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arXiv:1110.1220

Standard Quantum Teleportation of an Arbitrary N-Qubit State, Non-Existence of Magic Basis and Existence of Magic Partial Bases for 2N Entangled Qubit States with N>1

Hari Prakash, Vikram Verma

Published Oct 6, 2011• 1 citation
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Abstract

We present a simple and precise protocol for standard quantum teleportation of N-qubit state, considering the most general resource q-channel and Bell states. We find condition on these states for perfect teleportation and give explicitly the unitary transformation required to be done by Bob for achieving perfect teleportation. We discuss connection of our simple theory with the complicated related work on this subject and with character matrix, transformation, judgment and kernel operators defined in this context. We also prove that the magic basis discussed by Hill and Wootters [Phys. Rev. Lett. 78 (1997) 5022] does not exist for entangled 2N-qubit states with N > 1 but magic partial bases, similar to those discussed recently by Prakash and Maurya [Optics Commun. 284 (2011) 5024] do exist. We give explicitly all magic partial bases for N = 2.

DOI
10.48550/arxiv.1110.1220

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We present a simple and precise protocol for standard quantum teleportation of N-qubit state, considering the most general resource q-channel and Bell states. We find condition on these states for perfect teleportation and give explicitly the unitary transformation required to be done by Bob for achieving perfect teleportation. We discuss connection of our simple theory with the complicated related work on this subject and with character matrix, transformation, judgment and kernel operators defined in this context. We also prove that the magic basis discussed by Hill and Wootters [Phys. Rev. Lett. 78 (1997) 5022] does not exist for entangled 2N-qubit states with N > 1 but magic partial bases, similar to those discussed recently by Prakash and Maurya [Optics Commun. 284 (2011) 5024] do exist. We give explicitly all magic partial bases for N = 2.
We explicitly show a protocol in which an arbitrary two qubit state $\ensuremath{\mid}\ensuremath{\phi}⟩=a\ensuremath{\mid}00⟩+b\ensuremath{\mid}01⟩+c\ensuremath{\mid}10⟩+d\ensuremath{\mid}11⟩$ is faithfully and deterministically teleported from Alice to Bob. We construct the 16 orthogonal generalized Bell states that can be used to teleport the two qubits. The local operations Bob must perform on his qubits in order to recover the teleported state are also constructed. They are restricted only to single-qubit gates. This means that a controlled-NOT gate is not necessary to complete the protocol. A generalization where NNN qubits are teleported is also shown. We define a generalized magic basis, which possesses interesting properties. These properties help us to suggest a generalized concurrence from which we construct a measure of entanglement that has a clear physical interpretation: A multipartite state has maximum entanglement if it is a genuine quantum teleportation channel.
It is shown that standard quantum teleportation (SQT) with multi-qubit resource result in fidelity (2+C)/3(2+C)/3(2+C)/3 where CCC is concurrence of the resource in bipartite entanglement between qubit going to receiver and rest of the qubits. For perfect SQT, obviously, C=1C=1C=1. For a general 3-qubit resource, we find conditions for getting perfect SQT for state expressed in any basis states. Zha et al. [Mod. Phys. Lett. B 22, 2523-2528 (2008)], who studied perfect SQT using 3-qubit resource, reported conditions for perfect SQT for only those resource states which are given in the 3-qubit canonical form of Acin representation. We show that there is an alternative easily derivable representation which gives more generalized results. To illustrate the difference between the two schemes, we build an example of 3-qubit entangled state, giving perfect SQT and not included in Zha et al. results.
First, I show explicitly a scheme to {\it faithfully} and {\it deterministically} teleport an arbitrary 2-qubit state from Alice to Bob. In this scheme two same Bell states are sufficient for use. Bob can recover the 2-qubit state by performing at most 4 single-qubit unitary operations conditioned on Alice's 4-bit classical public message corresponding to her two Bell-state measurement outcomes. Then I generalize the 2-qubit teleportation scheme to an aribitrary N(N≥3)N(N\ge 3)N(N≥3)-qubit state teleportation case by using NNN same Bell states. In the generalized scheme, Alice only needs to identify NNN Bell states after quantum swapping and then publish her measurement outcomes (2N2N2N-bit classical message). Conditioned on Alice's 2N2N2N-bit classical message, Bob only needs to perform at most 2N2N2N single-qubit unitary operations to {\it fully} recover the arbitrary state. By comparing with the newest relevant work [Phys. Rev. A{\bf 71}, 032303(2005)], the advantages of the present schemes are revealed, respectively.
Quantum mechanics shows superiority than classical mechanics in many aspects and quantum entanglement plays an essential role in information processing and some computational tasks such as quantum teleportation (QT). QT was proposed to transmit the unknown states, in which EPR pairs, the entangled states, can be used as quantum channels. In this paper, we present two simple schemes for teleporting a product state of two arbitrary single-particle and an arbitrary two-particle pure entangled state respectively. Alice and Bob have shared an entangle state. Two Bell states are used as quantum channels. Then after Alice measuring her qubits and informing Bob her measurement results, Bob can perfectly reconstruct the original state by performing corresponding unitary operators on his qubits. It shown that a product state of two arbitrary single-particle and an arbitrary two-particle pure entangled state can be teleported perfectly, i.e. the success probabilities of our schemes are both 1.
A scheme of teleportation of an arbitrary three-qubit state is presented by using the generalized Bell-basis measurement and the Non-Bell-Pair quantum channel.Teleportation can be perfectly realized if Alice makes Bell-state measurement on her particles and Bob carries out appropriate unitary transformation(Toffoli transformation)on his particle.The unknown three-particle entangled state can be teleported perfectly,and the successful possibilities and fidelities of the two schemes both reach unity.Based on the transformation operator,the collapses states can easily obtained and Bob's unitary transformation can also easily given.
Two-qubit quantum teleportation is closely related to quantum computation,so a teleportation protocol in which an arbitrary bipartite quantum state is perfectly teleported probabilistically from sender to receiver is proposed. One of 16 generalized non-maximally entangled Bell states (G states for simplicity) functions as quantum channel. The teleportation can be successfully realized with a certain probability if sender performs generalized Bell state measurements (G measurements) and receiver introduces an auxiliary particle and operates appropriate unitary transformations and single-qubit measurements. The probability of successful teleportation is determined by the smallest one among the coefficients' absolute values of the quantum channel.