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R\'{e}nyi Markov length in one-dimensional non-trivial mixed state phases and mixed state phase transitions

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Pith's one-line read The second Rényi conditional mutual information is a mixed-state 'gap' measure: its Markov length stays short in average-SPT and trivial mixed phases, peaks at their transition, and becomes infinite in SWSSB, where it saturates at ln 2.

desk verdict Useful numerical method for second-Rényi CMI, but the conceptual 'gap' interpretation rests on an unproved proportionality that needs direct testing. read the letter →

arxiv 2505.02125 v1 pith:D4YUTDHH submitted 2025-05-04 quant-ph cond-mat.stat-mechcond-mat.str-el

Yoshihito Kuno , Takahiro Orito , Ikuo Ichinose This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-el
keywords RényiconditionalmutualinformationMarkovlengthmixed-statephasesaveragesymmetry-protectedtopologicalorderstrong-to-weakspontaneoussymmetrybreakingdoubledHilbertspacematrixproductstatesdecoherence
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 seeks to establish that the second Rényi conditional mutual information of a tripartite one-dimensional chain, and the Markov length extracted from its exponential decay, can serve as the mixed-state analogue of an energy gap: a diagnostic that classifies noisy quantum phases and signals the transitions between them. This matters because nontrivial mixed states such as average symmetry-protected topological (ASPT) states and strong-to-weak spontaneous symmetry breaking (SWSSB) states have no pure-state Hamiltonian description, so they currently lack a workable principle for 'gapped' versus 'gapless' classification. The authors propose an efficient numerical scheme—computing the second Rényi CMI as matrix-product-state norms in the doubled Hilbert space—and apply it to two decohered chains: a cluster model under odd-site Z noise, and a transverse-field Ising model under ZZ and X noise. Numerically, the Markov length stays short inside the ASPT and trivial mixed phases, grows sharply at the ASPT transition, and becomes infinite in the SWSSB phase, where the CMI saturates at the universal value $\ln 2$. If the paper is right, a single information-theoretic length scale organizes the mixed-state phase diagram of these systems and provides a computable, channel-based notion of mixed-state phase equivalence.

What carries the argument

The engine of the argument is the second Rényi conditional mutual information, $I^{(2)}(A:C|B)(r) = I^{(2)}(A,BC) - I^{(2)}(A,B)$, assembled from second Rényi entropies $S^{(2)}_X = -\log \operatorname{Tr}(\rho_X^2)$; the second Rényi Markov length $\xi^{(2)}_M$ is defined by the decay $I^{(2)}(A:C|B)(r) \sim e^{-r/\xi^{(2)}_M}$. Computationally, the paper vectorizes the density matrix into a supervector in the doubled Hilbert space and implements partial trace as a maximal depolarization operator $\hat{D}_{\bar X}$ obeying $\hat{D}_{\bar X}|\rho\rangle\rangle = |I_{\bar X}/d_{\bar X} \otimes \rho_X\rangle\rangle$, so every second Rényi entropy reduces to the logarithm of a matrix-product-state norm that filtering methods evaluate directly. Conceptually, the load-bearing link is the quoted proportionality $I(A:C|B) = \beta I^{(2)}(A:C|B)$, which carries the recovery-map bound of Ref. [33] into the Rényi-2 setting and gives $1/\xi^{(2)}_M$ its meaning as a gap controlling recoverability and mixed-state phase equivalence.

What would settle it

Compute the von Neumann conditional mutual information and the second Rényi CMI on the same decohered states—e.g., the cluster model at $p_z=0.1$ for $h_x = 0.78, 1.0, 1.22$, using the doubled-Hilbert-space matrix-product-state method with converged bond dimension—and test whether the ratio $I/I^{(2)}$ stays constant as $h_x$, $p_{zz}$, and $r$ vary. If the ratio drifts, or if an explicit recovery map built from the reduced state of $A\cup B$ fails to reproduce an error that decays with rate $1/\xi^{(2)}_M$, the identification of the Rényi Markov length with a mixed-state gap is falsified.

