Concurrence-based entanglement measures for isotropic states

Abstract

We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for the I-concurrence and tangle of isotropic states.

Concurrence-Based Entanglement Measure For Werner States

“…Another entanglement measure called tangle, was first proposed in [21]. Its generalization to generic mixed states and further properties were explored in [14,22]. The tangle τ (ρ) is by definition the squared concurrence for pure states, and can be similarly extended to mixed states…”

“…This problem has been advanced significantly in [12], providing an algebraic lower bound which can be optimized further by numerical approaches, and in [13] through an entirely analytical derivation of a complementary tightly lower bound. In addition, nice analytical results are also given for isotropic states [14] and rotationally symmetric states [15].An important class of quantum states are the Werner states [16,17], which appear in realistic quantum computing devices and quantum communication environments, e.g. transmitting perfect entangled states through a noisy depolarizing channel.…”

“…This problem has been advanced significantly in [12], providing an algebraic lower bound which can be optimized further by numerical approaches, and in [13] through an entirely analytical derivation of a complementary tightly lower bound. In addition, nice analytical results are also given for isotropic states [14] and rotationally symmetric states [15].…”

“…The key point of our idea is to apply the concavity properties of both τ (Φ) and C(Φ) with respect to the reduced density matrix ρ A = tr B (|ψ ψ|), as proved in [14], i.e.…”

“…To derive the tangle and concurrence for Werner states, we will use a technique developed in [14,17,23]. The EOF is defined to be E(ρ) ≡ min {pi,|ψi } i p i E(|ψ i ) for all possible ensemble realizations ρ = i p i |ψ i ψ i |, where…”