A limit theorem for quantum Markov chains associated to beam splittings (Q2765187)

A beam splitting as introduced in [\textit{W. Freudenberg}, On a class of quantum Markov chains on the Fock space. In: Accardi, L. (ed.), Quantum Probability \& Related Topics IX, World Scientific (1993)] is the isometry NEWLINE\[NEWLINEV_{\alpha\beta}: \Gamma(L^2(G))\rightarrow\Gamma(L^2(G))\otimes\Gamma(L^2(G)),\;\psi(x)\mapsto\psi(\alpha x)\otimes\psi(\beta x), NEWLINE\]NEWLINE (\(G\) some Polish space) where \(\alpha,\beta\in L^\infty(G)\) are functions (so-called splitting rates) such that \(|\alpha|^2+|\beta|^2=1\) and \(\psi(x)\) denotes the exponential vector to \(x\in L^2(G)\). Then \(V_{\alpha\beta}^*\bullet V_{\alpha\beta}\) defines a transition expectation \({\mathcal E}:{\mathcal B}(\Gamma(L^2(G)))\otimes{\mathcal B}(\Gamma(L^2(G)))\rightarrow{\mathcal B}(\Gamma(L^2(G)))\) (i.e. a unital CP-map) and every normal state \(\eta\) on \({\mathcal B}(\Gamma(L^2(G)))\) gives rise to a unique locally normal state \(\omega\) on the infinite \(C^*\)-tensor product \({\mathcal B}(\Gamma(L^2(G)))^{\otimes\infty}\) fulfilling \(\omega(a\otimes{\mathbf 1}\otimes\ldots)=\eta(a)\) and NEWLINE\[NEWLINE\omega(a_1\otimes\ldots\otimes a_n\otimes a_{n+1}\otimes{\mathbf 1}\otimes\ldots) =\omega(a_1\otimes\ldots\otimes{\mathcal E}(a_n\otimes{\mathcal E}(a_{n+1}\otimes{\mathbf 1}))\otimes{\mathbf 1}\otimes\ldots), NEWLINE\]NEWLINE i.e. to a quantum Markov chain in the sense of [\textit{L. Accardi}, Noncommutative Markov chains. Int. School of Mathematical Physics, Camerino, 268-295 (1974)]. The authors shows that \(\omega\) is, in fact, normal (in a natural representation space). He shows that the continuous time limit of the chain is a quantum Markov process and, furthermore, that this process is related to the solution of a quantum stochastic differential equation.