Deterministic submanifolds and analytic solution of the stochastic differential master equation describing a qubit

TL;DR

This paper characterizes the stochastic evolution of a qubit under certain measurements, showing that its state remains confined to deterministic surfaces or curves and providing explicit solutions and probability distributions.

Contribution

It explicitly solves the stochastic differential equation for a qubit under specific measurement regimes and links the existence of deterministic submanifolds to control theory accessibility criteria.

Findings

Qubit states remain on deterministic surfaces or curves under certain measurements.

Explicit solutions and probability distributions are derived for these submanifolds.

Weak measurements of the specified types are the only cases with such deterministic confinement.

Abstract

This paper studies the stochastic differential equation (SDE) associated to a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.