[Paper Review] Existence, uniqueness and approximation for stochastic Schrodinger equation: the Poisson case

Clément Pellegrini|ArXiv.org|Sep 24, 2007

Quantum Mechanics and Applications21 references3 citations

TL;DR

This paper establishes the existence, uniqueness, and weak convergence of solutions to a stochastic Schrödinger equation driven by a Poisson process, using random measure theory and a discrete quantum repeated interactions model. It proves that the continuous jump-type quantum trajectory emerges as the limit of a concrete discrete-time physical model, rigorously justifying the Belavkin jump equation via a Random Coupling Limit method.

ABSTRACT

In quantum physics, recent investigations deal with the so-called "quantum trajectory" theory. Heuristic rules are usually used to give rise to "stochastic Schrodinger equations" which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model.

Motivation & Objective

• To provide a rigorous mathematical foundation for stochastic Schrödinger equations with jump-type noise, particularly the Belavkin jump equation.
• To resolve ambiguities in the definition of solution for jump-type equations where the driving process depends on the solution path.
• To justify the continuous jump model as the limit of a physically realizable discrete-time quantum repeated measurement process.
• To establish weak solution existence and uniqueness using random Poisson measure theory and pathwise analysis.
• To prove convergence of discrete quantum trajectories to the continuous quantum trajectory in distribution on D([0,T]).

Proposed method

• Uses random Poisson measure theory to rigorously define the stochastic integral in the jump equation, overcoming issues with state-dependent intensity.
• Applies a Random Coupling Limit method to compare discrete and continuous quantum trajectories, enabling convergence analysis.
• Employs a renormalized interaction model in a quantum repeated interactions framework, where each system interacts with a chain of ancillas for time h.
• Develops a discrete Euler scheme for the jump equation and proves its convergence using a novel discrete Gronwall-type argument.
• Uses conditional expectation and moment bounds to control the difference between discrete and continuous processes.
• Establishes convergence in distribution in the Skorokhod space D([0,T]) by combining bounds on the Euler scheme and convergence of the limiting process.

Experimental results

Research questions

• RQ1 Can a rigorous probabilistic framework be constructed for stochastic Schrödinger equations with jump-type noise, given the state-dependent intensity of the driving process?
• RQ2 Does a weak solution exist and is it unique for the Poisson-driven Belavkin jump equation?
• RQ3 Can the continuous jump model be derived as the limit of a concrete discrete physical model of repeated quantum measurements?
• RQ4 What mathematical techniques are required to prove convergence of discrete quantum trajectories to the continuous solution when classical stochastic integration theory fails?
• RQ5 How can the convergence be established in the absence of Kurtz-Protter-type convergence results for jump processes?

Key findings

• The existence and uniqueness of a weak solution to the Poisson-driven stochastic Schrödinger equation are rigorously established using random Poisson measure theory.
• The discrete quantum trajectory model, based on repeated interactions with a spin chain and discrete measurements, converges in distribution to the continuous quantum trajectory in D([0,T]).
• A discrete Euler scheme for the jump equation is proven to converge, with the error bounded by O(1/n), using a novel discrete Gronwall-type inequality.
• The convergence result relies on a Random Coupling Limit method, which allows comparison between discrete and continuous dynamics despite the non-Markovian nature of the intensity.
• The solution to the jump equation is shown to be physically justified as the limit of a concrete physical model, resolving long-standing heuristic derivations.
• The convergence is uniform in time over any finite interval [0,T], with the error decaying as O(1/n) for large n.