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

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.

• The paper establishes a sharp equivalence between trace-norm memory loss and channel replacement using the trace-Dobrushin coefficient.
• It employs block-level contraction analysis to demonstrate exponential convergence in deterministic and random quantum channel products.
• Applications to inhomogeneous MPS reveal unique infinite-volume limits and explicit clustering bounds for complex quantum networks.

Asymptotic Replacement for Quantum Channel Products and Inhomogeneous MPS: A Technical Analysis

Introduction

This paper rigorously develops a trace-Dobrushin theory for finite-dimensional quantum channel products, focusing on quantitative memory loss and asymptotic replacement phenomena. The theory is subsequently applied to deterministic and stationary random inhomogeneous matrix product states (MPS), in left-canonical CPTP gauge. The methodology is grounded in product-level contraction coefficients, eschewing reliance on one-step criteria or projective metrics, thus enabling sharper characterization of trace-norm memory loss in inhomogeneous and random channel ensembles.

Core Trace-Dobrushin Framework

The primary object is the centered trace-Dobrushin coefficient $\kappa_{\rm tr}(\Phi)$ for a CPTP map $\Phi$ on $M_d$, defined as the sharp trace-norm contraction of self-adjoint trace-zero inputs. Its product form $\kappa_{\rm tr}(\Phi_{t:s})$ quantifies residual state memory after sequential application of non-identical channels. Crucially, the decay of this coefficient serves as the intrinsic criterion for trace-norm forgetting—uniform convergence of evolved states toward replacement channels—in both deterministic and random settings.

Notably, the coefficient is utilized at the block product level, not just for single-channel contraction, distinguishing the approach from classical Dobrushin or Doeblin estimates. The authors establish that memory loss and map convergence toward replacement channels are quantitatively equivalent, up to universal constants, to decay of the product-level trace-Dobrushin coefficient.

Deterministic Inhomogeneous Channel Products

For arbitrary sequences $(\Phi_n)$ of CPTP maps, the paper proves that vanishing of the product-level coefficient $\kappa_{t:s}$ is necessary and sufficient for strong channel-level asymptotic replacement. The equivalence is rigorously quantified: for every reference state $\tau$, the induced $1 \rightarrow 1$ norm between $\Phi_{t:s}$ and the reference replacement channel $R_{t:s}^{(\tau)}$ converges to zero as $\Phi$0, with a bound

$\Phi$1

Furthermore, in the two-sided setting, strong convergence from the remote past (pullback forgetting) produces a unique boundary state sequence $\Phi$2—a canonical replacement family determined endogenously by the channel sequence.

Block-level contraction phenomena, invisible to one-step criteria, are addressed via detailed submultiplicativity arguments, enabling rigorous detection of memory loss in systems lacking Markov-Doeblin minorization or strict positivity at each step.

Random Quantum Channel Cocycles

For stationary CPTP cocycles over an ergodic base $\Phi$3, the submultiplicativity of $\Phi$4 yields a trace-Dobrushin Lyapunov exponent $\Phi$5 via Kingman's theorem. The authors demonstrate that almost sure negativity of $\Phi$6 is equivalent to exponential trace-norm memory loss (quenched forgetting) and confirms convergence to a unique dynamically stationary random replacement channel.

The analysis provides exponential convergence bounds, with rates controlled by the Lyapunov exponent. Importantly, this regime subsumes—and extends—eventual strict positivity and Doeblin minorization: even channels lacking strict positivity or one-step minorization exhibit replacement mixing when block-level coefficients contract sufficiently. The random cocycle framework is strengthened further under decorrelation assumptions (e.g., $\Phi$7-mixing or independence of channel environments), yielding super-polynomial or exponential annealed convergence.

Applications to Inhomogeneous Matrix Product States

The paper translates the channel-level estimates directly to MPS, treating deterministic and stationary random inhomogeneous MPS in left-canonical CPTP gauge. The auxiliary channel products arising from MPS transfer maps inherit the replacement properties and correlation bounds proven for quantum channel products.

The results yield:

• Existence and uniqueness of infinite-volume limits of trace-closed finite-volume MPS states under decay of right-tail auxiliary trace-Dobrushin coefficients.
• Explicit quantitative convergence rates for local observable expectations: for fixed $\Phi$8 and $\Phi$9, the error in expectation for volume $M_d$0 decays as $M_d$1.
• Rigorous bounds on spatial clustering: correlations across separated regions are governed by auxiliary product coefficients, providing explicit exponential or super-polynomial decay in gap size.

The same mechanisms extend to random MPS with CPTP cocycles, with the random boundary state construction yielding quenched and annealed bounds for thermodynamic limits and correlation decay. The approach is robust to lack of block-level strict positivity—a marked departure from the assumptions underlying prior random MPS literature.

Numerical and Theoretical Strengths

• Sharp equivalence: The main theorems establish quantitative equivalence (up to tight constants) between trace-norm memory loss and channel-level replacement, for both deterministic and random products.
• General inhomogeneous setting: The results are valid for non-translation-invariant, site- and time-dependent channel ensembles, substantially broadening applicability compared to homogeneous ergodic scenarios.
• Block-level contractions: The analysis detects contraction mechanisms missed by classical minorization or strict positivity, via block counting arguments and submultiplicativity.
• Exponential convergence: Under almost sure negative Lyapunov exponent, forward and pullback products converge exponentially to dynamically stationary replacement channels and boundary states.
• Implications for MPS: Infinite-volume limits and clustering bounds for inhomogeneous and random MPS are governed by product-level coefficients, not by individual transfer map properties, enabling strong stability claims even when standard criteria fail.

Implications and Future Directions

Practically, these results provide rigorous, computable bounds for the stability and correlation length analysis of quantum Markov chains, quantum memories, and tensor network states (including inhomogeneous and random MPS). The framework is not restricted to translation-invariant or strictly positive environments, supporting analysis in driven, disordered, or adversarial settings.

Theoretically, the work lays foundational tools for sharp quantitative analysis of memory loss and boundary effects in noncommutative Markov processes, ergodic quantum dynamics, and tensor network models. Future research may extend this trace-Dobrushin methodology to infinite-dimensional or continuous-time settings, interacting many-body systems with spatial structure, or design quantum expander channels optimized for auxiliary contraction.

Conclusion

This paper provides a rigorous product-level trace-Dobrushin theory for quantum channels, delivering strong necessary and sufficient criteria for trace-norm memory loss and asymptotic replacement. These results are transferred quantitatively to deterministic and random inhomogeneous MPS, yielding explicit infinite-volume limits, boundary stability, and correlation bounds—all governed by auxiliary product contraction coefficients. The techniques elucidate mechanisms of quantum memory loss and tensor state stability beyond classical criteria, and suggest robust methodologies for analysis of complex quantum networks and stochastic channels.