arXiv:0803.4383 · Journal of Mathematical Physics

Discrete approximation of quantum stochastic models

Luc Bouten, Ramon van Handel

Published Oct 1, 2008• 13 citations

Abstract

We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.

DOI: 10.1063/1.3001109

Related papers

Stochastic limits of Quantum repeated measurements

Antoine Jacquier, Kostas Kardaras, Adeline Viot

We investigate quantum systems perturbed by noise in the form of repeated interactions between the system and the environment. As the number of interactions (aka time steps) tends to infinity, we show, following the works by Pellegrini, that this system converges to the solution of a Volterra stochastic differential equation. This development sets interesting future research paths at the intersection of quantum algorithms, stochastic differential equations, weak convergence and large deviations.

math.PRCross-listedarXiv:2512.11462Dec 15, 20250 citations

Quantum Stochastic Differential Equations

Mou-Hsiung Chang

Quite a few researchers have contributed to the development of quantum stochastic differential equations driven by quantum noises, initiated by Hudson and Parthasarathy [HP84a]. A nonexhaustive list of contributors includes Fagnola and Wills [FW00], [FW03], Obata [Oba97], and Linsay and Skalski [LS10]. Based on the results presented in Chapter 3, this chapter derives and considers a general form of linear (left) and (right) Hudson-Parthasarathy quantum stochastic differential equations driven by quantum noises in symmetric Fock space with operator-valued matrices as coefficients. Specifically, it studies the existence and uniqueness of the strong and mild solution processes for both the left and right quantum stochastic differential equations and conditions under which the solution processes are unitary, isometry, co-isometry, and regular. These results make extensive use of the properties of stochastic integral driven by quantum noise and its quantum stochastic calculus in the context of a symmetric Fock space, which was explored in Chapter 3. Various discrete approximation schemes, including Bouten et al. [BvH07], of the left Hudson-Parthasarathy QSDE are also explored for numerical computation. Specifically, it is shown that the solution of the Hudson- Parthasarathy QSDE can be approximated by a sequence of discrete interaction models with decreasing time step. In order to study this problem, discrete interaction models are embedded in a limiting space. This allows one to present strong convergence of the embedded discrete cocycles to the solution of the Hudson-Parthasarathy QSDE. It will be pointed out that the way in which the embedding is done does not affect the proof of the main result for discrete approximations. Examples of a QSDE that is analytically solvable and a QSDE that possesses a mild solution but not a strong solution are also given. QSDE of Hudson-Parthasarathy Type It has been mentioned earlier in Chapter 3 that an open quantum systems treated in this monograph are to be formulated as a quantum system in which the underlined (or the initial) quantum system is perturbed by quantum noises that represent the environment.

Cambridge University Press eBooksFeb 9, 20150 citations

A Trotter-Kato Theorem for Quantum Markov Limits

Luc Bouten, Rolf Gohm, John Edward Gough +1 more

Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Ito evolution.

arXiv:1409.2260Sep 8, 20140 citations

A Trotter-Kato theorem for quantum Markov limits

Luc Bouten, Rolf Gohm, John Edward Gough +1 more

EPJ Quantum TechnologyApr 29, 20152 citations

The convergence of unitary quantum random walks

Alexander C. R. Belton, Michał Gnacik, J. Martin Lindsay

We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.

Apr 25, 20141 citation

A Trotter Product Formula for quantum stochastic flows

B. Krishna Das, Debashish Goswami, Kalyan B. Sinha

We give a new method for proving the homomorphic property of a quantum stochastic ow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic ows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of [5]

Jan 1, 20102 citations