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REVIEW 4 major objections 5 minor 66 references

Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum Ising model under multiple decoherence

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Applying ZZ and X decoherence to a transverse-field Ising chain produces a mixed state close to the quantum Ashkin-Teller ground state, including a strong-to-weak Z2 symmetry-breaking phase.

desk verdict Careful numerics on a concrete decohered Ising chain; the SWSSB claim is carried by the correlators, while the qAT parent-model identification is a plausible but unverified interpretive layer. read the letter →

arxiv 2412.12738 v3 pith:YVRDI524 submitted 2024-12-17 quant-ph cond-mat.dis-nncond-mat.stat-mechcond-mat.str-el

Takahiro Orito , Yoshihito Kuno , Ikuo Ichinose This is my paper · ORCID
classification quant-phcond-mat.dis-nncond-mat.stat-mechcond-mat.str-el
keywords mixed-statequantumorderstrong-to-weakspontaneoussymmetrybreakingdoubledHilbertspaceformalismAshkin-Tellermodeltransverse-fieldIsingdecoherencechannelsmatrixproductstatesRenyi-2correlators
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

This paper tries to show that ordinary decoherence applied to the one-dimensional transverse-field Ising chain can create mixed-state quantum order with no pure-state counterpart. In the doubled-Hilbert-space picture, the two decoherence channels (nearest-neighbor $ZZ$ and single-site $X$) become local filtering operators on a two-copy ladder state, and the filtered state is argued to sit close to the ground state of the quantum Ashkin-Teller model. Matrix-product-state numerics then identify three mixed-state phases whose order parameters match the three Ashkin-Teller regimes. The central finding is a strong-to-weak $\mathbb{Z}_2$ symmetry-breaking (SWSSB) phase, where the strong-symmetry order parameter is finite while the weak-symmetry and single-chain correlators vanish. If correct, this gives a concrete, tunable protocol by which decoherence alone turns the simplest symmetry-breaking quantum chain into a genuinely mixed-state phase.

What carries the argument

The load-bearing machinery is the doubled Hilbert space vectorization: a density matrix $\rho$ becomes a vector $|\rho\rangle\rangle$ in a two-leg ladder $\mathcal{H}_u \otimes \mathcal{H}_\ell$, and each decoherence channel becomes a local operator. For the two channels used here, the combined action is the local filtering operator $\prod_j e^{\tau_{zz}\hat{h}^{zz}_{j,j+1}} e^{\tau_x \hat{h}^x_j}$ with $\tau_{zz,x} = \tanh^{-1}[p/(1-p)]$, applied to the decoupled two-copy TFIM ground state. The paper's key assumption is that this filtering deforms the MPS into a state close to the ground state of the quantum Ashkin-Teller model, whose phase diagram (paramagnetic, partially ordered/diagonal-$\mathbb{Z}_2$, and $\mathbb{Z}_2 \times \mathbb{Z}_2$ broken regimes) then labels the mixed-state phases. The order parameters are the Renyi-2 susceptibility $\chi^{II}_{ZZ}$ (strong-symmetry SSB), the strange-correlator susceptibility $\chi^{I}_{Z,\mathrm{st}}$ (weak-symmetry SSB), and the single-chain susceptibility $\chi^I_u$ (ordinary long-range order), with the doubled-space entanglement entropy marking the transitions.

What would settle it

Compute the overlap (or energy difference) between the normalized filtered matrix product state and the density-matrix renormalization group ground state of the quantum Ashkin-Teller model at the corresponding $\lambda$; if the overlap decays with system size, or if the correlator-extracted phase boundaries do not converge to the Ashkin-Teller boundaries as $L$ grows, the central correspondence is wrong.

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Extended reading notes

Core claim

The central claim is that the decohered density matrix $\rho_D = \mathcal{E}_{ZZ} \circ \mathcal{E}_X[\rho_0]$ is best understood through its doubled-Hilbert-space vector $|\rho_D\rangle\rangle$, which equals the two-copy TFIM ground state $|\psi_0^*\rangle|\psi_0\rangle$ acted on by local filters $\prod_j e^{\tau_{zz}\hat{h}^{zz}_{j,j+1}} e^{\tau_x \hat{h}^x_j}$. The authors claim that this filtered matrix product state approximates the ground state of the quantum Ashkin-Teller model, with couplings related by $J\lambda_{zz} = c\,\tau_{zz}(p_{zz})$ and $h\lambda_x = c\,\tau_x(p_x)$. On this basis they identify three regimes: a trivial paramagnetic mixed state (region I), a strong-to-weak $\mathbb{Z}_2$ SSB phase (region II) in which the Renyi-2 susceptibility $\chi^{II}_{ZZ}$ is $O(1)$ while the strange-correlator susceptibility $\chi^{I}_{Z,\mathrm{st}}$ and single-chain susceptibility $\chi^I_u$ vanish, and a strong-to-trivial $\mathbb{Z}_2$ SSB phase (region III) in which all three are $O(1)$. The entanglement entropy of the renormalized doubled vector peaks at $p_{zz}^c \approx 0.37$, $0.31$, and $0.39$ for $J/h = 0.8$, $1$, and $1.2$, marking phase transitions the authors take to match the Ashkin-Teller diagram. The conclusion is that decoherence, viewed as local filtering, is a concrete route to SWSSB from an ordinary Ising chain.

