Asymptotic spectral stability of the Gisin-Percival state diffusion

TL;DR

This paper investigates the long-term behavior of the spectral properties of quantum states evolving under Gisin-Percival state diffusion, demonstrating almost sure convergence of the spectrum and entropy for finite and infinite level systems.

Contribution

It provides a rigorous proof of the asymptotic spectral stability of the diffused quantum state using stochastic calculus and probabilistic methods.

Findings

Spectral and entropy convergence for finite systems.

Almost sure spectrum convergence for infinite systems.

Decomposition of moment processes into martingale and increasing parts.

Abstract

Starting from the Gisin-Percival state diffusion equation for the pure state trajectory of a composite bipartite quantum system and exploiting the purification of a mixed state via its Schmidt decomposition, we write the diffusion equation for the quantum trajectory of the mixed state of a subsystem $S$ of the bipartite system, when the initial state in $S$ is mixed. Denoting the diffused state of the system $S$ at time $t$ by $\rho_t(\mathbf{B})$ for each $t\geq 0$, where $\mathbf{B}$ is the underlying complex $n$-dimensional vector-valued Brownian motion process and using It{\^o} calculus, along with an induction procedure, we arrive at the stochastic differential of the scalar-valued moment process ${\rm Tr}[\rho_t^m( \mathbf{B})], \,\,\, m=2,3,\ldots$ in terms of $d\,\mathbf{B}$ and $d\,t$. This shows that each of the processes $\{{\rm Tr}[\rho_t^m( \mathbf{B})], t\geq 0\}$ admits a Doob-Meyer decomposition as the sum of a martingale $M^{(m)}_t(\mathbf{B})$ and a non-negative increasing process $S^{(m)}_t(\mathbf{B})$. This ensures the existence of $\underset{t\rightarrow\infty}{\lim}\, {\rm Tr}[\rho_t^m( \mathbf{B})]$ almost surely with respect to the Wiener probability measure $\mu$ of the Brownian motion $\mathbf{B}$, for each $m=2,\, 3,\, \ldots$. In particular, when $S$ is a finite level system, the spectrum and therefore the entropy of $\rho_t (\mathbf{B})$ converge almost surely to a limit as $t\rightarrow \infty$. In the Appendix, by employing probabilistic means, we prove a technical result which implies the almost sure convergence of the spectrum for countably infinite level systems.