Finding the Kraus decomposition from a master equation and vice versa

TL;DR

This paper presents a comprehensive method to derive Kraus decompositions from master equations and vice versa, applicable to both Markovian and non-Markovian dynamics, with explicit procedures and examples for qubits.

Contribution

It introduces a general procedure to construct the linear map and Kraus representations from any local-in-time master equation, and provides conditions for the existence of a master equation from a given map.

Findings

Explicit method to derive Kraus decompositions from master equations.

Necessary and sufficient conditions for the existence of a master equation from a linear map.

Examples demonstrating the methods with qubits.

Abstract

For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is reviewed for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N^2 x N^2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalising a related N^2 x N^2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a `best possible' master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.