Complete positivity for time-dependent qubit master equations

TL;DR

This paper analyzes conditions for complete positivity in time-dependent qubit master equations, linking them to oscillator equations and providing explicit criteria based on the decoherence matrix's structure.

Contribution

It introduces a novel approach connecting qubit master equations with oscillator equations, offering explicit conditions for complete positivity based on matrix block-diagonality.

Findings

Complete positivity conditions are characterized by oscillator Hamiltonian or Lagrangian.

Block-diagonal decoherence matrices simplify the positivity criteria.

A class of master equations where positivity reduces to positivity of the decoherence matrix.

Abstract

It is shown that if the decoherence matrix corresponding to a qubit master equation has a block-diagonal real part, then the evolution is determined by a one-dimensional oscillator equation. Further, when the full decoherence matrix is block-diagonal, then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator Hamiltonian or Lagrangian. When the solution of the oscillator equation is not known, an explicit sufficient condition for complete positivity can still be obtained, based on a Hamiltonian/Lagrangian inequality. A rotational form-invariance property is used to characterise the evolution via a single first-order nonlinear differential equation, enabling some further exact results to be obtained. A class of master equations is identified for which complete positivity reduces to the simpler condition of positivity.