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arXiv:quant-ph/0402133 · Physical Review A

Faithful teleportation with partially entangled states

Gilad Gour

Published Oct 4, 2004• 32 citations
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Abstract

We write explicitly a general protocol for faithful teleportation of a $d$-state particle (qudit) via a partially entangled pair of (pure) $n$-state particles. The classical communication cost (CCC) of the protocol is ${\mathrm{log}}_{2}(nd)$ bits, and it is implemented by a projective measurement performed by Alice, and a unitary operator performed by Bob (after receiving from Alice the measurement result). We prove the optimality of our protocol by a comparison with the concentrate and teleport strategy. We also show that if $d>n∕2$, or if there is no residual entanglement left after the faithful teleportation, the CCC of any protocol is at least ${\mathrm{log}}_{2}(nd)$ bits. Furthermore, we find a lower bound on the CCC in the process transforming one bipartite state to another by means of local operation and classical communication.

DOI
10.1103/physreva.70.042301

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We write explicitly a general protocol for faithful teleportation of a ddd-state particle (qudit) via a partially entangled pair of (pure) nnn-state particles. The classical communication cost (CCC) of the protocol is log2(nd){\mathrm{log}}_{2}(nd)log2​(nd) bits, and it is implemented by a projective measurement performed by Alice, and a unitary operator performed by Bob (after receiving from Alice the measurement result). We prove the optimality of our protocol by a comparison with the concentrate and teleport strategy. We also show that if d>n∕2d>n∕2d>n∕2, or if there is no residual entanglement left after the faithful teleportation, the CCC of any protocol is at least log2(nd){\mathrm{log}}_{2}(nd)log2​(nd) bits. Furthermore, we find a lower bound on the CCC in the process transforming one bipartite state to another by means of local operation and classical communication.
We propose a method to realize the teleportation of an unknown entangled state that consists of many qudits through a partially entangled-qudit quantum channel with the help of 2log 2 d -bit classical communication. The operations used in the teleportation process include a generalized Bell-state measurement and a series of single-qudit π-measurements performed by Alice, a series of generalized qudit-Pauli gates and two-level unitary gates, as well as a qubit measurement performed by Bob. For a maximally entangled quantum channel, the successful probability of the teleportation becomes unit.
Summary form only given. Characterizing the entanglement of a general bipartite quantum system is a difficult problem. Most measures involve difficult extremizations and their physical motivation is not always clear. In this paper we propose an experimental measure of entanglement based on a modified version of the familiar teleportation protocol of Bennett et al. (1993). Briefly, teleportation traditionally involves two parties, Alice and Bob. Initially, Alice has a qubit in an unknown quantum state which she would like to communicate to Bob and Bob has two qubits in an entangled Einstein-Podolsky-Rosen (EPR) pair. To accomplish this transfer, Bob shares one qubit of his EPR pair with Alice. She then performs a joint measurement or Bell measurement on her two qubits and relays this information back to Bob over a classical channel. Bob then uses the outcome of Alice's Bell measurement to transform his qubit, with a local unitary transformation, into the original quantum state, always with a fidelity of one.
We analyze approximate transformations of pure entangled quantum states by local operations and classical communication, finding explicit conversion strategies which optimize the fidelity of transformation. These results allow us to determine the most faithful teleportation strategy via an initially shared partially entangled pure state. They also show that procedures for entanglement manipulation such as entanglement catalysis [Jonathan and Plenio, Phys. Rev. Lett. 83, 3566 (1999)] are robust against perturbation of the states involved, and motivate the notion of nonlocal fidelity, which quantifies the difference in the entangled properties of two quantum states.
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.
We present many ensembles of states that can be remotely prepared by using minimum classical bits from Alice to Bob and their previously shared entangled state and prove that we have found all the ensembles in two-dimensional case. Furthermore we show that any pure quantum state can be remotely and faithfully prepared by using finite classical bits from Alice to Bob and their previously shared nonmaximally entangled state though no faithful quantum teleportation protocols can be achieved by using a nonmaximally entangled state.
It deserves mentioning that the quantum circuit, i.e. quantum logic network, is essential to the practical realization of teleportation in experiment. Using only one partially entangled pair, we first propose two novel strategies for probabilistically teleporting any partially entangled state of a bipartite system. The feature of the present protocol is to weaken the requirement for the quantum channel, and also to cut down the number of entangled particles initially shared by the sender and receiver. On the other hand, we explicitly construct the generalized measurement described by the positive operator-valued measure (POVM). Two kinds of efficient quantum circuits for implementing the teleportation are offered. In addition, we generalize the two-particle probabilistic teleportation to the system of many particles.