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Journal of Physics A Mathematical and Theoretical

Some entanglement features exhibited by two, three and four qubits interacting with an environment in a non-Markovian regime

A. Hamadou-Ibrahim, Angel Ricardo Plastino, Claudia Zander

Published Jan 14, 2010• 11 citations

Abstract

We explore various aspects of the quantum entanglement dynamics of systems of two, three and four qubits interacting with an environment at zero temperature in a non-Markovian regime, as described by the paradigmatic model recently studied by Bellomo et al (2007 Phys. Rev. Lett. 99 160502). We consider important families of initial states for the alluded systems. The average, typical entanglement evolution associated with each of these families is determined, and its relation with the evolution of the global degree of mixedness of the multi-qubit system is explored. For three and four qubits we consider the family of initial states equivalent under local unitary transformations to the |GHZ⟩ and | W ⟩ states, and compare their average behavior with the average behavior exhibited by initial maximally entangled two-qubit states. Furthermore, in the case of two qubits, the evolution of other manifestations of entanglement, related to measurable quantities, is also investigated. In particular, we consider the Mintert–Buchleitner concurrence lower bound and an entanglement indicator based upon the violation of local uncertainty relations.

DOI: 10.1088/1751-8113/43/5/055305

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We explore various aspects of the quantum entanglement dynamics of systems of two, three and four qubits interacting with an environment at zero temperature in a non-Markovian regime, as described by the paradigmatic model recently studied by Bellomo et al (2007 Phys. Rev. Lett. 99 160502). We consider important families of initial states for the alluded systems. The average, typical entanglement evolution associated with each of these families is determined, and its relation with the evolution of the global degree of mixedness of the multi-qubit system is explored. For three and four qubits we consider the family of initial states equivalent under local unitary transformations to the |GHZ⟩ and | W ⟩ states, and compare their average behavior with the average behavior exhibited by initial maximally entangled two-qubit states. Furthermore, in the case of two qubits, the evolution of other manifestations of entanglement, related to measurable quantities, is also investigated. In particular, we consider the Mintert–Buchleitner concurrence lower bound and an entanglement indicator based upon the violation of local uncertainty relations.

We investigate the decay of the global entanglement, due to decoherence, of multiqubit systems interacting with a reservoir in a non-Markovian regime. We assume that during the decoherence process each qubit of the system interacts with its own, independent environment. Most previous works on this problem focused on particular initial states or families of initial states amenable of analytical treatment. Here we determine numerically the typical, average behavior of the system corresponding to random initial pure states uniformly distributed (in the whole Hilbert space of n-qubit pure states) according to the Haar measure. We study systems consisting of 3, 4, 5, and 6 qubits. In each case we consider also the entanglement dynamics corresponding to important particular initial states, such as the GHZ states or multiqubit states maximizing the global entanglement, and determine in which cases any of these states is representative of the average behavior associated with general initial states.

We investigate the decay of the global entanglement, due to decoherence, of multiqubit systems interacting with a reservoir in a non Markovian regime. We assume that during the decoherence process each qubit of the system interacts with its own, independent environment. Most previous works on this problem focused on particular initial states or families of initial states amenable of analytical treatment. Here we determine numerically the typical, average behaviour of the system corresponding to random initial pure states uniformly distributed (in the whole Hilbert space of

n

-qubit pure states) according to the Haar measure. We study systems consisting of 3, 4, 5, and 6 qubits. In each case we consider also the entanglement dynamics corresponding to important particular initial states, such as the GHZ states or multiqubit states maximizing the global entanglement, and determine in which cases any of these states is representative of the average behaviour associated with general initial states.

Here is analyzed the dynamics of two-qubit entanglement, when the two qubits are initially in a mixed extended Werner-like state and each of them is in a zero temperature non-Markovian environment. The dependence of entanglement dynamics on the purity and degree of entanglement of the initial states and on the amount of non-Markovianity is also given. This extends the previous work about non-Markovian effects on the two-qubit entanglement dynamics for initial Bell-like states [Bellomo et al., Phys. Rev. Lett. 99, 160502 (2007)]. The effect on the two-qubit entanglement dynamics of nonzero temperature in Markovian environments is finally studied.

Quantum systems usually suffer from unavoidable interactions with the environment. In most experiments, it is virtually impossible to isolate the system under study from the environment. It is thus imperative to study and understand how quantum systems interact with their surroundings. In the present study, I consider the evolution of quantum entanglement in a two-qubit system interacting with an environment in a regime where the non-Markovian effects are important. The present thesis is organized as follows: Chapter one is a general introduction to the thesis. In chapter two, some preliminary concepts that will be used in the subsequent chapters will be introduced and defined. These concepts include qubits, density matrix, Quantum entanglement and Entropic measures in quantum information. Ways to detect or quantify entanglement in a quantum system will also be discussed in the section on quantum entanglement. Chapter three will be about open quantum systems in general: concepts like master equations, Markovian quantum systems and non-Markovian quantum systems will be briefly reviewed. Chapter four will deal with time dependent entanglement features of two-qubit and multi-qubit systems interacting with an environment, basically all the results obtained in my study will be presented there. Some general conclusions will be drawn in chapter five. Copyright