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Extended reading notes

Core claim

The paper's central claim is that the second Rényi conditional mutual information $I^{(2)}(A:C|B)(r)$, built from second Rényi entropies on a tripartition of a 1D spin chain, decays exponentially in the buffer size $r$ with a rate that defines the second Rényi Markov length $\xi^{(2)}_M$, and that $1/\xi^{(2)}_M$ behaves as a gap for mixed states: small inside gapped mixed phases, sharply peaked at mixed-state transitions, and effectively infinite inside the SWSSB phase. For the cluster model under odd-site $Z$ decoherence, the authors find $\xi^{(2)}_M \approx 2.8\text{–}2.9$ inside both the ASPT and trivial mixed phases and a pronounced peak (about 10 at $p_z=0.1$) at $h_x\approx 1$, the ASPT-to-trivial transition. For the transverse-field Ising model under $ZZ$ and $X$ decoherence, $\xi^{(2)}_M$ grows from about 1.6 toward the SWSSB transition, and in the deep SWSSB phase the CMI loses its $r$-dependence and takes the constant value $\ln 2$, which the paper verifies in a stabilizer limit; hence the Markov length is infinite throughout the SWSSB phase. The stated conclusion is that the second Rényi CMI and its Markov length are good measures to classify the regime of a nontrivial mixed state and to locate mixed-state phase transitions between them, mirroring the von Neumann CMI while being numerically far more tractable.

Load-bearing premise

The load-bearing premise is that the second Rényi conditional mutual information tracks the ordinary one up to a constant positive factor; if that factor varies from state to state, a short Rényi Markov length need not mean the noisy state can be recovered from its surroundings, and the phase classification built on it would slip.

Editorial extensions

If this is right

  • The ASPT-to-trivial transition in the decohered cluster model is detectable as a sharp peak in $\xi^{(2)}_M$ at $h_x \approx 1$ for every decoherence strength tested ($p_z = 0.1, 0.2, 0.3$), so the second Rényi Markov length serves as a numerical order parameter for that transition.
  • Inside the ASPT and trivial mixed phases the finite, short Markov length implies that the decohered states admit efficient recovery maps, so states within each regime belong to a common mixed-state phase in the sense of Ref. [33].
  • In the SWSSB phase the exponential decay of the second Rényi CMI disappears, the Markov length is infinite, and the local recovery bound no longer closes with $r$; the plateau at $\ln 2$ is the information-theoretic fingerprint of the phase.
  • The doubled-Hilbert-space matrix-product-state scheme is channel-agnostic, so the same computation applies to any local decoherence channel on a 1D spin chain, including the $ZZ$-only and $X$-only channels that make up the Ising-model noise studied here.

Reading between the lines

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

  • Because the paper never evaluates the von Neumann CMI on the same states, an immediate check of the proportionality $I = \beta I^{(2)}$ is still open: if the ratio drifts with $r$, $h_x$, or $p_{zz}$, then $\xi^{(2)}_M$ is a state-dependent proxy rather than a faithful gap, and the phase-equivalence statements inherit that caveat.
  • The growth of $\xi^{(2)}_M$ from about 2.8 to about 10 at $h_x \approx 1$ within a finite system suggests a critical scaling region; fitting $I^{(2)}(A:C|B)(r)$ to a power law $r^{-\alpha}$ at the transition would extract a Rényi-2 critical exponent and could connect the ASPT transition to known measurement-induced or percolation criticality.
  • The $\ln 2$ plateau is the Rényi-2 avatar of the topological entropy of the SWSSB state; extending the same norm-of-supervector technology to negativity or to 1-form-symmetric channels might expose intrinsic mixed-state topological order through the same computational route.
  • A practical extension would be to apply the method to monitored quantum circuits or to two-dimensional strips, where the matrix-product-state ansatz still works but the tripartition geometry changes, testing whether the Markov-length criterion survives beyond the quasi-1D setting.
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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

