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arXiv:quant-ph/0504126

Faithful and deterministic teleportation of an arbitrary $N$-qubit state using and identifying only Bell states

Zhanjun Zhang

Published Apr 17, 2005• 0 citations
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Abstract

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 $N(N\ge 3)$-qubit state teleportation case by using $N$ same Bell states. In the generalized scheme, Alice only needs to identify $N$ Bell states after quantum swapping and then publish her measurement outcomes ($2N$-bit classical message). Conditioned on Alice's $2N$-bit classical message, Bob only needs to perform at most $2N$ 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.

DOI
10.48550/arxiv.quant-ph/0504126

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Faithful and deterministic teleportation of an arbitrary NNN-qubit state using and identifying only Bell states
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.
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.
Recently, there is much interest in the study of quantum teleportation. Its marvelous security property makes it particularly attractive as potential means for future communication. Uncertain principal and no-cloning theories of quantum mechanics forbid exact copy of quantum states. To solve the problem, quantum teleportation devides message state into two parts: classical message and quantum message which are transmitted through relevant classical and quantum channels from a sender Alice to a distant receiver Bob, respectively. According to both message states, Bob can recover the original message state with certain possibility. Utilizing the general quantum teleportation protocol, in this scheme, a new method is presented by which multi-qubit quantum information can be teleported in a much easier way. It has been extended to teleportation of an arbitrary N-qubit state via entanglement swapping and probabilistic teleportation is investigated as well, which has a more general meaning. N non-maximally entangled triplet GHZ states (Greenberger-Home-Zeilinger), taking the place of the previous EPR states (Einstein-Podolsky-Rosen), serve as quantum channels, which avoids introducing extra auxiliary qubit. In order to complete the teleportation, Alice has to devide all qubits into N parts and make Bell-State measurement on qubit pair (i, x(subscript i)) and tell Bob the measurement result through classical channel, then Bob perform some simple local operations on qubits (y(subscript i), z(subscript i)) to finish teleporting the i-th qubit. Conditioned on N-times similar repeat, Bob can faithfully reconstruct the initial state. Results show that, in this method, the identification difficulty of entangled states is considerably reduced, the new method is more feasible in technique. And the total possibility of successful teleportation is (The equation is abbreviated). As a special case, teleportation of an arbitrary two-qubit state is proposed.
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.
A scheme to teleport an unknown two qubit state from Alice (the sender) to Bob (the receiver) using two Einstein Podolsky Rosen (EPR) pairs is presented, each EPR pair being shared by both Alice and Bob. Firstly, Alice combines each of the two particles in the teleported state with an EPR particle and makes Bell state measurement on each combination. Then she transmits the outcomes of her measurements to Bob classically. According to Alice′s measurement results, Bob can perform appropriate unitary operations on his two EPR particles to retrieve the initial state.
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.
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.