Quantum Stochastic Calculus and Quantum Gaussian Processes

TL;DR

This paper explores quantum stochastic calculus on boson Fock space, focusing on Gaussian processes, semigroups of quasifree maps, and their applications to quantum Gaussian Markov processes.

Contribution

It introduces and analyzes semigroups of quasifree completely positive maps, their generators, and the dilation to quantum Gaussian Markov processes, extending previous work in quantum information theory.

Findings

Semigroups of quasifree CP maps are not strongly continuous but preserve Gaussian states.

The generators of these semigroups are of Lindblad type.

An exact noisy Schrödinger equation for dilating these semigroups to quantum Gaussian Markov processes is derived.

Abstract

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space $\Gamma (\mathbb{C}^n)$ over $\mathbb{C}^n.$ These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn. They were recently investigated in the context of quantum information theory by Heinosaari, Holevo and Wolf. Here we present the exact noisy Schr\"odinger equation which dilates such a semigroup to a quantum Gaussian Markov process.