Completely positive trace-preserving maps for higher-order unraveling of Lindblad master equations

TL;DR

This paper develops a systematic method to assess and improve the accuracy of quantum trajectory maps derived from Lindblad master equations, introducing a second-order Kraus operator that minimizes errors in simulating quantum measurements.

Contribution

It proposes a new Kraus operator that satisfies the conditions for valid quantum evolution and matches Lindblad solutions to second order in time step, surpassing existing methods.

Findings

The new operator satisfies complete positivity, trace preservation, and convex linearity.

It matches Lindblad solutions up to second order in time increment.

It yields the smallest average trace distance to exact quantum trajectories in tested examples.

Abstract

Theoretical tools used in processing continuous measurement records from real experiments to obtain quantum trajectories can easily lead to numerical errors due to a non-infinitesimal time resolution. In this work, we propose a systematic assessment of the accuracy of a map. We perform error analyses for diffusive quantum trajectories, based on single-time-step Kraus operators proposed in the literature, and find the orders in time increment, $\Delta t$, to which such operators satisfy the conditions for valid average quantum evolution (completely positive, convex-linear, and trace-preserving), and the orders to which they match the Lindblad solutions. Given these error analyses, we propose a Kraus operator that satisfies the valid average quantum evolution conditions and agrees with the Lindblad master equation, to second order in $\Delta t$, thus surpassing all other existing approaches. In order to test how well our proposed operator reproduces exact quantum trajectories, we analyze two examples of qubit measurement, where exact maps can be derived: a qubit subjected to a dispersive ($z$-basis) measurement and a fluorescence (dissipative) measurement. We show analytically that our proposed operator gives the smallest average trace distance to the exact quantum trajectories, compared to existing approaches.