Entanglement of three-qubit pure states in terms of teleportation capability

Abstract

We define an entanglement measure, called the partial tangle, which represents the residual twoqubit entanglement of a three-qubit pure state. By its explicit calculations for three-qubit pure states, we show that the partial tangle is closely related to the faithfulness of a teleportation scheme over a three-qubit pure state. PACS numbers: 03.67.-a, 03.65.Ud, 03.67.Mn 03.67.Hk,Quantum entanglement has been considered to be one of the most crucial resources in quantum information processing, and hence has been studied intensively in various ways. Nevertheless, there are still a number of open problems for entanglement, such as what is the best way to quantify the amount of entanglement for bipartite or multipartite states.For two-qubit states, the Wootters' concurrence C [1, 2, 3], is known as a good measure of entanglement, since from it we can directly derive the explicit formula for the entanglement of formation as well as being readily calculable. On the other hand, in the multi-qubit cases, or even in the three-qubit case, no entanglement measure as good as the concurrence of two qubits has been found yet.Coffman et al. [4] presented an inequality to explain the relation between bipartite entanglement in a three-qubit pure state. The inequality is called the Coffman-Kundu-Wootters (CKW) inequality, which iswhere C 12 = C(tr 3 (Ψ 123 )), C 13 = C(tr 2 (Ψ 123 )), and C 1(23) = C(Ψ 1(23) ) = 2 det(tr 23 (Ψ 123 )) for a three-qubit pure state Ψ 123 = |ψ 123 ψ|. Here, the subscripts represent the indices of the qubits. From the CKW inequality, an entanglement measure for three-qubit pure states was naturally derived [4,5]. It is called the 3-tangle τ , which is defined asand represents the residual entanglement of the state.Here τ is invariant under any qubit taken as the focus qubit, that is, for any distinct i, j, and k in {1, 2, 3},Furthermore, it was shown that τ is an entanglement monotone [5], and it was also shown that τ can distinguish the Greenberger-Horne-Zeilinger (GHZ) class from * Electronic address: level@khu.ac.kr † Electronic address: jaewoo.joo@imperial.ac.uk ‡ Electronic address: jaewan@kias.re.kr the W class [5], where the GHZ class and the W class are the sets of all pure states with true three-qubit entanglement equivalent to the GHZ state [6],under stochastic local operations and classical communication (SLOCC), and equivalent to the W state,under SLOCC, respectively. Even though the 3-tangle τ is a useful entanglement measure for three-qubit pure states, in this paper, we investigate another quantity similar to τ , defined asfor distinct i, j, and k in {1, 2, 3}. We call the quantity the partial tangle. Then we clearly obtain the following equalities:τ 12 = C 2 1(23) − C 2 13 = τ + C 2 12 = τ 21 , τ 23 = C 2 2(31) − C 2 21 = τ + C 2 23 = τ 32 ,and hence τ 2 12 + τ 2 23 + τ 2 31 = 3τ + C 2 12 + C 2 23 + C 2 31 .We clearly remark that τ ij = C ij if and only if a given state is contained in the W class, that is, τ = 0.Observing the definition of τ ij in Eq. (7), τ ij seems to represent the re...

Entanglement Distribution and State Discrimination in Tripartite Qubit Systems

“…To such an end, we present a very simple toy model, reminiscent of a modified teleportation protocol over a three-qubit state. 41 Eve is able to introduce some noise in the form of a rather special spin-flip operation:…”

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

“…It includes various analytical curves from Eqs. (46), (53), (55). The curve corresponding to rank-1 is common to all the ranks.…”

“…In this section the measure R 12 is explored for such rank-2 density matrices with this intrinsic relationship to three-qubit states in mind. Three-qubit states (pure as well as mixed) have been actively explored in the recent past [29,[45][46][47][48][49][50][51][52][53][54][55][56][57], as they offer the simplest system that includes multiparty entanglement sharing. Construction of LU invariants was discussed in [47], however the quantity R 12 was not considered therein.…”

Minimal control power of the controlled teleportation

“…Then it was shown [7] that there is an interesting relation between the maximal teleportation fidelity F j CT and the partial tangle τ kl , that is,…”

“…We can also consider a teleportation scheme over a three-qubit pure state, called controlled teleportation (CT) [6][7][8], which is a variant of the splitting and reconstruction of quantum information over the GreenbergerHorne-Zeilinger (GHZ) state, introduced by Hillery et al [9]. In this scheme, a controller on one qubit of the state maximally assists the teleportation procedure on the other two qubits.…”