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arXiv:2206.04508 · Open Systems & Information Dynamics

Local Generation of Entanglement with Redfield Dynamics

Fabio Benatti, Dariusz Chruściński, Roberto Floreanini B

Published Mar 1, 2022• 5 citations
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Abstract

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.

DOI
10.1142/s1230161222500019

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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.
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.
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.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment was given. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state was described. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state was analyzed.