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arxiv: 2605.00157 · v1 · submitted 2026-04-30 · 🪐 quant-ph · math-ph· math.MP· math.PR

Asymptotic Replacement for Quantum Channel Products with Applications to Inhomogeneous Matrix Product States

Lubashan Pathirana This is my paper

Pith reviewed 2026-05-09 20:16 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPmath.PR
keywords quantum channel productstrace-Dobrushin coefficientLyapunov exponentquenched memory lossinhomogeneous matrix product statesasymptotic replacementrandom cocyclesboundary stability
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The pith

Negativity of the trace-Dobrushin Lyapunov exponent for quantum channel products is equivalent to exponential quenched memory loss and convergence to a unique replacement channel.

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

The paper establishes a product-level theory that links the decay of a centered coefficient measuring input-state dependence in finite-dimensional quantum channel products to trace-norm forgetting of memory. In the deterministic case this decay is equivalent to replacement by a moving target channel, while for stationary random sequences the almost-sure negativity of the associated Lyapunov exponent produces exponential forward and pullback convergence to a unique dynamically stationary replacement channel. The same estimates transfer directly to inhomogeneous matrix product states whose auxiliary maps are completely positive and trace-preserving, yielding infinite-volume limits, boundary stability, and correlation decay controlled by the product coefficients. A sympathetic reader cares because these results supply quantitative control on convergence and stability for open quantum dynamics and tensor-network states without requiring uniform bounds.

Core claim

For a product of completely positive trace-preserving maps the centered trace-Dobrushin coefficient quantifies residual dependence on the input state; its decay is equivalent to asymptotic replacement by a moving replacement channel. For stationary random CPTP cocycles, submultiplicativity produces a trace-Dobrushin Lyapunov exponent whose almost-sure negativity is equivalent to quenched trace-norm memory loss and yields exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the environment satisfies a vanishing rho-mixing profile the estimates become annealed super-polynomial; under independence they are annealed exponential. Thesame

What carries the argument

The centered trace-Dobrushin coefficient for a finite product of quantum channels, which measures how much the output trace-norm distance between two input states shrinks, together with its Lyapunov exponent formed by submultiplicative iteration over stationary random sequences.

If this is right

  • Deterministic products of channels asymptotically replace to a moving replacement channel once the coefficient decays.
  • Two-sided products with pullback forgetting admit a unique boundary state that fixes the canonical replacement family.
  • Under vanishing rho-mixing the annealed convergence is super-polynomial; under independence it is exponential.
  • Inhomogeneous matrix product states with CPTP transfer maps possess infinite-volume limits and quantitative boundary stability governed by the same coefficients.

Where Pith is reading between the lines

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

  • The same Lyapunov-exponent criterion could be used to certify stability of dissipative quantum circuits or error-corrected memories under random noise.
  • Numerical sampling of random channel products could directly test the predicted exponential rates by tracking trace-norm distances to the estimated replacement channel.
  • Classical Markov-chain mixing rates might admit quantum analogs via the same product-coefficient construction, opening a route to transfer techniques between the two settings.

Load-bearing premise

The auxiliary transfer maps must be completely positive and trace-preserving and the channel environment must obey suitable mixing conditions such as a rho-mixing profile tending to zero.

What would settle it

A stationary random sequence of CPTP channels for which the Lyapunov exponent is almost surely negative yet the products fail to converge in trace norm to a unique replacement channel would disprove the equivalence.

read the original abstract

We develop a product-level trace-Dobrushin theory for finite-dimensional quantum channel products and apply it to deterministic and stationary random inhomogeneous matrix product states in left-canonical CPTP gauge. For a product of channels, the centered trace-Dobrushin coefficient quantifies the residual dependence on the input state, and its decay is the criterion for trace-norm forgetting. In the deterministic setting, this decay is equivalent to asymptotic replacement by a moving replacement channel. For two-sided products, pullback forgetting produces a unique boundary state, which determines the canonical replacement family. For stationary random CPTP cocycles, submultiplicativity of the product coefficient yields a trace-Dobrushin Lyapunov exponent. We prove that the almost sure negativity of this exponent is equivalent to quenched trace-norm memory loss and gives exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the \(\varrho\)-mixing profile of the channel environment tends to zero, we obtain annealed super-polynomial estimates, while independence gives annealed exponential estimates. Finally, we transfer these estimates to inhomogeneous matrix product states whose auxiliary transfer maps are CPTP. These channel estimates transfer to deterministic and stationary random inhomogeneous MPS, giving infinite-volume limits of trace-closed finite-volume states, quantitative boundary stability, and correlation bounds governed by the same auxiliary product coefficients.

