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

TL;DR

This paper introduces a trace-Dobrushin theory for quantum channel products, establishing criteria for memory loss and convergence in inhomogeneous matrix product states with applications to quantum information.

Contribution

It develops a new product-level trace-Dobrushin framework for quantum channels and applies it to analyze inhomogeneous matrix product states, providing convergence and stability results.

Findings

Trace-Dobrushin coefficient decay characterizes trace-norm forgetting.

Negativity of the Lyapunov exponent is equivalent to quenched trace-norm memory loss.

Channel estimates lead to infinite-volume limits and correlation bounds in 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.