Max - Planck - Institut für Mathematik in den Naturwissenschaften Leipzig Complete Entanglement Witness for Quantum

We propose a set of linear quantum entanglement witnesses constituted by local quantum-mechanical observables with each two possible measurement outcomes. These witnesses detect all the entangled resources which give rise to a better fidelity than separable states in quantum teleportation and present both sufficient and necessary conditions in experimentally detecting the useful resources for quantum teleportation. Introduction. Quantum entanglement plays important roles in many quantum information processing such as quantum teleportation. However, not all entangled states are useful for quantum teleportation. The fidelity of optimal teleportation is determined by the fully entangled fraction (FEF) [1–3]. A bipartite n ⊗ n state ρ gives rise to a better fidelity of teleportation than separable states if its FEF is great than 1/n. For a known quantum state, analytical formula of FEF for two-qubit states has been derived by using the method of Lagrange multiplier [4]. The upper bounds of FEF for general high dimensional quantum states have been estimated [5]. Exact results of FEF are also obtained for some special quantum states like isotropic states and Werner states [6]. For a given unknown state, an important issue is to determine whether it is useful for quantum teleportation by experimental measurements. For the experimental detection of quantum entanglement, the Bell inequalities [7–11] and entanglement witness [12–21] have been extensively investigated. The Bell inequalities only involve measurements on local quantum mechanical observables. The entanglement witnesses are in general Hermitian operators with at least one negative eigenvalue. Since the set of the separable states is convex and compact, these witnesses give rise to inequalities (supersurfaces) separating a part of the entangled states from the rest ones including all the separable states. Different inequalities detect different entangled states. However, so far we do not have complete witnesses that detect all the entangled states in general. Recently in Ref. [22], the authors show that the set of entangled states which are useful for quantum teleporta-tion, i.e. their FEFs are great than 1/n, is also convex and compact. They presented a witness operator which detects some entangled states that are useful for telepor-tation. In this brief we give a general way to enquire how to determine experimentally whether an unknown entangled state could be used as a resource for quantum teleporta-tion. We present a linear witness operator which can detect all the entangled states that are useful for tele-portation. This witness operator gives rise to a Bell-like inequality …