Strong-to-Weak Symmetry Breaking Phases in Steady States of Quantum Operations

Michael Knap; Niklas Ziereis; Sanjay Moudgalya

REVIEW 2 major objections 4 minor 90 references

Strong-to-Weak Symmetry Breaking Phases in Steady States of Quantum Operations

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A trace-preserving quantum channel can never host a continuous strong-to-weak symmetry breaking transition; postselection, and only postselection, can drive one.

desk verdict Genuinely valuable analytic machinery for SW-SSB in steady states, but the headline no-go claim about trace-preserving channels outruns the proof; worth a serious referee with revisions. read the letter →

arxiv 2509.09669 v1 pith:2MLN4JQK submitted 2025-09-11 cond-mat.stat-mech cond-mat.str-elhep-thmath-phmath.MPquant-ph

Niklas Ziereis , Sanjay Moudgalya , Michael Knap This is my paper

classification cond-mat.stat-mechcond-mat.str-elhep-thmath-phmath.MPquant-ph

keywords strong-to-weaksymmetrybreakingmixed-statequantumphasessteadystateschannelspostselectionRenyi-2correlatorBrowniancircuitscommutantbound

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether a mixed quantum state evolving under symmetric noisy dynamics can change its symmetry character in the steady state: keep a strong symmetry only on average (SW-SSB) or restore it exactly. It proves that maximally mixed invariant states—infinite-temperature mixtures in a fixed symmetry sector—always show SW-SSB, with a closed-form long-distance Rényi-2 correlator. Next, it shows the SW-SSB phase survives weak measurement and postselection and that sufficiently strong postselection drives a sharp second-order transition to a strongly symmetric steady state, locating the Z2 transition at ps=4J via a mapping to the transverse-field Ising model. Finally, it argues that this kind of transition cannot occur in any strongly symmetric trace-preserving channel, because such channels always retain at least dim C steady states; only breaking trace preservation lowers the degeneracy.

What carries the argument

The doubled-Hilbert-space (vectorized density matrix) representation, in which a density matrix becomes a pure state and SW-SSB of a group G becomes ordinary spontaneous symmetry breaking G×G → G_diag, together with the commutant algebra C of the symmetric operators, whose dimension sum_λ d_λ^2 sets the lower bound on steady-state degeneracy. For the Z2 postselection protocol, the effective non-Hermitian Hamiltonian reduces to the transverse-field Ising model, whose ground state controls all steady-state correlators.

What would settle it

Find a strongly symmetric trace-preserving quantum channel whose steady-state manifold keeps the same (nonzero) degeneracy but whose long-distance Rényi-2 correlator C_sw crosses from nonzero to zero as a local coupling is tuned; the paper's argument predicts this cannot happen. Concretely, a numerical scan of strongly Z2-symmetric Lindbladians or unital channels with a two-dimensional steady-state manifold and a tunable coupling would settle it. Separately, the MMIS formula can be falsified by measuring the Rényi-2 correlator at large separation on small symmetric circuits and checking it equ

Watch

Extended reading notes

Core claim

Maximally mixed invariant states (MMIS), the uniform mixtures within a scalar symmetry sector, exhibit strong-to-weak spontaneous symmetry breaking (SW-SSB) for any on-site representation of a compact Lie or finite group whose scalar sectors grow exponentially; the long-distance Rényi-2 correlator is exactly ||O||_2^4 / ((dim H_loc)^2 |I_O|), independent of the distance between local order parameters. This SW-SSB is not an isolated point: it forms a phase stable to on-site measurements and a moderate amount of postselection. Sufficiently strong postselection drives a continuous transition to a strongly symmetric steady state, realized analytically for Z2 symmetry at ps = 4J, where the averag

Load-bearing premise

The whole no-go theorem rests on the stated but unproven premise, in Section VI A, that a transition from SW-SSB to a strongly symmetric phase must lower the steady-state degeneracy; if a phase change can occur without a degeneracy drop, trace-preserving channels could still host such transitions.

