Transmission losses in optical qubits for controlled teleportation

Abstract

In this work, we investigate the controlled teleportation protocol using optical qubits within the single-rail logic. The protocol makes use of an entangled tripartite state shared by the controller and two further parties (users) who will perform standard teleportation. The goal of the protocol is to guarantee that the teleportation is successful only with the permission of the controller. Optical qubits based on either superpositions of vacuum and single-photon states or superposition of coherent states are employed here to encode a tripartite maximal slice state upon which the protocol is based. We compare the performances of these two encodings under losses which are present when the qubits are guided through an optical fiber to the users. Finally, we investigate the non-locality of the shared tripartite state to see whether or not it impacts the efficiency of the protocol.

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Acknowledgements

IM acknowledges support by the Coordenao de Aperfeioamento de Pessoal de Nvel Superior (CAPES). FLS acknowledges partial support from CNPq (Grant No. 307774/2014-7).

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Correspondence to I. Medina.

Appendix: Teleportation fidelity

Appendix: Teleportation fidelity

Let us suppose that Alice and Bob share a general bipartite state described by the density operator \(\rho \), and they want to use this state as a channel for teleportation [6]. Assuming that Alice succeeds in projecting the local state of her two qubits in a Bell state, and that Bob can perform arbitrary rotations on his own qubit, it can be shown that the maximal fidelity for the standard teleportation protocol can be written as [63]

$$\begin{aligned} F(\rho )=\frac{2 f + 1}{3}, \end{aligned}$$
(28)

with f being the so-called fully entangled fraction defined as [64]

$$\begin{aligned} f=\underset{|{\phi }\rangle }{\mathrm{{max}}}\langle {\phi }|\rho |{\phi }\rangle , \end{aligned}$$
(29)

where the maximization is over all maximally entangled states \(|\phi \rangle \), i.e., all states that can be obtained from a singlet using local unitary transformations.

The fully entangled fraction can be found using a simple method whose steps we now briefly describe [65]. First , the density operator \(\rho \) must be written in a special basis usually referred to as the magic basis

$$\begin{aligned}&|{m_1}\rangle =|{\Phi ^+}\rangle =\frac{1}{\sqrt{2}}(|{00}\rangle +|{11}\rangle ),\nonumber \\&|{m_2}\rangle =i|{\Phi ^-}\rangle =\frac{i}{\sqrt{2}}(|{00}\rangle -|{11}\rangle ),\nonumber \\&|{m_3}\rangle =i|{\Psi ^+}\rangle =\frac{i}{\sqrt{2}}(|{01}\rangle +|{10}\rangle ),\nonumber \\&|{m_4}\rangle =|{\Psi ^-}\rangle =\frac{1}{\sqrt{2}}(|{01}\rangle -|{10}\rangle ). \nonumber \end{aligned}$$

The fully entangled fraction f is then evaluated simply as the biggest eigenvalue of the real part of \(\rho \) when written in the magic basis [65].

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Medina, I., Semião, F.L. Transmission losses in optical qubits for controlled teleportation. Quantum Inf Process 16, 235 (2017). https://doi.org/10.1007/s11128-017-1684-x

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Keywords

  • Teleportation
  • Coherent states
  • Photonic qubits