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arXiv:1204.0983

Generalized form of optimal teleportation witnesses

Kumar, Atul, Satyabrata Adhikari, Pankaj Agrawal

Published Apr 4, 2012• 0 citations
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Abstract

We propose a generalized form of optimal teleportation witness to demonstrate their importance in experimental detection of the larger set of entangled states useful for teleportation in higher dimensional systems. The interesting properties of our witness reveal that teleportation witness can be used to characterize mixed state entanglement using Schmidt numbers. Our results show that while every teleportation witness is also a entanglement witness, the converse is not true. Also, we show that a hermitian operator is a teleportation witness iff it is a decomposable entanglement witness. In addition, we analyze the practical significance of our study by decomposing our teleportation witness in terms of Pauli and Gell-Mann matrices, which are experimentally measurable quantities.

DOI
10.48550/arxiv.1204.0983

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We propose a generalized form of optimal teleportation witness to demonstrate their importance in experimental detection of the larger set of entangled states useful for teleportation in higher dimensional systems. The interesting properties of our witness reveal that teleportation witness can be used to characterize mixed state entanglement using Schmidt numbers. Our results show that while every teleportation witness is also a entanglement witness, the converse is not true. Also, we show that a hermitian operator is a teleportation witness iff it is a decomposable entanglement witness. In addition, we analyze the practical significance of our study by decomposing our teleportation witness in terms of Pauli and Gell-Mann matrices, which are experimentally measurable quantities.
Teleportation witnesses are Hermitian operators which can identify useful entanglement for quantum teleportation. Here we provide a systematic method to construct teleportation witnesses from entanglement witnesses corresponding to general qudit systems. The witnesses so constructed are shown to be optimal for qubit and qutrit systems and therefore detect the largest set of states useful for teleportation within a given class. We demonstrate the action of the witness pertaining to different classes of states in qubits and qutrits. Decomposition of the witness in terms of spin operators facilitates experimental identification of useful resources for teleportation.
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The ability of entangled states to act as a 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 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.
Not all entangled states are useful for quantum teleportation. We present a geometric method to construct optimal teleportation witnesses, which provide operational necessary and sufficient criteria for identifying the teleportation usefulness of arbitrary two-qudit entangled states. Specifically, by developing a two-layer iterative cutting-plane algorithm to solve the shortest distance problem from the target state ρρρ to the convex set SSS of useless states, we obtain the projection point σ∗∈Sσ^* \in Sσ∗∈S and then construct the optimal teleportation witness from the projection geometry. Moreover, the shortest distance D(ρ)D(ρ)D(ρ) obtained during this construction also serves as a necessary and sufficient criterion for usefulness. We apply our method to identify the teleportation usefulness of three classes of entangled states.