Editorial extensions

If this is right

The MMIS formula gives an exact, system-size-independent value for the SW-SSB order parameter for any on-site compact Lie or finite group with exponentially growing scalar sectors, so symmetric random circuits generically thermalize to SW-SSB states.
Weak measurements and weak postselection do not destroy SW-SSB; the phase occupies a finite region of parameter space, not just a single point.
At the critical postselection rate the averaged steady state undergoes a continuous transition, with critical point and universality set by the transverse-field Ising model for Z2 and the corresponding Potts model for S3; all Rényi-m and fidelity correlators cross at the same point because they are boundary correlators of the same bulk model.
No strongly symmetric trace-preserving channel can realize a Landau-type SW-SSB to strongly symmetric transition in its steady state, regardless of whether it is Lindbladian, unital, or Brownian.
A concrete trace-preserving feedback protocol that seemingly targets the same strongly symmetric state shows only smooth decay of the correlator and no critical point, consistent with the general degeneracy argument.
If the paper is right, postselection is not just one way to prepare strongly symmetric steady states from SW-SSB attractors; it is the only way, making it a necessary resource for such state preparation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: The paper's no-go result is only as strong as the stated Landau-type premise that leaving SW-SSB requires lowering the steady-state degeneracy; the authors themselves leave open the possibility that other, non-Landau mechanisms—such as dynamical freezing—could produce transitions without a degeneracy drop.
Editorial inference: Equation (12) suggests a direct experimental test: measure the Rényi-2 correlator of a symmetric random circuit's steady state (or a finite-depth symmetric channel) and compare with ||O||_2^4 / ((dim H_loc)^2 |I_O|); exact agreement at large separation would confirm MMIS as the universal SW-SSB attractor.
Editorial inference: The bulk-boundary reasoning implies that any observable acting as a boundary insertion in the same effective statistical-mechanics model—including randomized-measurement estimators—should locate the same transition, which could make experiments cheaper than full fidelity or full state tomography.
Editorial inference: The combination of the MMIS formula and the trace-preservation no-go suggests that symmetries alone dictate a floor on steady-state complexity; breaking that floor requires information loss, and postselection is the physical knob that encodes that information loss.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper before you read it: the technical core is strong, and the abstract oversells one of its conclusions. The authors derive a clean formula (Eq. 12) for the Renyi-2 correlator of maximally mixed invariant states for arbitrary finite or compact Lie groups with exponential sector growth, prove that SW-SSB of these states is independent of the correlator used (Appendix D resolves a conjecture from Ref. [10]), and construct postselection-driven transitions for Z2 and S3 Brownian circuits. The Z2 transition maps exactly to the transverse-field Ising model, and the critical point at ps = 4J comes out of that mapping. Numerical ED and DMRG data support the analytic claims. This is real, reproducible work: the main theorem is proven in detail, the no-postselection limit reproduces known results, and the MMIS formula is derived rather than assumed.

The soft spot is the Sec. VI A no-go argument. The proof establishes a lower bound on the full steady-state degeneracy of strongly symmetric trace-preserving channels: at least dim C = Σλ dλ^2. That bound is correct. But the claim that continuous SW-SSB transitions are absent in all such channels requires the additional premise that any transition to a strongly symmetric phase must lower the steady-state degeneracy. The authors state this as an 'expectation,' not a theorem, and the lower bound is on the full manifold while the relevant transitions are sector-restricted. A channel could have a unique strongly symmetric steady state in one sector and an unrelated steady state in another without violating the bound. The stress-test note is right about this. The authors themselves walk the claim back in Sec. VII, mentioning freezing transitions. So the abstract's unqualified sentence about absence of transitions is not supported by the proof. That is a fixable overreach: either soften the claim to 'Landau-type transitions' at the stated level, or prove the sector-local statement.

Two minor points: the S3 critical point at ps ≈ 3.75 is numerical only, and the tensor-network-plus-machine-learning fidelity estimates for N = 32, 64 lack error bars. Neither shakes the core results.

Who should read this: people working on mixed-state phases, monitored circuits, and open quantum systems. The formula and the Z2 mapping are worth having, and the equivalence theorem in Appendix D is a genuine advance. Send it to referees. I would ask them to focus on the scope of the no-go claim and whether the sector-local loophole can be closed or should be acknowledged.

Referee Report

2 major / 4 minor

Summary. The paper studies strong-to-weak spontaneous symmetry breaking (SW-SSB) in steady states of quantum operations. It proves that maximally mixed invariant states (MMIS) exhibit SW-SSB for on-site compact Lie or finite groups, giving a closed-form Rényi-2 correlator in Eq. (12) and bounding fidelity and other correlators. It then introduces a postselection protocol for Brownian circuits: for a Z_2 symmetric circuit the averaged steady state maps exactly onto the transverse-field Ising model, Eq. (31), predicting a transition at ps = 4J between a SW-SSB phase and a strongly symmetric phase; an S_3 Potts circuit is studied numerically with a transition at ps ≈ 3.75. Finally, Section VI argues that continuous Landau-type SW-SSB transitions are absent in the steady states of general strongly symmetric, trace-preserving quantum channels, because the steady-state degeneracy is bounded below by dim C = Σ_λ d_λ^2. The Z_2 analytic mapping, the MMIS formula, and the equivalence theorems in Appendices A–D are the main technical contributions.