Load-bearing premise

Everything rests on the assumption that the local filtering deforms the doubled-space matrix product state into a state close to the quantum Ashkin-Teller ground state, with the coupling correspondence $J\lambda_{zz} = c\,\tau_{zz}(p_{zz})$ and $h\lambda_x = c\,\tau_x(p_x)$ holding at least qualitatively; if that proximity fails, the SWSSB identification loses its parent-model justification.

Editorial extensions

If this is right

  • Decoherence strengths $p_{zz}$ and $p_x$ act as a tunable coupling $\lambda$ in the Ashkin-Teller parent model, so the mixed-state phase diagram can be navigated by changing noise rates rather than Hamiltonian parameters.
  • The middle regime is a genuine strong-to-weak $\mathbb{Z}_2$ SSB phase: finite $\chi^{II}_{ZZ}$ with vanishing $\chi^{I}_{Z,\mathrm{st}}$ and $\chi^I_u$ means the strong symmetry is broken while the weak (diagonal) symmetry is restored, an order with no pure-state analogue.
  • The phase boundaries obtained from entanglement-entropy peaks ($p_{zz}^c \approx 0.37$ for $J/h=0.8$, $0.31$ for $J/h=1$, and $0.39$ for $J/h=1.2$ in the thermodynamic limit) locate the decoherence-induced transitions.
  • At the critical point $J/h=1$, the state remains critical up to $p_{zz} \approx 0.3$ with effective central charge $c_{\mathrm{eff}} = 1$ and is then driven into the gapped SWSSB region, showing a noise-induced transition out of criticality.
  • The same filtering formalism is expected to apply to other spin models such as the $XXZ$ chain, where the doubled-space ladder picture can reveal mixed-state phases not yet known.

Reading between the lines

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

  • Because regime II of the Ashkin-Teller model is a spin-glass-type phase, the SWSSB region likely reflects a glassy, non-commuting order in the original density matrix; a direct test would be to check whether $\rho_D$ shows Edwards-Anderson-type freezing of local Renyi-2 correlators, not just the doubled-space $\chi^{II}_{ZZ}$.
  • The strong symmetry of the decoherence channel suggests the SWSSB phase should be robust to weak-symmetric perturbations of the channel; adding a small weak-symmetric noise channel and checking whether $\chi^{II}_{ZZ}$ stays finite would test that robustness.
  • The numerics at $J/h=1$ show $c_{\mathrm{eff}}=1$ before the transition; one testable extension is to measure the central charge at the extrapolated $p_{zz}^c$ and check whether it flows to the $\mathbb{Z}_2$-orbifold boson CFT expected at the Ashkin-Teller transition.
  • The filtering/parent-Hamiltonian logic could be inverted: choose any ladder Hamiltonian whose ground state is a doubled pure state, identify the local filters that deform it, and the corresponding single-chain decoherence protocol should produce the same mixed-state phase diagram.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the one-dimensional transverse-field Ising model (TFIM) under two types of decoherence, ZZ and X channels, using the doubled Hilbert space formalism. The decohered density matrix is vectorized and the decoherence channels become local filtering operators acting on the doubled-space MPS. The authors argue that the resulting filtered state is close to the ground state of the quantum Ashkin-Teller (qAT) model on a ladder, with parameter correspondence J λ_zz = c τ_zz and h λ_x = c τ_x. They numerically compute Rényi-2 correlator susceptibilities, a strange correlator susceptibility, single-chain ZZ correlator susceptibilities, and entanglement entropies for three parameter sweeps (J/h = 0.8, 1.0, 1.2), identifying three phases: a trivial paramagnetic regime, a strong-to-weak symmetry breaking (SWSSB) regime, and a strong-to-trivial SSB regime. The paper concludes that multiple decoherence applied to the ordinary Ising chain produces a SWSSB mixed state whose phase boundaries resemble those of the qAT model.

