Entanglement Witness Operator for Quantum Teleportation

Abstract

The ability of entangled states to act as resource for teleportation is linked to a property of the fully entangled fraction. We show that the set of states with their fully entangled fraction bounded by a threshold value required for performing teleportation is both convex and compact. This feature enables for the existence of hermitian witness operators the measurement of which could distinguish unknown states useful for performing teleportation. We present an example of such a witness operator illustrating it for different classes of states.PACS numbers: 03.67.-a, 03.67.Mn A. Introduction.-Quantum information processing is now widely recognized as a powerful tool for implementing tasks that cannot be performed using classical means [1]. A large number of algorithms for various information processing tasks such as super dense coding[2], teleportation [3] and key generation [4] have been proposed and experimentally demonstrated. At the practical level information processing is implemented by manipulating states of quantum particles, and it is well known that not all quantum states can be used for such purposes. Hence, given an unknown state, one of the most relevant issues here is to determine whether it is useful for quantum information processing.The key ingredient for performing many information processing tasks is provided by quantum entanglement. The experimental detection of entanglement is facilitated by the existence of entanglement witnesses [5,6] which are hermitian operators with at least one negative eigenvalue. The existence of entanglement witnesses is a consequence of the Hahn-Banach theorem in functional analysis [7,8] providing a necessary and sufficient condition to detect entanglement. Motivated by the nature of different classes of entangled states, various methods have been suggested to construct entanglement witnesses [9][10][11][12]. Study of entanglement witnesses [13] has proceeded in directions such as the construction of optimal witnesses [9,11], Schmidt number witnesses [14], and common witnesses [15]. The possibility of experimental detection of entanglement through the measurement of expectation values of witness operators for unknown states is facilitated by the decomposition of witnesses in terms of Pauli spin matrices for qubits [16] and Gell-Mann matrices in higher dimensions [17]. For macroscopic systems the properties of thermodynamic quantities provide a useful avenue for detection of entanglement [18].Teleportation [3] is a typical information processing task where at present there is intense activity in extending the experimental frontiers [19]. However, it is well known that not all entangled states are useful for teleportation. For example, while the entangled Werner state [20] in 2 ⊗ 2 dimensions is a useful resource [21], another class of maximally entangled mixed states [22], as well as other non-maximally entangled mixed states achieve a fidelity higher than the classical limit only when their magnitude of entanglement exceeds a certain value [23]. The problem of det...

Detection of the ideal resource for multiqubit teleportation

“…(ii) there exists at least one entangled state ρ which is useful for teleportation such that T r(W ρ) < 0 [11,13]. For entangled state ρ, if T r(W ρ) < 0, then teleportation witness W could detect ρ as a useful resource for teleportation.…”

“…Recently in Ref. [11], the authors show that the set of states which are not useful for quantum teleportation, i.e. their fully entangled fractions are no more than 1 n , is also convex and compact.…”

“…For bipartite case, it has been shown that the quantum states which are not useful for quantum teleportation compose a compact convex set and a teleportation witness has been presented for the first time in Ref. [11]. Then a complete set of teleportation witness to detect all the ideal resource for teleportation is constructed in Ref.…”

Some applications of uncertainty relations in quantum information

“…An observational scheme which can detect mixedness of qutrit systems unambiguously, requires less resources compared to tomography, and is implementable through the measurement of Hermitian witness-like operators 28 . It may be relevant to note here though that the set of pure states is not convex, and hence, such witness-like operators do not arise from any geometrical separability criterion inherent to the theory of entanglement witnesses 37 , that has been applied more recently to the cases of teleportation witnesses 38 , as well as for witnesses of absolutely separable states 39 . The operational determination of purity using the RS relation requires a few additional steps.…”

Complete entanglement witness for quantum teleportation

“…[22], the authors show that the set of entangled states which are useful for quantum teleportation, i.e. their FEFs are great than 1/n, is also convex and compact.…”