Significance. If the main claims hold, the paper provides a rigorous and useful characterization of SW-SSB in steady states: the MMIS formula Eq. (12) is exact and widely applicable, the Z_2 postselection model is solved exactly via the transverse-field Ising mapping, and the numerical S_3 results extend the framework to non-abelian symmetries. The appendices contain detailed proofs of the MMIS correlator and of the equivalence of fidelity, trace-distance, and Rényi-divergence definitions of SW-SSB, which are valuable independent of the phase-transition claims. However, the no-go statement in Section VI A is the load-bearing part of the abstract's strongest claim, and, as detailed below, the proof does not support the claim in the form stated.

major comments (2)

[Sec. VI A, paragraph beginning 'Our argument is based on the expectation that...'] The no-go claim is stronger than what is proven. The lower bound dim C = Σ_λ d_λ^2 applies to the total steady-state manifold of the channel, but it does not constrain the steady-state degeneracy within a fixed scalar symmetry sector. The positive transition in Sec. V B (Eq. (31)) occurs within a single parity sector, and the 'degeneracy change' there is in the effective TFI ground-state manifold, not in the full fixed-point dimension of the physical channel. A strongly symmetric trace-preserving channel could have, say, one steady state in the even sector and one in the odd sector, satisfying dim C = 2, while the even-sector steady state changes from an MMIS to a strongly symmetric pure state. Thus the abstract's unqualified statement that continuous SW-SSB transitions are absent in general strongly symmetric trace-preserving channels is not established. The paper should either prove th
[Sec. VI B] The concrete Lindbladian example does not fill the gap left by the general argument. It demonstrates the absence of a transition in one specific all-to-all adaptive model, but the reasoning that any trace-preserving protocol targeting |↑...↑> must be all-to-all is a heuristic locality argument, not a general theorem. The numerical smooth decay of C_sw in Fig. 6 is consistent with the no-go but does not prove it. The general claim in the abstract therefore rests entirely on the unproven degeneracy-lowering premise in Sec. VI A.

minor comments (4)

[Sec. V A] The phrase 'Assis the conditional probability' should read 'As s is the conditional probability'.
[Sec. VI A] Typo: 'stronlgy symmetric phase' should be 'strongly symmetric phase'.
[Sec. VII] Typo: 'Morevoer' should be 'Moreover'.
[Fig. 4 caption] The 'TN+ML' method is described in the text, but the caption would benefit from a one-sentence statement that the fidelity for N=32,64 is obtained by a neural-network extrapolation from exact diagonalization data, since the method is not a controlled numerical technique.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MMIS formula and the postselection-induced TFIM transition are derived rather than fitted; the Sec VI A no-go relies on an explicitly stated but unproven Landau premise, which is a logical gap, not a circular reduction.

full rationale

The paper's central derivations are self-contained. Eq. (12) for the Rényi-2 correlator of MMIS is proven in App. A from the Haar-integral form of the MMIS, Schur orthogonality, and Lemma 1; it is not an assumed ansatz. The Brownian-circuit effective Hamiltonian Eq. (21) and Eq. (26) are imported from Refs. [4,5]; Ref. [5] shares an author (Moudgalya), but it is a published, parameter-free commutant-algebra result with stated assumptions and externally checkable content, so under the review rules it is independent evidence and does not raise the circularity score. The Z2 postselection transition is obtained by deriving Eq. (31) (transverse-field Ising model) and using the standard Pfeuty critical point ps/(4J)=1; the S3 transition is obtained numerically from Eq. (38). No fitted parameter is renamed as a prediction: even the neural-network fidelity estimate is trained on small-N ED data and evaluated at larger N, with an independent analytic argument in App. G. The genuine weakness is the Sec VI A no-go: after proving the lower bound dim C = Σ_λ d_λ^2, the text says 'Our argument is based on the expectation that a phase transition from a SW-SSB phase to a strongly symmetric phase requires a change in the steady-state degeneracy.' This premise is not proven, so the abstract's unqualified claim that continuous SW-SSB transitions are absent in all trace-preserving strongly symmetric channels is stronger than the proof supports (the authors partially concede this in Sec VII). That is a correctness/scope gap, not circularity: the conclusion is not identical to the inputs, no equation is defined in terms of the target result, and the proof's lower bound is a genuine theorem. Hence score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard representation theory (Haar integration, Schur orthogonality), the exponential-growth condition for symmetry sectors, the Brownian-circuit-to-commutant mapping from prior work [4,5], and the physically standard but unproven Landau-degeneracy assumption. No free parameters are fitted to data; the model parameters (ps, J, U) are protocol inputs.

assumptions (4)

standard math Compact Lie group integration: Schur's lemma, orthogonality of matrix elements, and Fubini's theorem on quotient G/H
Used in App A to evaluate the group integrals defining the MMIS and to prove Theorem 1 and Eq. (12).
domain assumption The scalar symmetry sector grows exponentially: dim V_theta(N) ~ d^N / poly(N)
Assumed in Theorem 1 (App A) and stated in Sec III B to restrict validity of the correlator formula; excludes sub-exponentially growing sectors (Hilbert space fragmentation) which the paper leaves open.
domain assumption Averaged Brownian circuit dynamics equals imaginary-time evolution with P_B = sum_alpha J_alpha (B_{alpha,f} - B_{alpha,b}^T)^2, whose ground states span the commutant C
Imported from [4,5]; used in Sec IV D to identify MMIS as steady states of unital channels and in Sec V to derive the effective transverse-field Ising Hamiltonian.
domain assumption A Landau-type phase transition from SW-SSB to strong symmetry requires a decrease in steady-state degeneracy
Stated in Sec VI A as an 'expectation'; the entire no-go argument for trace-preserving channels depends on it, and it is not proven.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Strong-to-Weak Symmetry Breaking Phases in Steady States of Quantum Operations." pith.science (2026). https://pith.science/paper/2MLN4JQK

