A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement
Owen Ekblad
TL;DR
This work establishes a multiplicative ergodic theorem for bistochastic ergodic quantum processes, yielding an invariant splitting governed by the stabilized multiplicative domain $\mathcal{A}_\Phi$ and its complement tied to the second Lyapunov exponent $\lambda_2$. The authors show that $\mathcal{A}_\Phi$ is invariant under the random channels and is generated by unimodular eigenmatrices of a finite-time product $\Phi^{(\tau)}$, while the orthogonal complement aligns with the Lyapunov subspace $V^{\le\lambda_2}$. They apply this structure to entanglement dynamics, proving a sharp 0–1 law: the event that the process is asymptotically entanglement breaking (a.e.b.) occurs with probability 0 or 1, and it equals 1 precisely when $\mathcal{A}_\Phi$ is abelian; this yields that occasionally PPT maps drive a.e.b. Moreover, if $\mathcal{A}_\Phi$ is trivial ($\mathbb{C}I$), the process is eventually entanglement breaking, with a quantitative exponential convergence toward the replacement channel. The results connect noncommutative MET to entanglement theory under disorder, offering a framework to analyze when random quantum dynamics erase entanglement and how PPT occurrences influence long-time behavior; extensions to unital-only dynamics and a random Kuperberg-type limit are discussed as promising directions.
Abstract
We prove a multiplicative ergodic theorem for bistochastic completely positive (bcp) linear cocycles acting on finite-dimensional matrix algebras, giving an invariant splitting described explicitly in terms of the multiplicative domains of the underlying bcp maps. As an application of our theorem, we classify when compositions of random bcp maps are asymptotically entanglement breaking, and use this classification to show that occasionally PPT bcp maps are asymptotically entanglement breaking. We conclude by demonstrating a certain class of bcp linear cocycles are almost surely entanglement breaking in finite time.