Prediction and Characterization of Multiple Extremal Paths in Continuously Monitored Qubits

TL;DR

This paper investigates the most-likely paths in continuously monitored qubits, revealing the existence of multiple extremal paths, analyzing their formation via caustics, and applying the framework to specific quantum measurement scenarios.

Contribution

It introduces a Hamiltonian dynamical system approach to characterize multipaths and develops methods to determine optimal traversal times and states in monitored qubit systems.

Findings

Multiple most-likely paths can connect the same states in monitored qubits.

Multipaths are caused by caustics and winding numbers in phase space.

Methods for finding optimal measurement times and states are demonstrated.

Abstract

We examine most-likely paths between initial and final states for diffusive quantum trajectories in continuously monitored pure-state qubits, obtained as extrema of a stochastic path integral. We demonstrate the possibility of "multipaths" in the dynamics of continuously-monitored qubit systems, wherein multiple most-likely paths travel between the same pre- and post-selected states over the same time interval. Most-likely paths are expressed as solutions to a Hamiltonian dynamical system. The onset of multipaths may be determined by analyzing the evolution of a Lagrange manifold in this phase space, and is mathematically analogous to the formation of caustics in ray optics or semiclassical physics. Additionally, we develop methods for finding optimal traversal times between states, or optimal final states given an initial state and evolution time; both give insight into the measurement dynamics of continuously-monitored quantum states. We apply our methods in two systems: a qubit with two non-commuting observables measured simultaneously, and a qubit measured in one observable while subject to Rabi drive. In the two-observable case we find multipaths due to caustics, bounded by a diverging Van-Vleck determinant, and their onset time. We also find multipaths generated by paths with different "winding numbers" around the Bloch sphere in both systems.