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arXiv:1707.08157

Asymptotic spectral stability of the Gisin-Percival state diffusion

Kalyanapuram Rangachari Parthasarathy, A. R. Usha Devi

Published Jul 25, 2017• 0 citations
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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 $ρ_t(\mathbf{B})$ for each $t\geq 0$, where $\mathbf{B}$ is the underlying complex $n$-dimensional vector-valued Brownian motion process and using It{ô} calculus, along with an induction procedure, we arrive at the stochastic differential of the scalar-valued moment process ${\rm Tr}[ρ_t^m( \mathbf{B})], \,\,\, m=2,3,\ldots$ in terms of $d\,\mathbf{B}$ and $d\,t$. This shows that each of the processes $\{{\rm Tr}[ρ_t^m( \mathbf{B})], t\geq 0\}$ 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 $\underset{t\rightarrow\infty}{\lim}\, {\rm Tr}[ρ_t^m( \mathbf{B})]$ almost surely with respect to the Wiener probability measure $μ$ 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 $ρ_t (\mathbf{B})$ 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.

DOI
10.48550/arxiv.1707.08157

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Quantum state diffusion unravels a Lindblad master equation into stochastic trajectories of pure states. The ensemble of trajectories is described by a probability density on the manifold of pure states, which contains much more information than the density matrix, its first moment. We derive the time reversal of this diffusion. Writing the reversal of a general diffusion on a manifold in Stratonovich form, and passing through the manifold of pure states, we obtain a backward stochastic differential equation for the state vector, valid for any Hilbert space dimension and any Lindbladian. The backward stochastic equation provides a physical, nonlinear unravelling of the inverse Lindbladian that holds only for the ensemble from which it was constructed. Following score-based generative models, we then use the backward equation to generate a new ensemble close to the original. For the depolarizing channel the unravelling is a Brownian motion on complex projective space, so the uniform distribution of pure states is its stationary law in any dimension. Starting the backward equation from that prior, we bound the error of the resulting distribution compared to the original one in different settings: exact and learned scores, continuous and discrete time.
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"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schrödinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories.