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Acta Physica Sinica

Concurrence of the mixed state of two non-orthogonal pure states

Di Yao-Min, Baolin Hu, Liu Dong-Dong, Yan Shi-Ming, 徐州师范大学物理系，徐州 221116

Published Jan 1, 2006• 7 citations
Publisher

Abstract

In this article an obvious expression of concurrence of the mixed state of two non-orthogonal pure states in two qubits is derived. The result reveals that the expression of concurrence of the mixed state of two orthogonal states can be directly generalized to the case of the mixed state of two non-orthogonal states. Then the maximally entangled states mixed by two pure states and the concurrences in some important special cases are discussed.

DOI
10.7498/aps.55.3869

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In this article an obvious expression of concurrence of the mixed state of two non-orthogonal pure states in two qubits is derived. The result reveals that the expression of concurrence of the mixed state of two orthogonal states can be directly generalized to the case of the mixed state of two non-orthogonal states. Then the maximally entangled states mixed by two pure states and the concurrences in some important special cases are discussed.
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Abstract We investigate the entanglement of a bipartite partly non-orthogonal state of the form | ϕ ⟩ = u | ξ ⟩ 1 ⊗ | 1 ⟩ 2 + v | ξ ⟩ 1 ⊗ | 0 ⟩ 2 + w | ξ ′ ⟩ 1 ⊗ | 1 ⟩ 2 + z | ξ ′ ⟩ 1 ⊗ | 0 ⟩ 2 through 1 2 -spin coherent states by adopting the concurrence as a measure to quantify the degree of entanglement of the two-qubit system. After, we study the amount of entanglement of mixed states represented by a statistical mixture of partly non-orthogonal bipartite pure states. We also express the lower and upper bounds of the squared concurrence of mixed states in terms of the amplitudes of spin coherent states.
In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.
In the present paper, we examine the quantum entanglement for more general states of two-qubit system in the context of spin coherent states (SCSs). We consider the concurrence as a quantifier of entanglement and express it in terms of SCSs. We determine new set of maximally entangled conditions that provide the maximal amount of entanglement for certain values of the amplitudes of SCSs for the case of pure states. Finally, we examine the entanglement of a class of mixed states of the two qubits and provide the range in which the entanglement value is maximal with respect to the values of the amplitudes of SCSs.
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.
Entanglement in incoherent mixtures of pure states of two qubits is considered via the concurrence measure. A set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states. When two or three pure real states are mixed, it is shown that 28.5% or 5.12% of the cases, respectively, are optimal. Conditions that are obeyed by the pure states generating such optimally entangled mixtures are derived. For four or more pure states, it is shown that there are no such sets of real states. The implication of these on the superposition of two or more dimerized states is discussed. A corollary of these results also show in how many cases rebit concurrence can be the same as that of qubit concurrence.