Quantum random walks with general particle states
The paper under review is concerned with the convergence of quantum random walks to quantum stochastic (QS) cocycles. The general setup is as follows: given a family of maps \(\Psi_{\tau}:V \to V \otimes_M B(K)\) indexed by \(\tau >0\) we investigate the discrete evolution given by repeated applications of a given \(\Psi_{\tau}:V \to V \otimes_M B(K)\), representing a single step of a quantum random walk, embedded into a Fock space, as \(\tau\) tends to \(0\). Here \(V\) is an initial space (taken to be an operator space), \(\otimes_M\) is the matrix tensor product and \(K\) is a Hilbert space representing the particle space. As we are interested in the convergence in some fixed particle state on the particle space, say \(\rho \in B(K)_*\), we in fact consider the modification of \(\Psi_{\tau}\) related to the GNS representation of \(B(K)\) with respect to \(\rho\). This means that the corresponding embedding lands in the symmetric Fock space over \(L^2(\mathbb{R}_+;k)\), where \(k\) is a Hilbert space related to the GNS representation space. In the earlier work of the author and Attal, Gough, Joye and Pautrat convergence of such quantum random walks to certain quantum stochastic cocycles on the Fock space was studied in some specific cases, i.e.\ for \(\rho\) being a faithful state or a vector state. The novelty here lies in that arbitrary states on the particle algebra \(B(K)\) are allowed. This influences the scaling required from maps \(\Phi_{\tau}\) to obtain the desired convergence; different `parts' of the maps \(\Phi_{\tau}\), depending on the decomposition of \(K\) into the kernel of the density matrix corresponding to \(\rho\) and its orthogonal complement need to converge after dividing after different normalising factors depending on \(\tau\). The author shows that under correct scaling and the assumption that the `limit' map \(\Phi:V \to V \otimes_M B(k \oplus \mathbb{C})\) is completely bounded (or just bounded if \(K\) is finite dimensional) one obtains a convergence of the quantum random walks induced by \(\Psi_{\tau}\) to the QS cocycle induced by \(\Phi\). Several concrete examples are presented, in particular such in which the limit cocycle is implemented by unitary QS cocycles of Hudson and Parthasarathy. The author also points out the connection between the thermalisation phenomenon -- naively speaking meaning that the number of independent dimensions of noise featuring in the QS differential equation satisfied by the limit cocycle is smaller than expected -- and the fact that the considered state \(\rho\) is not a vector state.
scientific article; zbMATH DE number 6305255
scientific article