@misc{pith2026250909669,
  author       = {Pith},
  title        = {Pith review of: Strong-to-Weak Symmetry Breaking Phases in Steady States of Quantum Operations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2MLN4JQK}},
  note         = {Machine review of arXiv:2509.09669}
}

abstract

Mixed states can exhibit two distinct kinds of symmetries, either on the level of the individual states (strong symmetry), or only on the level of the ensemble (weak symmetry). Strong symmetries can be spontaneously broken down to weak ones, a mechanism referred to as Strong-to-Weak Spontaneous Symmetry Breaking (SW-SSB). In this work, we first show that maximally mixed symmetric density matrices, which appear, for example, as steady states of symmetric random quantum circuits have SW-SSB when the symmetry is an on-site representation of a compact Lie or finite group. We then show that this can be regarded as an isolated point within an entire SW-SSB phase that is stable to more general quantum operations such as measurements followed by weak postselection. With sufficiently strong postselection, a second-order transition can be driven to a phase where the steady state is strongly symmetric. We provide analytical and numerical results for such SW-SSB phases and their transitions for both abelian $\mathbb{Z}_2$ and non-abelian $S_3$ symmetries in the steady state of Brownian random quantum circuits with measurements. We also show that such continuous SW-SSB transitions are absent in the steady-state of general strongly symmetric, trace-preserving quantum channels (including unital, Brownian, or Lindbladian dynamics) by analyzing the degeneracies of the steady states in the presence of symmetries. Our results demonstrate robust SW-SSB phases and their transitions in the steady states of noisy quantum operations, and provide a framework for realizing various kinds of mixed-state quantum phases based on their symmetries.

Figures

Figures reproduced from arXiv: 2509.09669 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Both order correlation functions appear to indicate the same phase transition. Let us now compare our formula for the R´enyi-2 correlator Eq. (12) with the numerical data at s = 0. The local Hilbert space dimension is d = 3 and σ transforms under a two-dimensional re…

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

90 extracted references · 1 canonical work pages

[10]

Y. Kuno, T. Orito, and I. Ichinose, Strong-to-weak sym- metry breaking states in stochastic dephasing stabilizer circuits (2024)

2024
[1]

With probabilityp dt N(N−1) 2 measure the operators Zi andZ j for a randomly uniform chosen pairi̸=j
[2]

freezing

If the measurement outcomes correspond to↓for both measurements, apply the gateX iXj, thereby flipping both spins. This increases the proba- bility of flowing towards the target steady state |↑ · · · ↑⟩ ⟨↑ · · · ↑|. For any other measurement outcome, do nothing. The normalization of the probability is such thatphas the interpretation of a measurement rate...

2020
[3]

Buˇ ca and T

B. Buˇ ca and T. Prosen, A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains, New Journal of Physics , 073007 (2012)

2012
[4]

V. V. Albert and L. Jiang, Symmetries and conserved quantities in lindblad master equations, Phys. Rev. A 89, 022118 (2014)

2014
[5]

Y. Bao, S. Choi, and E. Altman, Symmetry enriched phases of quantum circuits, Annals of Physics435, 168618 (2021)

2021
[6]

Ogunnaike, J

O. Ogunnaike, J. Feldmeier, and J. Y. Lee, Unify- ing Emergent Hydrodynamics and Lindbladian Low- Energy Spectra across Symmetries, Constraints, and Long-Range Interactions, Physical Review Letters131, 220403 (2023)

2023
[7]

Moudgalya and O

S. Moudgalya and O. I. Motrunich, Symmetries as ground states of local superoperators: Hydrodynamic implica- tions, PRX Quantum5, 040330 (2024)

2024

Show all 90 references

[8]

L. Chen, N. Sun, and P. Zhang, Strong-to-weak symme- try breaking and entanglement transitions, Phys. Rev. B 111, L060304 (2025)

2025
[9]

Huang, M

X. Huang, M. Qi, J.-H. Zhang, and A. Lucas, Hydro- dynamics as the effective field theory of strong-to-weak spontaneous symmetry breaking (2024)

2024
[11]

J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4, 030317 (2023)

2023
[12]

L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-weak spontaneous symmetry break- ing in mixed quantum states, PRX Quantum6, 010344 (2025)

2025
[13]

