  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.
