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arXiv:0902.4524 · International Journal of Quantum Information

ENTANGLEMENT AND TELEPORTATION IN BIPARTITE SYSTEM

Satyabrata Adhikari

Published Oct 1, 2010• 0 citations
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Abstract

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.

DOI
10.1142/s0219749910006411

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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.
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In this paper, we propose a protocol to deterministically teleport an unknown mixed state of qubit by utilizing a maximally bipartite entangled state of qubits as quantum channel. If a non-maximally entangled bipartite pure state is employed as quantum channel, the unknown mixed quantum state of qubit can be teleported with 1−(1− C 2 ) 1/2 probability, where C is the concurrence of the quantum channel. The protocol can also be generalized to teleport a mixed state of qudit or a multipartite mixed state. More important purpose is that, on the basis of the protocol, the teleportation of an arbitrary multipartite (pure or mixed) quantum state can be decomposed into the teleportation of each subsystem by employing separate entangled states as quantum channels. In the case of deterministic teleportation, Bob only needs to perform unitary transformations on his single particles in order to recover the initial teleported multipartite quantum state.
The bipartite quantum teleportation with a three-qubit mixture of GHZ and W states as a quantum channel is discussed. When the quantum channel is a mixed state consisting of the GHZ state plus small perturbed W state, the entanglement of the channel becomes zero when the average fidelity FˉGHZ\bar{F}_{GHZ}FˉGHZ​ is less than 0.775. This means that the mixed state cannot play an any role as a quantum channel at FˉGHZ≤0.775\bar{F}_{GHZ} \leq 0.775FˉGHZ​≤0.775. For the case of the mixed state consisting of the W state plus small perturbed GHZ state it turn out that this mixed state cannot play a role as a quantum channel at FˉW≤0.833\bar{F}_W \leq 0.833FˉW​≤0.833.
We study the efficiency of two-qubit mixed entangled states as resources for quantum teleportation. We first consider two maximally entangled mixed states, viz., the Werner state\cite{werner}, and a class of states introduced by Munro {\it et al.} \cite{munro}. We show that the Werner state when used as teleportation channel, gives rise to better average teleportation fidelity compared to the latter class of states for any finite value of mixedness. We then introduce a non-maximally entangled mixed state obtained as a convex combination of a two-qubit entangled mixed state and a two-qubit separable mixed state. It is shown that such a teleportation channel can outperform another non-maximally entangled channel, viz., the Werner derivative for a certain range of mixedness. Further, there exists a range of parameter values where the former state satisfies a Bell-CHSH type inequality and still performs better as a teleportation channel compared to the Werner derivative even though the latter violates the inequality.
The efficacies of maximally and that of non-maximally entangled mixed states as teleportation channels have been studied. A new class of non-maximally entangled mixed states have been proposed also. Their advantages as quantum teleportation channels over existing non-maximally entangled mixed states have been verified. The mixed states can also be generated using quantum cloning machines. We have also studied how one can utilize the mixed entangled states obtained as output from a state independent quantum cloning machine in teleportation and dense coding. Mixed states can also be used in secret sharing. A new protocol of secret sharing has been devised where four parties are involved. One of them is honest by nature and the other is dishonest. The dishonest one has two associates who are also involved in the protocol. It has been shown how the honest party, by using quantum cloning machine, can prevent the dishonest party from communicating the secret message to his accomplices. A relation has been established among entanglement of the initially prepared state, entanglement of the mixed state received by the recipients of the dishonest one after honest party applies the cloning scheme and the cloning parameters of the cloning machine. The usefulness of tri-partite and four-partite entangled states in controlled dense coding has been discussed. The thesis concludes with the summary and possible courses of future works.