Ma and C

R. Ma and C. Wang, Average Symmetry-Protected Topo- logical Phases, Physical Review X13, 031016 (2023)

2023
[14]

P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open sys- tems: Purification perspective, Phys. Rev. B110, 155150 17 (2024)

2024
[15]

Moharramipour, L

A. Moharramipour, L. A. Lessa, C. Wang, T. H. Hsieh, and S. Sahu, Symmetry enforced entanglement in maxi- mally mixed states, PRX Quantum5, 040336 (2024)

2024
[16]

Y. Li, F. Pollmann, N. Read, and P. Sala, Highly- entangled stationary states from strong symmetries (2024), arXiv:2406.08567 [quant-ph]

2024 arXiv
[17]

J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, Quantum9, 1607 (2025)

2025
[18]

Agrawal, A

U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits, Physical Review X12, 041002 (2022)

2022
[19]

Barratt, U

F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Physical Review Letters129, 120604 (2022)

2022
[20]

Singh, R

H. Singh, R. Vasseur, A. C. Potter, and S. Gopalakr- ishnan, Mixed-state learnability transitions in moni- tored noisy quantum dynamics (2025), arXiv:2503.10308 [quant-ph]

2025
[21]

Zerbaet al., (2025), in preparation

C. Zerbaet al., (2025), in preparation

2025
[22]

A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, inEntanglement in Spin Chains: From Theory to Quantum Technology Applica- tions, edited by A. Bayat, S. Bose, and H. Johannes- son (Springer International Publishing, Cham, 2022) pp. 211–249

2022
[23]

M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vi- jay, Random Quantum Circuits, Annual Review of Con- densed Matter Physics14, 335 (2023)

2023
[24]

D. Gu, Z. Wang, and Z. Wang, Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak (2024), arXiv:2406.19381 [quant-ph]

2024 arXiv
[25]

M. A. Nielsen and I. L. Chuang, Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion (2010)

2010
[26]

Mitsuhashi, R

Y. Mitsuhashi, R. Suzuki, T. Soejima, and N. Yosh- ioka, Unitary designs of symmetric local random circuits, Phys. Rev. Lett.134, 180404 (2025)

2025
[27]

N. Sun, P. Zhang, and L. Feng, Scheme to detect the strong-to-weak symmetry breaking via randomized mea- surements, Phys. Rev. Lett.135, 090403 (2025)

2025
[28]

Biane, Asymptotic estimates for the multiplicities in tensor powers of a g-module, C

P. Biane, Asymptotic estimates for the multiplicities in tensor powers of a g-module, C. R. Acad. Sci., Paris, S´ er. I316, 849 (1993)

1993
[29]

Tate and S

T. Tate and S. Zelditch, Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers, J. Funct. Anal.217, 402 (2004)

2004
[30]

Hauser, Y

J. Hauser, Y. Li, S. Vijay, and M. P. A. Fisher, Continu- ous symmetry breaking in adaptive quantum dynamics, Phys. Rev. B109, 214305 (2024)

2024
[31]

Breuer, F

H.-P. Breuer, F. Petruccione, H.-P. Breuer, and F. Petruccione,The Theory of Open Quantum Systems (Oxford University Press, Oxford, New York, 2007)

2007
[32]

Watrous,The Theory of Quantum Information(Cam- bridge University Press, Cambridge, 2018)

J. Watrous,The Theory of Quantum Information(Cam- bridge University Press, Cambridge, 2018)

2018
[33]

Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

1976
[34]

Lashkari, D

N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, Jour- nal of High Energy Physics2013, 22 (2013)

2013
[35]

Bauer, D

M. Bauer, D. Bernard, and T. Jin, Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics, SciPost Physics3, 033 (2017)

2017
[36]

S.-K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang, SYK meets non-Hermiticity II: measurement-induced phase transition, Physical Review Letters127, 140601 (2021)

2021
[37]

S¨ underhauf, L

C. S¨ underhauf, L. Piroli, X.-L. Qi, N. Schuch, and J. I. Cirac, Quantum chaos in the Brownian SYK model with large finite n: OTOCs and tripartite information, Journal of High Energy Physics2019, 38 (2019)

2019
[38]

Agarwal and S

L. Agarwal and S. Xu, Emergent symmetry in Brownian SYK models and charge dependent scrambling, Journal of High Energy Physics2022, 45 (2022)

2022
[39]

Zhou and X

T. Zhou and X. Chen, Operator dynamics in a Brownian quantum circuit, Physical Review E99, 052212 (2019)

2019
[40]

Agarwal, S

L. Agarwal, S. Sahu, and S. Xu, Charge transport, infor- mation scrambling and quantum operator-coherence in a many-body system with U(1) symmetry, Journal of High Energy Physics2023, 37 (2023)

2023
[41]

Knap, Entanglement production and information scrambling in a noisy spin system, Physical Review B 98, 10.1103/physrevb.98.184416 (2018)

