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Scientific Reports

Localizable entanglement as a necessary resource of controlled quantum teleportation

Artur Barasiński, Ievgen I. Arkhipov, Jiŕí Svozilík

Published Oct 8, 2018• 26 citations
Publisher

Abstract

We analyze the controlled teleportation protocol through three-qubit mixed states. In particular, we investigate the relation between the faithfulness of the controlled teleportation scheme and entanglement. While our knowledge concerning controlled teleportation and entanglement in pure states is well established, for mixed states it is considerably much harder task and very little has been done in this field. Here, we present counterintuitive results that provide a new light on controlled teleportation protocol. It is shown that even mixed biseparable states are useful for this protocol along with genuine entangled three-qubit states.

DOI
10.1038/s41598-018-33185-5

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We analyze the controlled teleportation protocol through three-qubit mixed states. In particular, we investigate the relation between the faithfulness of the controlled teleportation scheme and entanglement. While our knowledge concerning controlled teleportation and entanglement in pure states is well established, for mixed states it is considerably much harder task and very little has been done in this field. Here, we present counterintuitive results that provide a new light on controlled teleportation protocol. It is shown that even mixed biseparable states are useful for this protocol along with genuine entangled three-qubit states.
We consider the controlled quantum teleportation performed on a single copy of entangled three-qubit pure states and show how the fidelity of teleportation and the control power are related to the tripartite entanglement measures, namely, three-tangle and genuine concurrence. We characterize the states with extreme properties, and use them to derive tight lower and upper bounds on both the teleportation fidelity and control power for a given amount of entanglement. Furthermore, we discuss the usefulness of these two quantities for experimental detection and/or quantification of tripartite entanglement. In particular, we present the results which imply that the control power cannot be considered as a candidate for a degree of three-qubit entanglement complementary to the three-tangle as conjectured recently [K. Jeong, J. Kim, and S. Lee, Phys. Rev. A 93, 032328 (2016)]. Furthermore, we prove a mutual relation between teleportation fidelity and control power which provides attainable limits of the controller's authority for existing controlled teleportation schemes. Although those relations are derived for pure states, their extensions for mixed states are also examined.
Three-qubit mixed states are used as a channel for controlled quantum teleportation (CQT) of single-qubit states. The connection between different channel parameters to achieve maximum controlled teleportation fidelity is investigated. We show that for a given multipartite entanglement and mixedness a class of non-maximally-entangled mixed XXX states (XXX-NMEMS) achieves optimum controlled quantum teleportation fidelity; interestingly a class of maximally entangled mixed XXX states (XXX-MEMS) fails to do so. This demonstrates, for a given spectrum and mixedness, that XXX-MEMS are not sufficient to attain optimum controlled quantum teleportation fidelity, which is in contradiction with the traditional quantum teleportation of single qubits. In addition, we show that biseparable XXX-NMEMS, for a certain range of mixedness, are useful as a resource to attain high controlled quantum teleportation fidelity, which essentially lowers the requirements of quantum channels for CQT.
We study the general case of controlled quantum teleportation and give analytic expressions for the maximal probabilities of successful controlled teleportation of an unknown qubit via every kind of three-qubit state. We also give an explicit expression for the three-qubit states that can be used as quantum channel for controlled teleportation of an unknown state with unit probability and with unit fidelity. In addition, an exact value of localizable entanglement is determined. We obtain the necessary and sufficient condition that a three-qubit state can be collapsed to an EPR pair by a measurement on one qubit.
We study faithful teleportation systematically with arbitrary entangled states as resources. The necessary conditions of mixed states to complete perfect teleportation are proved. Based on these results, the necessary and sufficient conditions of faithful teleportation of an unknown state |ϕ⟩ in with an entangled resource ρ in and are derived. It is shown that for ρ in , ρ must be a maximally entangled state, while for ρ in , ρ must be a pure maximally entangled state. Moreover, we show that the sender's measurements must be all projectors of maximally entangled pure states. The relations between the entanglement of the formation of the resource states and faithful teleportation are also discussed.
Teleporting an unknown qubit state is a paradigmatic quantum information processing task revealing the advantage of quantum communication protocols over their classical counterpart. For a teleportation protocol using a Bell state as quantum channel, the resource has been identified to be the concurrence. However, for mixed multipartite states the lack of computable entanglement measures has made the identification of the quantum resource responsible for this advantage more challenging. Here, by building on previous results showing that localizable concurrence is the necessary resource for controlled quantum teleportation, we show that any teleportation protocol using an arbitrary multipartite state, that includes a Bell measurement, requires a nonvanishing localizable concurrence between two of its parties to perform better than the classical protocol. By first analyzing Greenberger–Horne–Zeilinger (GHZ) channel and GHZ measurement teleportation protocol, in the presence of GHZ-symmetric-preserving noise, we compare different multipartite entanglement measures with the fidelity of teleportation, and we find that the protocol performs better than the classical protocol when all multipartite entanglement measures vanish, except for the localizable concurrence. Finally, we extend our proof to an arbitrary teleportation protocol with an arbitrary multipartite entangled channel.