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arXiv:quant-ph/0303027 · Physical Review A

Nonpositive evolutions in open system dynamics

Fabio Benatti, Roberto Floreanini B, Marco Piani

Published Apr 22, 2003• 23 citations
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Abstract

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.

DOI
10.1103/physreva.67.042110

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The long-time evolution of a system in interaction with an external environment is usually described by a family of linear maps \ensuremathγt,{\ensuremath{\gamma}}_{t},\ensuremathγt​, 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 \ensuremathγt{\ensuremath{\gamma}}_{t}\ensuremathγt​ 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 \ensuremathγt\ensuremath⨂\ensuremathγt{\ensuremath{\gamma}}_{t}\ensuremath{\bigotimes}{\ensuremath{\gamma}}_{t}\ensuremathγt​\ensuremath⨂\ensuremathγt​ starting from an entangled initial state.
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.
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.
We study the role of correlations with the environment as the source of non-Markovian quantum evolutions. We first focus on the impact that correlations with the environment can have on the dynamical map that evolve the system. We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We discuss the implications and limitations of the Markov approximation necessary to derive the Kossakowski-Lindblad master equation. A generalized non-Markovian master equation is derived from the dynamical map of systems correlated with their environment. The physical meaning of not completely positive maps is studied to obtain a consistent theory of non-Markovian quantum dynamics. These are associated to inverse maps necessary to establish correlations and they give rise to a canonical embedding map that is local in time. This master equation goes beyond the Kossakowski-Lindblad master equation. Non-equilibrium quantum thermodynamics can be be studied within this theory. Through out this discussion, the general dynamics of two interacting qubits is used as an example for illustrations.
In the simplest approximation, the reduced dynamics of a quantum system SSS interacting with its environment EEE is considered to be given by a completely positive map. But, in general, this is not the case. In fact, the reduced dynamics of the system in not even linear, in general. Whether the reduced dynamics is linear or not is determined by two factors: the set of possible initial states of the system-environment $\mathcal{S}=\left\lbrace ρ_{SE} \right\rbrace $, and the joint system-environment unitary evolution UUU. When UUU is factorisable as U=US⊗UEU=U_S\otimes U_EU=US​⊗UE​, then we can choose S=D\mathcal{S}=\mathcal{D}S=D, where D\mathcal{D}D is the set of all system-environment density operators. In other words, when UUU is factorisable, the reduced dynamics of the system SSS is linear (in fact unitary) for arbitrary initial state of the system-environment ρSEρ_{SE}ρSE​. We show that this result cannot be generalized to any non-factorisable UUU: For any non-factorisable unitary evolution of the whole system-environment UUU, the set S\mathcal{S}S must be chosen as a proper subset of D\mathcal{D}D to achieve linear reduced dynamics. As a byproduct, considering a convex set of possible initial states of the system-environment S\mathcal{S}S such that TrE S=TrE D\mathrm{Tr}_{E} \ \mathcal{S}=\mathrm{Tr}_{E} \ \mathcal{D}TrE​ S=TrE​ D, we show that when the reduced dynamics of the system, for one system-environment unitary evolution U1U_1U1​, is positive, but not completely positive, this implies that reduced dynamics is not linear for another U2U_2U2​.