3 major / 5 minor

Summary. The paper proposes the second Rényi conditional mutual information (CMI), I^(2)(A:C|B), and its associated Markov length ξ_M^(2) as diagnostic quantities for non-trivial mixed-state phases and mixed-state phase transitions in one-dimensional spin systems. The authors introduce a doubled-Hilbert-space numerical scheme, based on applying depolarizing channels to compute reduced density-matrix purities via matrix product states. They apply this method to two models: (i) the cluster model in a transverse field under odd-site Z decoherence, where they report exponential decay of I^(2) away from h_x ≈ 1, a growing Markov length near the ASPT-to-trivial transition, and finite Markov lengths in both phases; and (ii) the transverse-field Ising model under ZZ and X decoherence, where they report growth of ξ_M^(2) approaching the SWSSB transition and an r-independent CMI saturating at ln 2 in the deep SWSSB phase. Appendix C gives an exact stabilizer calculation showing that the ln 2 value is obtained for a prototypical SWSSB state. The central claim is that the second Rényi CMI and its Markov length are good measures for classifying non-trivial mixed states and locating mixed-state phase transitions.

Significance. If the main claims are established, the paper would provide a numerically tractable proxy for the von Neumann CMI/Markov-length framework of Ref. [33] and a new diagnostic for strong-to-weak spontaneous symmetry breaking. The doubled-Hilbert-space computational scheme is simple and likely useful for future studies of decohered states. The ln 2 saturation in the SWSSB phase is a concrete, falsifiable prediction, and the exact stabilizer calculation in Appendix C is a genuine strength. However, the paper's conceptual claim that ξ_M^(2) is a mixed-state 'gap' and the phase-equivalence interpretation depend on an unproved proportionality between von Neumann CMI and second Rényi CMI, and the numerical evidence currently lacks error bars and finite-size control. The work is therefore a promising but not yet fully established contribution.

