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arxiv: 2605.00088 · v1 · submitted 2026-04-30 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

Recognition: unknown

A Unified Framework for Locally Stable Phases

Zhi Li , Raz Firanko , Timothy H. Hsieh
Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:53 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP
keywords local stabilityshort-range correlationsquantum phases of mattermixed statesnon-equilibrium phasesconditional mutual informationreversible quantum channels
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The pith

Local stability of quantum states is equivalent to short-range correlations, unifying phases for pure and mixed states.

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

The paper introduces locally stable states as those for which any local operation, including post-selection, can be reversed by a local quantum channel. It proves this property is equivalent to the state having short-range correlations, where both two-point correlations and conditional mutual information decay with distance. These properties remain unchanged under locally reversible channels, allowing the definition of phases that apply equally to equilibrium, non-equilibrium, and metastable regimes. The framework also connects mixed states to pure states by showing decay of correlations in the canonical purification and of nonlinear observables.

Core claim

We prove that local stability is equivalent to a state being short range correlated, defined by the decay of both correlations and conditional mutual information. We demonstrate that these properties are invariant under locally reversible channels, thus defining locally stable phases. Furthermore, we prove that local stability implies both the decay of a family of nonlinear correlators, including the fidelity correlator, and the decay of correlations in the canonical purification, thus bridging the gap between mixed and pure states.

What carries the argument

The equivalence between operational local reversibility (any local operation reversible by a local channel) and short-range correlations (decay of correlations and conditional mutual information).

If this is right

  • Local stability defines phases in non-equilibrium and metastable regimes.
  • Post-selection on locally stable states can be realized by local channels.
  • Quantum Markov chains can be identified by local computability of nonlinear observables.
  • Nonlinear correlators such as the fidelity correlator decay in locally stable states.
  • Correlations decay in the canonical purification of locally stable mixed states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This may allow phase classification in open quantum systems without assuming equilibrium.
  • Experimental tests could involve applying local operations and checking reversibility in quantum simulators.
  • The local computability result might simplify numerical simulations of Markovian systems.

Load-bearing premise

The assumption that reversibility by local channels, including post-selection, fully captures the stability of phases in general quantum systems.

What would settle it

Finding a quantum state where correlations decay but some local operation including post-selection cannot be reversed by a local channel, or vice versa.

Figures

Figures reproduced from arXiv: 2605.00088 by Raz Firanko, Timothy H. Hsieh, Zhi Li.

Figure 1
Figure 1. Figure 1: FIG. 1. (left) Summary of main results, which provides a unified framework for the (right) generalized landscape of phases [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The partition of the lattice used in Theorem [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic proof of Theorem [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Lattice partition in Theorem [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The light cone structure in ( [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

We propose a unifying framework for characterizing pure and mixed state phases of matter across equilibrium, non equilibrium, and metastable regimes. We introduce the concept of locally stable states, defined by the operational property that any local operation (including post selection) can be reversed by a local channel. We prove that local stability is equivalent to a state being short range correlated, defined by the decay of both correlations and conditional mutual information. We demonstrate that these properties are invariant under locally reversible channels, thus defining locally stable phases. Furthermore, we prove that local stability implies both the decay of a family of nonlinear correlators, including the fidelity correlator, and the decay of correlations in the canonical purification, thus bridging the gap between mixed and pure states. Along the way, we establish two results which may be of independent interest: we show that post-selection on locally stable (short range correlated) states can be implemented via local channels and that quantum Markov chains can be characterized by the local computability of nonlinear observables.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a unifying framework for pure and mixed state phases of matter across equilibrium, non-equilibrium, and metastable regimes. It defines locally stable states operationally as those for which any local operation (including post-selection) is reversible by a local quantum channel. It proves equivalence to short-range correlated states (via decay of correlations and conditional mutual information), shows invariance of these properties under locally reversible channels (thereby defining phases), and extends the framework to mixed states via decay of nonlinear correlators (including the fidelity correlator) and correlations in the canonical purification. Two results of independent interest are established: post-selection on locally stable states can be implemented by local channels, and quantum Markov chains are characterized by local computability of nonlinear observables.

