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arXiv:0809.2332

Entanglement of a mixed state of two qubit system

L. Chotorlishvili

Published Sep 13, 2008• 0 citations
arXivHTMLPDF

Abstract

Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of random variable.

DOI
10.48550/arxiv.0809.2332

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Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of random variable.
Adding the maximally mixed state with some weight to the entanglement system leads to disentanglement of the latter. For each predefined entangled state there exists a minimal value of this weight for which the system loses its entanglement properties. These values were proposed to be used as a quantitative measure of entanglement called robustness [G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999)]. Using the concurrence, we propose the derivation of this measure for the system of two-qubit. Namely, for a two-qubit pure state, an exact expression of robustness is obtained. Finally, in the same way, the robustness of special cases of mixed two-qubit states is calculated.
Entanglement of quantum systems refers to the situation where two or more systems interact and the state of the whole system cannot be seen as a superposition of states of the single systems. This chapter is devoted to the definition of entanglement both for pure states and mixed states. It also discusses methods to detect entanglement. The chapter is devoted to the study of the dynamics of entanglement. It is also devoted to the problem of local equivalence between states, i.e., how to determine when, given two states, it is possible to evolve from one to the other by transformations which act on systems separately.
We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We find a similarly simple rule for average concurrence when creating a Greenberger-Horne-Zeilinger state from three entangled pairs. We look at examples of mixed qubit states and find that the relation for pure states gives an upper bound on what is possible with mixed states. We then move on to qudits, where we make use of the I-concurrence. Here the situation is not as simple as for qubits, but in some cases relatively straightforward results can be obtained.
Quantum information and computation is one of the new fields of physics and information science. Entanglement is one of the most significant resources in quantum information processing. The most basic quantitative measure of entanglement, particularly for mixed states of a bipartite system is the entanglement of formation. This entanglement of formation is used in the concurrence, another measurement of entanglement. Here I have briefly reviewed the entanglement of formation and the concurrence of bipartite systems. I have discussed that for a three qubit pure state, if A, B and C are the three qubits, then sum of the squared concurrence between A and B and the squared concurrence between A and C never exceeds the squared concurrence between A and BC pair. The residual entanglement, obtained from the aforementioned inequality, is also discussed here. Keywords- Quantum information, entanglement, qubit.