$

with $d_\lambda$ the dimensions of irreducible components. MMIS saturate this degeneracy; only non-trace-preserving operations (like postselection) can reduce it. Nonlinear transitioning thus inherently depends on measurement-induced postprocessing. In contrast, for pure Lindbladian dynamics or unital channels, the system remains in robust SW-SSB as long as the symmetry is preserved.

5. Abelian and Non-Abelian Examples, Universality, and Critical Behavior

The SW-SSB framework applies to both Abelian (e.g., $\mathbb{Z}_2$) and non-Abelian (e.g., $S_3$) symmetries. For $\mathbb{Z}_2$, disorder-averaged and measurement-driven steady states precisely reproduce all order parameter and critical phenomena associated with the TFIM, including the continuous transition at a critical postselection rate. For $S_3$, the corresponding Potts model acts as the minimal example, with nonlinear correlators built from operators carrying the fundamental representations.

Explicit numerics confirm that both Rényi-2 and fidelity-based correlators display nontrivial critical points in these models, marking sharp boundaries between robust SW-SSB and trivial strongly symmetric phases.

Symmetry Group	Model	SW-SSB Phase Characterization
$\mathbb{Z}_2$	Brownian Circuit, TFIM	LRO in nonlinear correlator, vanishing in linear
$S_3$	Random Potts Model	Similar SW-SSB transition, nonlinear diagnostics

6. Physical and Experimental Implications

SW-SSB fundamentally expands the landscape of phases available to open quantum systems and noise-dominated steady states. Key points include:

• Hidden order in noise-robust steady states: Even infinite-temperature steady states (MMIS) of symmetric circuits can organize into distinct phases distinguished only by higher-order (nonlinear) observables.
• Nontrivial quantum information structure: SW-SSB phases evade classical description and require nontrivial informational processing or purification for detection.
• Measurement and postselection as tuning parameters: Only via measurement and postselection can one continuously control SW-SSB order and access strongly symmetric phases in open quantum circuits.
• Framework for classification: SW-SSB offers a scheme for classifying quantum phases in dissipative, noisy, or non-unitary quantum evolutions.

7. Key Formulas and Theoretical Summary

Central mathematical statements underlying SW-SSB in steady states of quantum operations include:

• Rényi-2 correlator for SW-SSB:

$$C^{sw}(i,j)[\rho] = \frac{ \operatorname{Tr}[ O_i \rho O_i^\dagger O_j \rho O_j^\dagger ] }{ \operatorname{Tr} (\rho^2) }$$

• Maximally mixed invariant state (MMIS):

$$\rho_N^\infty = \frac{1}{\dim(V_\theta(N))} \int_G e^{-i\theta_g} U_g\, dg$$

with asymptotic order parameter

$$\lim_{|i-j| \to \infty} C^{sw}(i,j)[\rho_N^\infty] = \frac{ \|O\|_2^4 }{ (\dim \mathcal{H}_\mathrm{loc})^2 | \mathcal{I}_O | }$$

• Lower bound on steady-state degeneracy for strong symmetry:

$\dim \mathcal{C} = \sum_\lambda d_\lambda^2$

which ensures that trace-preserving channels cannot reduce the SW-SSB phase degeneracy.

• Analytical and numerical demonstrations of these mechanisms and conclusions appear in (Ziereis et al., 11 Sep 2025), with phase diagrams, critical exponents, and construction for both Abelian and non-Abelian symmetry. The classification and detection of SW-SSB remain at the frontier of open quantum systems research, with implications for both experimental realizability and conceptual advances in many-body quantum theory.