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arXiv:quant-ph/0701157 · Journal of Physics A Mathematical and Theoretical

Redfield reduced dynamics and entanglement

Sebastiano Anderloni, Fabio Benatti, Roberto Floreanini B

Published Jan 30, 2007• 13 citations
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Abstract

In phenomenological applications, the time evolution of subsystems immersed in an external environment is sometimes described by Markovian semigroups of the Redfield type that have non-positive results: the appearance of negative probabilities is avoided by restricting the admissible initial conditions to those states that remain positive under the action of the dynamics. We show that this often adopted procedure may lead to physical inconsistencies in the presence of entanglement.

DOI
10.1088/1751-8113/40/7/013

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In phenomenological applications, the time evolution of subsystems immersed in an external environment is sometimes described by Markovian semigroups of the Redfield type that have non-positive results: the appearance of negative probabilities is avoided by restricting the admissible initial conditions to those states that remain positive under the action of the dynamics. We show that this often adopted procedure may lead to physical inconsistencies in the presence of entanglement.
Abstract. In phenomenological applications the time evolution of subsystems immersed in an external environment are sometimes described by Markovian semigroups of Redfield type that result non-positive: the appearence of negative probabilities is avoided by restricting the admissible initial conditions to those states that remain positive under the action of the dynamics. We show that this often adopted procedure may lead to physical inconsistencies in presence of entanglement. Redfield Reduced Dynamics and Entanglement 2 1.
In phenomenological applications, time evolutions of Bloch-Redfield type are widely adopted for modelling open system dynamics, despite their nonpositive preserving character: this physical inconsistency, that in general shows up at small times, is usually cured by suitably restricting the space of allowed initial states. Nevertheless, additional problems may arise in relation to entanglement: specifically, we show that Redfield dynamics, though being a semigroup, can generate entanglement through a purely local action; moreover, this unphysical effect can persist on long time-scales.
Non-positive, Markovian semigroups are sometimes used to describe the time evolution of subsystems immersed in an external environment. A widely adopted prescription to avoid the appearance of negative probabilities is to eliminate from the admissible initial conditions those density matrices that would not remain positive by the action of the semigroup dynamics. Using a continuous variable model, we show that this procedure leads to physical inconsistencies when two subsystems are considered and their initial state is entangled.
A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.
The long-time evolution of a system in interaction with an external environment is usually described by a family of linear maps ${\ensuremath{\gamma}}_{t},$ generated by master equations of Block-Redfield type. These maps are, in general, nonpositive; a widely adopted cure for this physical inconsistency is to restrict the domain of definition of the dynamical maps to those states for which ${\ensuremath{\gamma}}_{t}$ remains positive. We show that this prescription has to be modified when two systems are immersed in the same environment and evolve with the factorized dynamics ${\ensuremath{\gamma}}_{t}\ensuremath{\bigotimes}{\ensuremath{\gamma}}_{t}$ starting from an entangled initial state.
Nonpositive Markov approximations are sometimes used to describe the dynamics of qubits in weak interaction with suitable environments; the appearance of negative probabilities is avoided by assuming that the transient regime eliminates from the possible initial conditions those qubit states which would otherwise be mapped out of the Bloch sphere by the subsequent Markovian time evolution. By means of a simple model, we discuss some physical inconsistencies of this approach in relation to entanglement; in particular, we show that slipped nonpositive reduced dynamics might create entanglement through a purely local action.