Quantum trajectory framework for general time-local master equations

Abstract

Master equations are one of the main avenues to study open quantum systems. When the master equation is of the Lindblad–Gorini–Kossakowski–Sudarshan form, its solution can be “unraveled in quantum trajectories” i.e., represented as an average over the realizations of a Markov process in the Hilbert space of the system. Quantum trajectories of this type are both an element of quantum measurement theory as well as a numerical tool for systems in large Hilbert spaces. We prove that general time-local and trace-preserving master equations also admit an unraveling in terms of a Markov process in the Hilbert space of the system. The crucial ingredient is to weigh averages by a probability pseudo-measure which we call the “influence martingale”. The influence martingale satisfies a 1d stochastic differential equation enslaved to the ones governing the quantum trajectories. We thus extend the existing theory without increasing the computational complexity.

Unraveling-paired dynamical maps can recover the input of quantum channels

“…We now show that the proof of the unraveling holds true even under the more restrictive hypothesis than in [45] that the expectation value…”

“…A straightforward application of Itô lemma as done in [45] completes the proof. For readers' convenience we reproduce the steps of the calculation in appendix C. Some remarks are here in order.…”

“…If we now insert (12) into the drift (7b) and use (3) on the Bloch hyper-sphere we recover the form of the drift hypothesized in [45]…”

“…The time dependence in general cannot be removed by unitary transformations unless d = 2; iv the time dependent functions {w l;t } L l=1 in (1b) are uniquely determined by the canonical representation. They are referred to in [29] as canonical decoherence rates a terminology that we adopt here too at variance with [45] where we called them "weights"; v the canonical decoherence rates in (1b) are not sign definite. In the present work we also surmise that canonical rates are sufficiently regular bounded functions.…”

“…Very recently [45], we introduced a framework for unraveling (solutions of) the general time-local and sufficiently regular master equation. The conceptual point of the unraveling is to enforce the Wittstock-Paulsen canonical form of a completely bounded map [46,47].…”