Symmetric form of the Hudson-Parthasarathy stochastic equation

Wilhelm von Waldenfels

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.

Wilhelm von Waldenfels

The solution of the quantum stochastic differential equation of Hudson and Partasarathy with constant coefficients can easily be formulated with the help of the normal ordered product. Introducing time ordering we obtain a new representation of the solution. This new representation gives an easy access to the Evans-Hudson flow and the unitary condition of Hudson and Parthasarathy. We use the measure theoretical white noise calculus.

Yuri E. Gliklikh, Mikhail Yu. Ryazantsev

Lingaraj Sahu, Preetinder Singh

We consider the GNS Hilbert space H of a uniformly hyper-finite C * -algebra and study a class of unbounded Lindbladian arises from commutators.Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution.The vacuum expectation of homomorphic cocycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian.We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.

Gerard McCaul, Christian D. Lorenz, Lev N Kantorovich