major comments (3)
  1. [Section II A, Eq. (6)] The proportionality I(A:C|B) = β I^(2)(A:C|B) with a positive factor β is introduced as an expectation, and Eq. (7) then uses it to convert exponential decay of the second Rényi CMI into a Petz-map recovery bound. This is load-bearing for the interpretation of ξ_M^(2) as a mixed-state 'gap' and for importing the phase-equivalence framework of Ref. [33]. The relation is not proven for the models studied, and the cited Refs. [39–41] are numerical and analytical studies of different settings. Because the second Rényi CMI does not in general inherit the information-theoretic properties of the von Neumann CMI (nonnegativity, strong subadditivity, data processing), a universal state-independent positive β is a nontrivial assumption. If β is state-dependent or if the relation fails, exponential decay of I^(2) does not imply recoverability, and the central conceptual claim is not established. I ask the authors to either prove or directly test Eq. (6) on the actual decohered states considered here, for example by computing the von Neumann CMI alongside I^(2) at the same parameters, or to explicitly restrict the claims to a phenomenological diagnostic and remove the recoverability/gap interpretation.
  2. [Section IV A and Section IV B, fitting of ξ_M^(2)] The Markov-length estimates are obtained by fitting I^(2)(r) = e^{-c0 r} + c1 using a small number of r values, with no error bars, no stated r range, no fit residuals, and no extrapolation in system size. Because the system size is tied to r through |C| = r, finite-size effects from the complementary region C can masquerade as exponential decay or as a constant offset. The central distinction between a finite Markov length and an infinite one is exactly what the fits are used to establish, so the absence of quantitative fit quality and of any finite-size scaling analysis is a serious gap. For example, at p_zz = 0.28 the reported ξ_M^(2) ≈ 3.825 is still finite, yet the text concludes that the Markov length becomes infinite throughout the SWSSB phase; that conclusion rests on an extrapolation to p_zz → 1/2 for which no r-dependence data are shown. I request that the authors report full parameters (L, r values, bond dimensions, truncation errors), provide error bars or at least show the fit curves together with the data, and include a finite-size scaling analysis at the purported transition points.
  3. [Section IV B and Appendix C, SWSSB saturation claim] The statement that the second Rényi CMI takes the universal value ln 2, and hence that the Markov length is infinite in the whole SWSSB phase, extrapolates from data at p_zz ≲ 0.28 plus an exact calculation for a single stabilizer state. The stabilizer state in Appendix C has only the global stabilizer S = {∏_j X_j}, and while the ln 2 value is derived correctly in that limit, the claim that this value is universal across the entire SWSSB phase is not proven. The numerical saturation in Fig. 5(a) is supportive, but the r-dependence in the deep SWSSB region is not displayed. I recommend either presenting r-dependence data at larger p_zz values (e.g., p_zz = 0.4, 0.45, 0.5) or softening the universality claim to a prediction consistent with the data.
minor comments (5)
  1. [Appendix B] In the p_z = 0.3 paragraph of Appendix B, the extracted Markov lengths are reported 'at pzz = 0.11, 0.19 and 0.28', but the context is the cluster model and the correct parameters should be h_x = 0.78, 1.0, 1.22. This copy-paste from Section IV B should be corrected.
  2. [Appendix C heading] The heading 'Second Rényi CMI for Jzz→0 and pzz→∞' is inconsistent with the definition 0 ≤ p_zz ≤ 1/2 elsewhere in the paper; it should presumably read p_zz → 1/2.
  3. [Section I] The introductory paragraph refers to 'Section VIII' for the summary and conclusion, but the paper only has five numbered sections plus appendices; the reference should be corrected.
  4. [Equations (18) and (19)] The Choi operators in Eqs. (18) and (19) include complex-conjugated identity and Pauli operators on the upper chain (e.g., Ê_ZZ contains to Z^∗_{j,u}); since the Pauli matrices are Hermitian this is harmless, but it may confuse readers and should be explained or simplified.
  5. [Figure captions and numerical settings] The main-text figures do not specify the system sizes L, the values of r used in the fits, or the bond dimensions used for the mixed-state MPS. Reporting these in the captions or in the text would substantially improve reproducibility, especially because estimates of ξ_M^(2) are the central quantitative output.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the second-Rényi CMI is computed directly, and the unproven proportionality Eq. (6) is an explicit assumption, not a circular reduction.

full rationale

The numerical quantities I(2)(A:C|B) are evaluated directly from the decohered density matrices via the doubled-Hilbert filtering method; no fitted parameter is renamed as a prediction, and the exponential-decay Markov lengths are extracted from the same directly computed CMI curves. The paper's phase boundaries for the TFIM case are taken from the authors' prior work [22], but this is used as a reference phase diagram against which the new CMI diagnostic is tested, not as an input that forces the CMI values. The central load-bearing assumption is Eq. (6), I(A:C|B)=β I(2)(A:C|B), which the paper itself introduces with 'we expect' and attributes to Refs. [39-41]. This relation is unproved and possibly state-dependent, so the recovery-map interpretation and the identification of ξ_M^(2) with a mixed-state gap remain conditional. However, this is an openly stated assumption on which the physical interpretation rests, not a circularity: the paper does not define I(2) in terms of I, nor does it fit β from the data it then 'predicts.' The ln 2 saturation in the SWSSB phase is checked against an independent stabilizer-limit calculation in Appendix C. Self-citations to [22], [23], [32], and [50] are present but not load-bearing for the main claim; they supply methods or previously mapped phase diagrams rather than the CMI results. Overall, no step reduces to its own input by construction, so the paper receives a low circularity score, with the unproved proportionality Eq. (6) noted as a correctness risk rather than a circular step.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central observable is a direct computation on the decohered state, so no invented entities appear. The main unpaid inputs are the assumed proportionality between von Neumann and second Rényi CMI (Eq. 6), the posited exponential decay form (Eq. 9), and the cited Petz-recovery framework of Ref. [33]. The only tuned quantity is the unknown factor beta; the exponential fits also include a constant offset that is not tabulated.