Significance. If the central identification with the qAT ground state is correct, the paper provides a concrete and tunable decoherence protocol that produces SWSSB from a conventional pure-state model, which is a valuable addition to the growing literature on mixed-state quantum orders. The numerical work is careful in several respects: the MPS implementation uses bond dimension D=200 and truncation 1e-6, the DMRG convergence criterion is stated, and Appendix C compares MPS results with exact diagonalization for L=8 and finds exact agreement for the tested observables. The data and code are deposited on Zenodo, which supports reproducibility. However, the phase identification—and in particular the SWSSB label for region II—is inherited from the qAT phase diagram through an unverified correspondence, so the significance of the numerical transitions depends on strengthening that correspondence with direct evidence.

major comments (4)
  1. [Sec. IV and Sec. VIII] The load-bearing claim, stated in Sec. VIII, is that the filtered state is 'close to the ground state of the qAT model.' This is supported only by the expectation that the filtering construction, previously validated for frustration-free models, also holds for the non-frustration-free TFIM. No direct numerical evidence is provided that the filtered MPS resembles a qAT ground state: the paper computes correlators and entanglement entropies of the filtered state alone, and matches phase boundaries by eye. I ask the authors to provide a quantitative check, for example the overlap or fidelity between the filtered MPS and a DMRG ground state of H_qAT at the corresponding parameters, or a comparison of local observables and correlation lengths. Without such a check, the assignment of the three phases to qAT regimes I, II, and III—and hence the identification of region II as SWSSB—remains a conjecture.
  2. [Sec. IV and Sec. V] The parameter correspondence J λ_zz = c τ_zz and h λ_x = c τ_x contains an unspecified positive constant c. The protocol in Sec. V chooses p_x = 1/2 - (1/2)(1 - 2p_zz)^{1/J} to enforce λ_zz = λ_x, but the actual path in the qAT phase diagram depends on the ratio λ/J, which is fixed only up to the unknown c. The paper does not explain how c is determined or why the selected path in the qAT phase diagram is the relevant one. The authors should either fix c independently or demonstrate that the observed transition points are compatible with qAT phase boundaries for a range of c values.
  3. [Sec. VI, Figs. 3 and 4] The transition points p_c^zz are estimated from peaks of the entanglement entropy. The peak locations are obtained by fitting a sixth-order polynomial and then extrapolated linearly in 1/L. This procedure is sensitive to the choice of fitting function, and no scaling collapse or finite-size scaling of the order parameters is shown to confirm that the EE peaks correspond to genuine phase transitions rather than crossover behavior. A finite-size scaling analysis for χ_II_ZZ, χ_I_Z,st, and χ_I_u, or a collapse of the EE data, would strengthen the claim that the observed changes are true mixed-state phase transitions.
  4. [Sec. VI and Appendix D] For J/h=1, the initial state is the critical Ising state, and the filtered state is claimed to remain critical up to the transition. Appendix D reports c_eff = 1 from a fit of the entanglement entropy, but the paper does not discuss how robust this value is to the fitting range or to finite-size effects. Since the qAT model has a continuously varying central charge, c_eff alone does not identify a specific critical theory; additional evidence, such as a scaling collapse or comparison with known qAT critical correlators, is needed before assigning the critical line to the qAT criticality.
minor comments (5)
  1. [Sec. V] There is a typo in the paragraph defining χ_I_u: 'sightly different' should read 'slightly different.'
  2. [Secs. IV and VII and Table I] The terms 'regime' and 'region' are used inconsistently; for example, the text refers to 'regime I' in Sec. VI but 'Region I' in Table I and Fig. 2. This should be harmonized.
  3. [Sec. V and Fig. 5] After defining χ_I_Z,st, the text then writes 'χ_I_ZZ,st ~ 0' in one sentence; the subscript is inconsistent with the definition. Please check all subscripts for the strange correlator.
  4. [Appendix C] The exact-diagonalization comparison is performed only for L=8 and for J=0.1 or J=0.01, i.e., in the paramagnetic phase. It would be useful to state whether the reported exact agreement also holds for the regimes relevant to the SWSSB transition, where the initial state is critical or ferromagnetic.
  5. [Fig. 3] The figure caption states L=28, but the text in Sec. VI does not explicitly list the system sizes for the correlator data in Fig. 3. Please state the system sizes used for each panel in the main text or caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the qAT parent-model identification is a conjecture tested against independently computed observables, not a fitted input.

