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Journal of Physics A Mathematical and Theoretical

Complete positivity for time-dependent qubit master equations

Michael J. W. Hall

Published Apr 24, 2008• 34 citations
Publisher

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 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.

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

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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.
We study a non-Markovian spin-1∕21∕21∕2 master equation with exponential memory. We derive the conditions under which the dynamical map describing the reduced system dynamics is completely positive, i.e., the nonunitary evolution of the system is compatible with a description in terms of a closed total spin-reservoir system. Our results show that for a zero-TTT reservoir, the dynamical map of the model here considered is never completely positive. For moderate- and high-TTT reservoirs, on the contrary, positivity is a necessary and sufficient condition for complete positivity. We also consider the Shabani-Lidar master equation recently introduced [A. Shabani and D.A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and we demonstrate that such a master equation is always completely positive.
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given 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×N 2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalizing a related N 2×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.
A post-Markovian quantum master equation is derived, which includes bath memory effects via a phenomenologically introduced memory kernel k(t)k(t)k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t)k(t)k(t) is presented, in addition to a prescription for the experimental determination of k(t)k(t)k(t). The formalism is illustrated with examples.
Master equations are typically adopted to describe the dynamics of open quantum systems. Such equations are either in integro-differential or in time-local form, with the latter class more frequently adopted due to the simpler numerical methods developed to obtain the corresponding solution. Here we show that any time-local master equation with positive rates in the generator, i.e., any completely positive divisible quantum process, admits a microscopic model whose reduced dynamics is well described by the given equation.
Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalization procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)] is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0t>0t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.
The main aim of this work is to provide a criterion for conservativity of minimal solutions of time-dependent master equations with unbounded coefficients (Quantum Dynamical Evolution System, QDES). In Section 2 we prove a criterion for conservativity of a QDES without using resolvents and in Section 3 we give sufficient conditions for conservativity of the class of QDES whose completely positive part is a time-dependent bounded sesquilinear form on a Hilbert space K t densely and continuously embedded in H and satisfies an extension property.