free parameters (2)
  • beta in Eq. (6) = not determined
    Assumed positive constant relating von Neumann CMI to second Rényi CMI; its state-independence is needed for the Markov-length interpretation.
  • fit offset c1 in I=e^{-c0 r}+c1 = not tabulated
    Ad hoc background constant included in every Markov-length fit; the extracted c0 depends on this choice, so the reported Markov lengths are fit-dependent.
assumptions (4)
  • domain assumption Proportionality I(A:C|B)=β I^(2)(A:C|B) (Eq. 6)
    Assumed from Refs. [39-41]; used to connect the second Rényi CMI to recovery-map bounds and the notion of mixed-state gap.
  • domain assumption Exponential decay form Eq. (9) for the second Rényi CMI
    The paper posits exponential decay away from criticality with a possible power-law at criticality; this form is not derived and underpins the definition of the second Markov length.
  • standard math Petz map recovery bounds (Eq. 7)
    Taken from Ref. [33]; the framework of mixed-state phase equivalence via recoverability rests on these cited theorems.
  • domain assumption The SWSSB phase is characterized by saturation of CMI at ln 2
    The paper verifies this in a stabilizer limit (Appendix C) and then assumes it extends to the full SWSSB phase.

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Cite this review

Pith. "Pith review of R\'{e}nyi Markov length in one-dimensional non-trivial mixed state phases and mixed state phase transitions." pith.science (2026). https://pith.science/paper/D4YUTDHH

@misc{pith2026250502125,
  author       = {Pith},
  title        = {Pith review of: R\'enyi Markov length in one-dimensional non-trivial mixed state phases and mixed state phase transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4YUTDHH}},
  note         = {Machine review of arXiv:2505.02125}
}
abstract

Discovering and classifying non-trivial mixed states and mixed state phase transitions are some of the most important current issues in condensed matter and quantum information. In this study, we investigate some non-trivial mixed states and phase transitions between them by using the second R\'{e}nyi conditional mutual information (CMI). The CMI can measure mixed state ``gap'', estimated by the exponential decay rate of the second R\'{e}nyi CMI under a tripartition of system, which provides the second R\'{e}nyi version of the Markov length. We introduce an efficient numerical scheme for the calculation of the second R\'{e}nyi CMI based on the doubled Hilbert space formalism, and study the classification of non-trivial mixed states and the emergence of mixed state phase transitions for (i) the cluster model under odd-site local $Z$ decoherence and (ii) transverse field Ising model under both $ZZ$ and $X$ decoherence. The second R\'{e}nyi CMI is a powerful measure to study non-trivial mixed ``gapped" quantum matters and mixed phase transitions. In addition to this, the present study shows that the second R\'{e}nyi CMI exhibits specific behavior for the transition to strong-to-weak spontaneous symmetry breaking mixed phase.

Figures

Figures reproduced from arXiv: 2505.02125 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic image of the system: (a) Tripartite for the con [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. (a) displays hx-dependence of the second Renyi ´ CMI with pz = 0.1 for various r’s. The peaks appear at a specific value of hx for each r. This indicates the existence of a mixed state phase transition from the ASPT to the trivial mixed phase with increasing hx. To elucidate this result, we further observe r-dependence of the second Renyi CMI for ´ hx = 0.78, 1.0 and 1.22 around peaks. The data is displayed in [PIT… view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic image of the phase diagram on [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: (a) shows the pzz-dependence of the second Renyi ´ CMI for various r’s. The value increases as pzz increases, and the r dependence is getting larger in the region 0.1 ⪅ pzz ⪅ 0.2. This implies that in the vicinity of pz = 0.2, the drastic change of the second Renyi Mar…
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic image of the phase diagram on [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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