On a Class of Quantum Markov Chains on the Fock Space

Kalyanapuram Rangachari Parthasarathy

Hans Maassen

Franco Fagnola

Franco Fagnola

Volkmar Liebscher

We study special quantum Markov chains on a Fock space related to iterated beam splittings as introduced in [23]. Besides a characterization of the position distributions of the chain, we show some kind of weak convergence of such discrete time quantum Markov chains to a kind of continuous time quantum Markov process. Furthermore, we provide existence and uniqueness for the solution of a quantum stochastic differential equation related to this quantum Markov process both on an exponential domain and, on a larger domain, in a pointwise sense.

Yuan Feng, Ernst Moritz Hahn, Andrea Turrini +1 more

Quantum Markov chains are an extension of classical Markov chains which are labelled with super-operators rather than probabilities. They allow to faithfully represent quantum programs and quantum protocols. In this paper, we investigate model checking omega-regular properties, a very general class of properties (including, e.g., LTL properties) of interest, against this model. For classical Markov chains, such properties are usually checked by building the product of the model with a language automaton. Subsequent analysis is then performed on this product. When doing so, one takes into account its graph structure, and for instance performs different analyses per bottom strongly connected component (BSCC). Unfortunately, for quantum Markov chains such an approach does not work directly, because super-operators behave differently from probabilities. To overcome this problem, we transform the product quantum Markov chain into a single super-operator, which induces a decomposition of the state space (the tensor product of classical state space and the quantum one) into a family of BSCC subspaces. Interestingly, we show that this BSCC decomposition provides a solution to the issue of model checking omega-regular properties for quantum Markov chains.