Stochastic processes with values in a Hilbert space and information channels

Shan Gao

Forrest G. Hall

O. Nicrosini, A. Rimini

G. C. Ghirardi, O. Nicrosini, A. Rimini

The relationship between discontinuous and continuous stochastic processes in Hilbert space is investigated. It is shown that for any continuos process there is a parent discontinuous process, that becomes the continuous one in the proper infinite frequency limit. From the point of view of solving the quantum measurement problem, what really matters is the choice of the set of operators whose value distributions are made sharp. In particular, the key role of position sharping is emphasized.

Jason Doukas

The stochastic--quantum correspondence reinterprets quantum dynamics in terms of an underlying stochastic process on a configuration space. We generalize its lifting prescription, overcoming restrictions on the reachable density matrices without enlarging the configuration space or its associated Hilbert space. A compatibility relation and Kraus dictionary connect stochastic kernels to completely positive, trace-preserving (CPTP) maps: every finite-dimensional kernel admits a compatible lift, and these lifts collectively encompass all quantum channels as the kernels vary. We describe sufficiently regular composable CPTP dynamics through time-local Lindblad generators. We establish that every rooted quantum dynamical family admits a stochastic realizer on the same configuration space, and that a revival of configuration-space distinguishability rules out every Markovian realizer on that space. We show how the predictive information carried by off-diagonal degrees of freedom in a compatible composable lift is encoded in the continuation probabilities of a family of context-dependent stochastic processes. We also analyze physical measurement by separating active basis rotations from configuration recording. The rooted quantum probabilities alone do not enforce faithfulness in idealized recording interactions; nevertheless, we show that a faithful stochastic realizer always exists.

Toshio Fukumi

The power of quantum communication channel arises from additively of Holevo information, which certainly depends on the fact that the Hilbert space grows exponentially. The additively will be be shown to be achieved by stochastically maximizing a Hilbert space. In the conjecture we introduce operator statistics which is a natural extension of functions of operator, and employing it we discuss that the information is additive in probability.