M. Knap, Entanglement production and information scrambling in a noisy spin system, Physical Review B 98, 10.1103/physrevb.98.184416 (2018)

2018 doi
[42]

Bernard and L

D. Bernard and L. Piroli, Entanglement distribution in the quantum symmetric simple exclusion process, Phys- ical Review E104, 014146 (2021)

2021
[43]

Swann, D

T. Swann, D. Bernard, and A. Nahum, Spacetime pic- ture for entanglement generation in noisy fermion chains, Phys. Rev. B112, 064301 (2025)

2025
[44]

Vardhan and S

S. Vardhan and S. Moudgalya, Entanglement dynamics from universal low-lying modes (2024), arXiv:2407.16763

2024
[45]

Sahu, S.-K

S. Sahu, S.-K. Jian, G. Bentsen, and B. Swingle, Entan- glement phases in large- N hybrid Brownian circuits with long-range couplings, Physical Review B106, 224305 (2022)

2022
[46]

Moudgalya and O

S. Moudgalya and O. I. Motrunich, Hilbert Space Frag- mentation and Commutant Algebras, Physical Review X 12, 011050 (2022)

2022
[47]

Moudgalya and O

S. Moudgalya and O. I. Motrunich, From symmetries to commutant algebras in standard Hamiltonians, Annals of Physics455, 169384 (2023)

2023
[48]

Pfeuty, The one-dimensional Ising model with a trans- verse field, Annals of Physics57, 79 (1970)

P. Pfeuty, The one-dimensional Ising model with a trans- verse field, Annals of Physics57, 79 (1970)

1970
[49]

Zhang, T

Y. Zhang, T. H. Hsieh, Y. B. Kim, and Y. Zou, Probing mixed-state phases on a quantum computer via Renyi correlators and variational decoding (2025), arXiv:2505.02900 [quant-ph]

2025 arXiv
[50]

M¨ uller, A simplified expression for quantum fidelity (2023), arXiv:2309.10565 [quant-ph]

A. M¨ uller, A simplified expression for quantum fidelity (2023), arXiv:2309.10565 [quant-ph]

2023 arXiv
[51]

Hauschild and F

J. Hauschild and F. Pollmann, Efficient numerical sim- ulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Physics Lecture Notes , 5 (2018)

2018
[52]

R. J. Baxter, Exactly solved models in statistical me- chanics (1982)

1982
[53]

F. Y. Wu, The potts model, Rev. Mod. Phys.54, 235 (1982)

1982
[54]

A. J. Friedman, R. Vasseur, A. C. Potter, and S. A. Parameswaran, Localization-protected order in spin chains with non-Abelian discrete symmetries, Physical Review B98, 064203 (2018)

2018
[55]

Zhang, J

Z. Zhang, J. Tindall, J. Mur-Petit, D. Jaksch, and B. Buˇ ca, Stationary state degeneracy of open quantum systems with non-abelian symmetries, Journal of Physics 18 A: Mathematical and Theoretical53, 215304 (2020)

2020
[56]

Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, arXiv e-prints 10.48550/arXiv.2308.00747 (2023), arXiv:2308.00747 [hep-th]

S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, arXiv e-prints 10.48550/arXiv.2308.00747 (2023), arXiv:2308.00747 [hep-th]

2023 doi arXiv
[57]

P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Poll- mann, Ergodicity breaking arising from hilbert space fragmentation in dipole-conserving hamiltonians, Phys. Rev. X10, 011047 (2020)

2020
[58]

Khemani, M

V. Khemani, M. Hermele, and R. Nandkishore, Local- ization from hilbert space shattering: From theory to physical realizations, Phys. Rev. B101, 174204 (2020)

2020
[59]

Rakovszky, P

T. Rakovszky, P. Sala, R. Verresen, M. Knap, and F. Poll- mann, Statistical localization: From strong fragmenta- tion to strong edge modes, Phys. Rev. B101, 125126 (2020)

2020
[60]

Y. Li, P. Sala, and F. Pollmann, Hilbert space fragmenta- tion in open quantum systems, Phys. Rev. Res.5, 043239 (2023)

2023
[61]

Moudgalya, B

S. Moudgalya, B. A. Bernevig, and N. Regnault, Quan- tum many-body scars and Hilbert space fragmentation: a review of exact results, Reports on Progress in Physics 85, 086501 (2022)

2022
[62]

Moudgalya and O

S. Moudgalya and O. I. Motrunich, Exhaustive charac- terization of quantum many-body scars using commutant algebras, Phys. Rev. X14, 041069 (2024)

2024
[63]

Serbyn, D

M. Serbyn, D. A. Abanin, and Z. Papi´ c, Quantum many- body scars and weak breaking of ergodicity, Nature Physics17, 675 (2021)

2021
[64]

Papi´ c, Weak Ergodicity Breaking Through the Lens of Quantum Entanglement, inEntanglement in Spin Chains: From Theory to Quantum Technology Applica- tions, edited by A

