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arXiv:1212.2294

Quantum entanglement and teleportation assisted by multiple parties

Hayashida, Takeshi

Published Dec 11, 2012• 0 citations
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Abstract

Quantum teleportation can be performed if and only if a message is combined with a maximally entangled state. Bennett et al. originally claimed this feasibility condition. In this paper, I give a proof of that condition. Moreover, I specify the condition for teleportation when a single-qubit message is combined with a three-qubit state distributed among three parties. I also consider the situation that a single-qubit message is combined with an $n$-qubit state distributed among $n$ parties.

DOI
10.48550/arxiv.1212.2294

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Quantum teleportation can be performed if and only if a message is combined with a maximally entangled state. Bennett et al. originally claimed this feasibility condition. In this paper, I give a proof of that condition. Moreover, I specify the condition for teleportation when a single-qubit message is combined with a three-qubit state distributed among three parties. I also consider the situation that a single-qubit message is combined with an nnn-qubit state distributed among nnn parties.
We present a mathematical formulation of old teleportation protocol (original teleportation protocol introduced by Bennett et al.) for mixed state and study in detail the role of mixedness of the two-qubit quantum channel in a teleportation protocol. We show that maximally entangled mixed state described by the density matrix of rank-4 will be useful as a two-qubit teleportation channel to teleport a single qubit mixed state when the teleportation channel parameter p1 > 1/2. Also we discuss the case when p1 ≤ 1/2.
We provide an alternative simple proof of the necessity of entanglement in quantum teleportation by using the no-disentanglement theorem. We show that this is true even when the state to be teleported is known to be among two noncommuting qubits. We further show that to teleport any set of commuting qubits, it is sufficient to have a classically correlated channel. Using this result we provide a simple proof of the fact that any set of bipartite entangled states can be exactly disentangled if the single-particle density matrices of any one party commute.
We provide an alternative simple proof of the necessity of entanglement in quantum teleportation by using the no-disentanglement theorem. We show that this is true even when the state to be teleported is known to be among two noncommuting qubits. We further show that to teleport any set of commuting qubits, it is sufficient to have a classically correlated channel. Using this result we provide a simple proof of the fact that any set of bipartite entangled states can be exactly disentangled if the single particle density matrices of any one party commute.
It is shown that a maximally seven-qubit state can be utilized for perfect teleportation of arbitrary three qubit systems and controlled teleportation of an arbitrary three qubit state. For perfect teleportation, the sender first performs a Von-Neumann measurement on her qubits, then the sender(Alice) tells the measurement result to the receiver(Bob) via a classical channel, the receiver can reconstruct the original state by performing appropriate unitary transformations on his qubits at hand. For controlled teleportation, the sender firstly carries out proper Bell-state measurements and the controller(charlie) needs to operate a single-qubit measurement, the receiver can reconstruct the original state by applying appropriate unitary transformations on his qubits after he knows the measurement results of both the sender and controller. Besides, the two teleportation schemes can be realized with the maximal probability.
A total of seven qubits are in a maximally entangled state. Using such an entangled state as quantum channel is based on the construction requirements of quantum long-distance communication [Pan et al., Nature 488, 185 (2012)]. Multi-party quantum channel (QC) should be studied. We put forward three deterministic bidirectional quantum controlled teleportation (BQCT) schemes. To be specific, BQCT can be realized between any two parties in a deterministic manner with another as the control. Alternatively, the BQCT capacity of such state in the given qubit distribution is thus essentially revealed by virtue of the schemes.