full rationale

The paper's central claim is that the filtered doubled-space state (Eq. 1) resembles the ground state of the quantum Ashkin-Teller model. This is explicitly presented as an expectation, not as a definition: Sec. IV says the parameter relations J λzz ↔ τzz and h λx ↔ τx are 'expected to qualitatively hold,' and the same section concedes that the starting TFIM is not frustration-free, unlike the toric-code cases in Refs. [36,37]. No qAT parameter is fitted to the data; the numerical sweeps are defined by the independent condition (1/J)τzz(pzz) = τx(px), i.e. px = 1/2 − (1/2)(1−2pzz)^{1/J}, and the resulting order parameters χII_ZZ, χI_Z,st, χI_u and the entanglement entropy are computed directly on |ρD⟩⟩. The phase boundaries (pzz ≈ 0.372, 0.308, 0.393) are then compared with the qAT diagram by eye and by polynomial extrapolation, which is a test of the conjecture rather than a tautology. The qAT phase diagram itself is imported from external references [39,47,48], not from the authors' prior work; the authors' self-citations [16,26] are background on mixed-state topological order and SWSSB and are not load-bearing for the main derivation. The main vulnerability is evidential rather than circular: the paper never computes a direct overlap or energy comparison between |ρD⟩⟩ and the qAT ground state, so the 'close to the qAT ground state' claim in Sec. VIII remains an unproven interpretive hypothesis. That gap concerns correctness and verification, not circularity of the derivation chain. No quantity used as an output is also used as an input by construction, no fitted parameter is relabeled as a prediction, and no uniqueness theorem from the authors is invoked to force the choice of the parent model. Therefore no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The main ledger entry is the qAT correspondence, which is an expectation rather than a derivation. Standard Choi vectorization and MPS truncation are conventional tools. No new particles, forces, or entities are introduced. The free parameter c is arbitrary but cancels in the equal-lambda sweep, and the EE-peak fits are analysis parameters rather than physical inputs.

free parameters (2)
  • coupling constant c = not fixed (positive)
    Introduced in Sec. IV as J λzz = c τzz and h λx = c τx; the equality λzz = λx is imposed via (1/J) τzz = τx, so c does not enter the central parameter sweep, but the quantitative match to the qAT phase boundary is never computed.
  • EE peak polynomial fit coefficients = not reported
    A sixth-order polynomial is used to locate SA maxima and a linear fit a/L + pc_zz extrapolates transition points; these fitted coefficients are analysis parameters, not physical, and no errors are quoted.
assumptions (5)
  • standard math Choi-Jamiołkowski vectorization maps density matrices to states in doubled Hilbert space and maps the Pauli channels to local filtering operators.
    Used in Sec. III to derive Eq. (1); this is a standard quantum information construction.
  • domain assumption The vectorized initial pure state is the product of upper and lower TFIM ground states, with the J/h > 1 ground state chosen as a Z2-symmetric cat state.
    Sec. V and VII; the cat-state choice is needed so the initial state is strong symmetric and SSB can be assessed in a finite system.
  • domain assumption After filtering, the state remains well-approximated by an MPS at bond dimension D=200 with singular values below 1e-6 discarded.
    Sec. VI; a standard truncation assumption, validated in Appendix C only for an L=8 exact-diagonalization comparison.
  • ad hoc to paper The filtered state is close to the qAT ground state with coupling correspondence J λzz = c τzz and h λx = c τx.
    Sec. IV states this as 'we expect parameter relations... are expected to qualitatively hold'. This is the load-bearing nontrivial premise; without it the phase labels do not follow.
  • domain assumption The known ground-state phase diagram of the quantum Ashkin-Teller model (Refs. 39, 46, 47, 48) is accurate and applies to the ladder with periodic boundary conditions.
    Used throughout Sec. VI to identify regimes and estimate where transitions should occur in the filtered state.

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Cite this review

Pith. "Pith review of Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum Ising model under multiple decoherence." pith.science (2026). https://pith.science/paper/YVRDI524

@misc{pith2026241212738,
  author       = {Pith},
  title        = {Pith review of: Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum Ising model under multiple decoherence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YVRDI524}},
  note         = {Machine review of arXiv:2412.12738}
}
abstract

Discovering and categorizing quantum orders in mixed many-body systems are currently one of the most important problems. Specific types of decoherence applied to typical quantum many-body states can induce a novel kind of mixed state accompanying characteristic symmetry orders, which has no counterparts in pure many-body states. We study phenomena generated by interplay between two types of decoherence applied to the one-dimensional transverse field Ising model (TFIM). We show that in the doubled Hilbert space formalism, the decoherence can be described by filtering operation applied to matrix product states (MPS) defined in the doubled Hilbert system. The filtering operation induces specific deformation of the MPS, which approximates the ground state of a certain parent Hamiltonian in the doubled Hilbert space. In the present case, such a parent Hamiltonian is the quantum Ashkin-Teller model, having a rich phase diagram with a critical lines and quantum phase transitions. By investigating the deformed MPS, we find various types of mixed states emergent from the ground states of the TFIM, and clarify phase transitions between them. In that study, strong and weak $Z_2$ symmetries play an important role, for which we introduce efficient order parameters, such as R\'{e}nyi-2 correlators, entanglement entropy, etc., in the doubled Hilbert space.

Figures

Figures reproduced from arXiv: 2412.12738 by the authors.

Figure 1
Figure 1. FIG. 1. (a) 1D quantum TFIM under two types of decoherence. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic phase diagram of the doubled system and pa [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Entanglement entropy for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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