Z. Papi´ c, Weak Ergodicity Breaking Through the Lens of Quantum Entanglement, inEntanglement in Spin Chains: From Theory to Quantum Technology Applica- tions, edited by A. Bayat, S. Bose, and H. Johannes- son (Springer International Publishing, Cham, 2022) pp. 341–395

2022
[65]

S. Sahu, Y. Li, and P. Sala, Entanglement cost hierar- chies in quantum fragmented mixed states, arXiv e-prints , arXiv:2506.04637 (2025), arXiv:2506.04637 [quant-ph]

2025 arXiv
[66]

Moudgalya, A

S. Moudgalya, A. Prem, R. Nandkishore, N. Regnault, and B. A. Bernevig, Thermalization and Its Absence within Krylov Subspaces of a Constrained Hamiltonian, inMemorial Volume for Shoucheng Zhang, Chap. 7, pp. 147–209
[67]

Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola, Hilbert-space fragmentation from strict confinement, Phys. Rev. Lett.124, 207602 (2020)

2020
[68]

Morningstar, V

A. Morningstar, V. Khemani, and D. A. Huse, Kinetically constrained freezing transition in a dipole-conserving sys- tem, Phys. Rev. B101, 214205 (2020)

2020
[69]

Feldmeier and M

J. Feldmeier and M. Knap, Critically slow operator dy- namics in constrained many-body systems, Phys. Rev. Lett.127, 235301 (2021)

2021
[70]

Elben, R

A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bij- nen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, and B. Vermersch, Mixed-State Entanglement from Local Randomized Measurements, Physical Review Letters125, 200501 (2020), publisher: American Physical Society

2020
[71]

Brydges, A

T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing R´ enyi entanglement entropy via random- ized measurements, Science364, 260 (2019)

2019
[72]

K. J. Satzinger, Y.-J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth,et al., Realizing topologically ordered states on a quantum processor, Science374, 1237 (2021)

2021
[73]

Ziereis, S

N. Ziereis, S. Moudgalya, and M. Knap, Zenodo entry for: Strong-to-Weak Symmetry Breaking Phases in Steady States of Quantum Operations (2025)

2025
[74]

Appel and E

W. Appel and E. Kowalski,Mathematics for Physics and Physicists, Mathematical notes (Princeton Univer- sity Press, 2007)

2007
[75]

L. H. Loomis,An Introduction To Abstract Harmonic Analysis(D.Van Nostrand Company Inc., 1953)

1953
[76]

Br¨ ocker and T

T. Br¨ ocker and T. Tom Dieck,Representations of Com- pact Lie Groups, Graduate Texts in Mathematics, Vol. 98 (Springer, Berlin, Heidelberg, 1985)

1985
[77]

Fulton and J

W. Fulton and J. Harris,Representation Theory, Gradu- ate Texts in Mathematics, Vol. 129 (Springer, New York, NY, 2004)

2004
[78]

M. R. Sepanski, ed.,Compact Lie Groups, Graduate Texts in Mathematics, Vol. 235 (Springer, New York, NY, 2007)

2007
[79]

R. L. Frank and E. H. Lieb, Monotonicity of a rela- tive r´ enyi entropy, Journal of Mathematical Physics54, 122201 (2013)

2013
[80]

M¨ uller-Lennert, F

M. M¨ uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum r´ enyi entropies: A new generalization and some properties, Journal of Mathe- matical Physics54, 122203 (2013)

2013
[81]

R. Iten, J. M. Renes, and D. Sutter, Pretty good mea- sures in quantum information theory, 2017 IEEE Interna- tional Symposium on Information Theory (ISIT) , 3195 (2016). Appendix A: Proof of SW-SSB of MMIS

2017
[82]

We first need to know for whichNthe MMISρ ∞ N (θ) is well defined (i.e., the scalar symmetry sectorV θ(N) is not empty)

Main Theorem Before we proceed to the main definition, we need to address a small technical point about taking theN→ ∞limit correctly. We first need to know for whichNthe MMISρ ∞ N (θ) is well defined (i.e., the scalar symmetry sectorV θ(N) is not empty). For example for qubit...
[83]

1, we need to collect a few tools that will aid in evaluating the expressions forC sw and C w∅ (see Eq

Preliminary Lemmas Before we can prove Thm. 1, we need to collect a few tools that will aid in evaluating the expressions forC sw and C w∅ (see Eq. (8a) and (8b) for their definitions) in the limitN L → ∞. Crucially, we will be using the fact that the maximally mixed invariant...
[84]

Notation:We will frequently make use of the notation AdM (g)O=u ⊗M g Ou†⊗M g (A12) for an operatorOonH ⊗M loc