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

0 major / 2 minor

Summary. The paper develops a product-level trace-Dobrushin theory for finite-dimensional quantum channel products. It defines the centered trace-Dobrushin coefficient to quantify residual input-state dependence and shows its decay is equivalent to asymptotic replacement by a moving replacement channel in the deterministic setting. For stationary random CPTP cocycles, submultiplicativity yields a trace-Dobrushin Lyapunov exponent whose almost-sure negativity is equivalent to quenched trace-norm memory loss, implying exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. Under vanishing rho-mixing or independence, annealed estimates follow. These channel results transfer to inhomogeneous MPS with CPTP auxiliary transfer maps, yielding infinite-volume limits of trace-closed states, quantitative boundary stability, and correlation bounds governed by the same coefficients.

Significance. If the equivalences hold, the work supplies a coherent quantitative framework for memory loss and convergence in quantum channel products and inhomogeneous MPS. Strengths include the self-contained finite-dimensional treatment, explicit use of submultiplicativity and ergodic arguments for the random case, and direct transfer of estimates to MPS applications under the stated CPTP gauge and mixing conditions. The introduction of the centered trace-Dobrushin coefficient and Lyapunov exponent provides reusable tools for open quantum systems and many-body physics. No machine-checked proofs or reproducible code are present, but the deterministic-to-random passage and boundary-state uniqueness claims appear internally consistent.

minor comments (2)
  1. The abstract is dense with new terminology; a one-sentence parenthetical definition of the centered trace-Dobrushin coefficient would improve immediate readability.
  2. Notation for the family of replacement channels and the rho-mixing profile is introduced gradually; an early consolidated table or diagram in §2 would aid cross-reference when the estimates are transferred to MPS.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the accurate summary of our results on trace-Dobrushin coefficients, Lyapunov exponents, and their application to inhomogeneous MPS. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no points requiring point-by-point rebuttal or clarification at this time. We will address any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops a trace-Dobrushin coefficient from the product of channels and proves its decay equivalent to trace-norm forgetting and asymptotic replacement via submultiplicativity and ergodic arguments under CPTP gauge and mixing conditions. The Lyapunov exponent is constructed directly from the product coefficients rather than from the target convergence or replacement channel; the equivalences are shown in both directions without reducing one to a definition or fit of the other. No self-citations appear load-bearing, no ansatz is smuggled, and no renaming of known results occurs. The derivations remain self-contained within the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on finite-dimensionality of the channels, the CPTP property of transfer maps, and the existence of a rho-mixing profile for the random environment; no free parameters or new physical entities are introduced.

axioms (2)
  • domain assumption All channels are completely positive and trace-preserving (CPTP) maps on finite-dimensional spaces.
    Stated in the abstract for both deterministic and random settings.
  • domain assumption The rho-mixing profile of the channel environment tends to zero or the channels are independent.
    Used to obtain annealed super-polynomial or exponential estimates.
invented entities (2)
  • centered trace-Dobrushin coefficient no independent evidence
    purpose: Quantifies residual dependence on input state for channel products
    Defined in the paper as the key quantity whose decay implies forgetting.
  • trace-Dobrushin Lyapunov exponent no independent evidence
    purpose: Characterizes almost-sure decay rate for random products
    Introduced for stationary random CPTP cocycles.

pith-pipeline@v0.9.0 · 5538 in / 1385 out tokens · 39008 ms · 2026-05-09T20:16:43.526935+00:00 · methodology

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