Significance. If the equivalences hold rigorously, the work supplies an operational, channel-theoretic definition of stability that unifies short-range correlation concepts with phase invariance under local reversibility, applying to both pure and mixed states without introducing free parameters or self-referential definitions. The proofs build directly on established quantum channel and correlation theory, and the two independent-interest results (local post-selection implementation and Markov-chain characterization) add concrete value beyond the main framework.

major comments (2)
  1. [Abstract and introduction (non-equilibrium/metastable regimes)] Abstract and introduction (non-equilibrium/metastable regimes): the central claim that local stability defines phases across non-equilibrium and metastable regimes rests on the static operational definition (reversibility by time-independent local channels). The proof that post-selection on short-range correlated states is implementable by local channels assumes closed-system locality; this risks failing for metastable states with slow relaxation or long-lived correlations not detected by static CMI decay, as noted in the stress-test concern. A concrete example or additional dynamical assumption is needed to confirm the equivalence extends without further conditions.
  2. [Section on mixed-state extensions (canonical purification and nonlinear correlators)] Section on mixed-state extensions (canonical purification and nonlinear correlators): the claim that local stability implies decay of correlations in the canonical purification and of the family of nonlinear correlators is load-bearing for bridging pure and mixed states. The derivation should explicitly address whether the equivalence to short-range correlations (via CMI decay) carries over when the purification introduces additional degrees of freedom, and whether any edge cases for non-pure states are covered.
minor comments (2)
  1. [Throughout] Ensure uniform terminology between 'locally stable states' and 'locally stable phases' to prevent reader confusion when the invariance result is stated.
  2. [Abstract] The abstract would benefit from a single sentence highlighting the two independent-interest results to better foreground their contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the scope and presentation of our framework. We respond to each major comment below.

read point-by-point responses

  1. Referee: Abstract and introduction (non-equilibrium/metastable regimes): the central claim that local stability defines phases across non-equilibrium and metastable regimes rests on the static operational definition (reversibility by time-independent local channels). The proof that post-selection on short-range correlated states is implementable by local channels assumes closed-system locality; this risks failing for metastable states with slow relaxation or long-lived correlations not detected by static CMI decay, as noted in the stress-test concern. A concrete example or additional dynamical assumption is needed to confirm the equivalence extends without further conditions.

    Authors: We agree that the framework is defined statically via reversibility under time-independent local channels, and this is intentional to provide an operational characterization applicable to metastable regimes when the static short-range correlation condition holds. The local post-selection implementation follows directly from the CMI decay property without invoking full closed-system dynamics. We acknowledge that metastable states exhibiting slow relaxation or correlations beyond static CMI decay may require supplementary dynamical assumptions for full equivalence. We will revise the introduction to explicitly note the static scope of the claims and indicate that dynamical extensions lie beyond the present work. revision: partial

  2. Referee: Section on mixed-state extensions (canonical purification and nonlinear correlators): the claim that local stability implies decay of correlations in the canonical purification and of the family of nonlinear correlators is load-bearing for bridging pure and mixed states. The derivation should explicitly address whether the equivalence to short-range correlations (via CMI decay) carries over when the purification introduces additional degrees of freedom, and whether any edge cases for non-pure states are covered.

    Authors: The proofs establish that short-range correlations (via CMI decay) imply the desired decay properties in the canonical purification and for the family of nonlinear correlators. The ancillary degrees of freedom in the purification do not participate in the local operations or measurements on the physical system, so locality is preserved and the equivalence carries over. The derivations are formulated for general density operators and thus cover non-pure states, including edge cases. We will expand the mixed-state section with an explicit remark addressing the role of the purification's extra degrees of freedom and confirming applicability to non-pure states. revision: yes

Circularity Check

0 steps flagged

No circularity: operational definition of local stability proved equivalent to short-range correlations via independent mathematical steps.

full rationale

The paper defines locally stable states operationally as those where any local operation (including post-selection) is reversible by a local channel. It then proves equivalence to short-range correlations (decay of correlations and conditional mutual information) through explicit theorems, without the equivalence being imposed by definition or by fitting parameters. Invariance under locally reversible channels follows directly from the proved equivalence rather than self-reference. Extensions to mixed states via canonical purification and nonlinear correlators are derived consequences, not inputs. No load-bearing self-citations, ansatzes smuggled via prior work, or renamings of known results appear in the derivation chain; the framework is self-contained against external quantum information benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on standard quantum information axioms for channels, states, and correlations; the new concept is introduced by definition rather than derived from prior equations.

axioms (1)
  • standard math Standard axioms of quantum mechanics, quantum channels, and correlation functions including conditional mutual information.
    Invoked to define local operations, reversibility, and decay of correlations throughout the framework.
invented entities (1)
  • locally stable state no independent evidence
    purpose: Operational definition to unify phases across regimes and state types.
    Newly introduced concept based on reversibility property; no independent evidence outside the definition itself.

pith-pipeline@v0.9.0 · 5473 in / 1441 out tokens · 27320 ms · 2026-05-09T20:53:02.336887+00:00 · methodology

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Reference graph

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