Asymptotic spectral stability of the Gisin-Percival state diffusion

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 $t\ge 0$, 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$intermsof$d,$\mathbf{B}$$and$d,t$. This shows that each of the processes $\{\mathrm{T}\mathrm{r}[{\rho }_t^m(\mathbf{B})],t\ge 0\}$ admits a Doob-Meyer decomposition as the sum of a martingale $M_t^{(m)}(\mathbf{B})$ and a non-negative increasing process $S_t^{(m)}(\mathbf{B})$. This ensures the existence of $\underset{t\rightarrow\infty}{\lim}, {\rm Tr}[\rho_t^m( $\mathbf{B}$)]$almostsurelywithrespecttotheWienerprobabilitymeasure$\mu$oftheBrownianmotion$$\mathbf{B}$$,foreach$m=2,, 3,, \ldots$. 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 $t\to \mathrm{\infty }$. 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.