Proof Strategy a. Notation:We will frequently make use of the notation AdM (g)O=u ⊗M g Ou†⊗M g (A12) for an operatorOonH ⊗M loc . This is the adjoint representation. We will also used= dimH loc interchangeably. b. Proof strategy:To show Thm. 1 we will first consider sequences ...
[85]

lim N→∞ Tr h ρ∞ NL Oα† ΩiNL Oα ΩjNL i = 0∀α∈ I(A17) We will pick a sequence of system sizes of the formN L =N 0 +L·n

Evaluating Correlation F unctions Lemma 5(Linear correlations at infinite temperature).Ordinary correlation function vanishes for the ∞−temperature states. lim N→∞ Tr h ρ∞ NL Oα† ΩiNL Oα ΩjNL i = 0∀α∈ I(A17) We will pick a sequence of system sizes of the formN L =N 0 +L·n. Fro...
[86]

Proof of Theorem Proof.Let us introduce the notation: C sw OαP (iNL , jNL )[ρ∞ N0+L·n] = Tr h ρ∞ N0+L·nOα† ΩiNL Oα ΩjNL ρ∞ N0+L·nP † ΩjNL PΩiNL i Tr (ρ∞ N0+L·n)2 .(A23) Sinceρ ∞ N0+L·n is a projection normalized to Tr ρ∞ N0+L·n = 1, we find immediately that Tr (ρ∞ N0+L·n)2 = T...
[87]

1 we need to show that dimV Q(N)∼ 1 poly(N) 2N whenever it is nonzero

Application of the formula To be able to apply Thm. 1 we need to show that dimV Q(N)∼ 1 poly(N) 2N whenever it is nonzero. Using Sterling’s approximationk!∼ √ 2πk k e k we find that forN→ ∞: dimV Q(N) = N N↑ = N! N↑!(N−N ↑)! = N! ( N 2 + Q 2 )!( N 2 − Q 2 )! ∼ N! N 2 ! 2 ∼ 2√ ...
[88]

r √ρβO† Ω2 OΩ1 √ρβ † √ρβO† Ω2 OΩ1 √ρβ # =|| √ρβO† Ω1 OΩ2 √ρβ||1 = max n Tr h√ρβO† Ω2 OΩ1 √ρβA i ,||A|| ∞ ≤1 o ≥Tr

Direct calculation To confirm the result of Eq. (B3), we can also calculateC sw U(1) directly. We use thatS i ·S j = 1 4 (Pij −1), where Pij swaps sitesiandj. In the symmetry sector withQas the eigenvalue of P i Zi on a chain ofLqubits, there are N↑ = Q+L 2 ↑spins. Let{|b⟩}be ...
[89]

We consider theZ 2 symmetric circuits and defineρ(t→ ∞) = lim t→∞ E[ρ(t)], whereE[

Fidelity as series of R´ enyi correlators We will sketch an analytical argument for why we expect the fidelity correlator to behave similarly to the R´ enyi correlators. We consider theZ 2 symmetric circuits and defineρ(t→ ∞) = lim t→∞ E[ρ(t)], whereE[. . .] denotes the 34 ave...
[90]

V B and VI B respectively

Simulation data In this section we provide additional data for the fidelity correlator of theZ 2 symmetric circuit both in the postse- lection model and in the adaptive model studied in Secs. V B and VI B respectively. For models with local interactions, 35 0 0.5 1 time t 0.0 ...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [10]

Y. Kuno, T. Orito, and I. Ichinose, Strong-to-weak sym- metry breaking states in stochastic dephasing stabilizer circuits (2024)

2024

[2] [1]

With probabilityp dt N(N−1) 2 measure the operators Zi andZ j for a randomly uniform chosen pairi̸=j

[3] [2]

freezing

If the measurement outcomes correspond to↓for both measurements, apply the gateX iXj, thereby flipping both spins. This increases the proba- bility of flowing towards the target steady state |↑ · · · ↑⟩ ⟨↑ · · · ↑|. For any other measurement outcome, do nothing. The normalization of the probability is such thatphas the interpretation of a measurement rate...

2020

[4] [3]

Buˇ ca and T

B. Buˇ ca and T. Prosen, A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains, New Journal of Physics , 073007 (2012)

2012

[5] [4]

V. V. Albert and L. Jiang, Symmetries and conserved quantities in lindblad master equations, Phys. Rev. A 89, 022118 (2014)

2014

[6] [5]

Y. Bao, S. Choi, and E. Altman, Symmetry enriched phases of quantum circuits, Annals of Physics435, 168618 (2021)

2021

[7] [6]

Ogunnaike, J

O. Ogunnaike, J. Feldmeier, and J. Y. Lee, Unify- ing Emergent Hydrodynamics and Lindbladian Low- Energy Spectra across Symmetries, Constraints, and Long-Range Interactions, Physical Review Letters131, 220403 (2023)

2023

[8] [7]

Moudgalya and O

S. Moudgalya and O. I. Motrunich, Symmetries as ground states of local superoperators: Hydrodynamic implica- tions, PRX Quantum5, 040330 (2024)

2024