Quantum filter for a non-Markovian single qubit system

TL;DR

This paper develops a quantum filtering method for non-Markovian qubit systems by embedding them into an augmented Markovian framework with ancillary systems, enabling estimation under arbitrary colored noise.

Contribution

It introduces a quantum filter for non-Markovian qubits using ancillary systems to model colored noise, extending filtering techniques to arbitrary noise spectra.

Findings

Successfully models non-Markovian dynamics with Lorentzian noise

Constructs a quantum filter for arbitrary colored noise

Demonstrates the filter with an illustrative Lorentzian noise example

Abstract

In this paper, a quantum filter for estimating the states of a non-Markovian qubit system is presented in an augmented Markovian system framework including both the qubit system of interest and multi-ancillary systems for representing the internal modes of the non-Markovian environment. The colored noise generated by the multi-ancillary systems disturbs the qubit system via a direct interaction. The resulting non-Markovian dynamics of the qubit is determined by a memory kernel function arising from the dynamics of the ancillary system. In principle, colored noise with arbitrary power spectrum can be generated by a combination of Lorentzian noises. Hence, the quantum filter can be constructed for the qubit disturbed by arbitrary colored noise and the conditional state of the qubit system can be obtained by tracing out the multi-ancillary systems. An illustrative example is given to show the non-Markovian dynamics of the qubit system with Lorentzian noise.