Quasistationary quaternionic Hamiltonians and complex stochastic maps

TL;DR

This paper demonstrates that under a quasistationarity condition, quaternionic Hamiltonian dynamics project onto complex stochastic dynamics, linking quaternionic and complex quantum evolutions.

Contribution

It establishes a necessary and sufficient condition for quaternionic Hamiltonian dynamics to produce complex stochastic maps, specifically when the metric operator is time-independent and Hermitian.

Findings

Quaternionic Hamiltonian dynamics project to complex stochastic maps under quasistationarity.

The quasistationarity condition requires the metric operator to be Hermitian, positive, and time-independent.

An example illustrating the theory is provided.

Abstract

We show that the complex projections of time-dependent $\eta $-quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi-Hermitian density matrices if and only if a quasistationarity condition is fulfilled, i. e., if and only if $\eta $ is an Hermitian positive time-independent complex operator. An example is also discussed.