TY - JOUR

T1 - Complete positivity for time-dependent qubit master equations

AU - Hall, Michael J.W.

PY - 2008/5/6

Y1 - 2008/5/6

N2 - 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 characterize 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.

AB - 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 characterize 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.

UR - http://www.scopus.com/inward/record.url?scp=44449179856&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/41/20/205302

DO - 10.1088/1751-8113/41/20/205302

M3 - Article

SN - 1751-8113

VL - 41

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 20

M1 - 205302

ER -