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International Journal of Modern Physics B

DESCRIBING QUANTUM TELEPORTATION OF MIXED STATES WITH CONTINUUM VARIABLES BY THE ENTANGLED STATE REPRESENTATION

Liyun Hu, Hong-Yi Fan

Published Mar 23, 2010• 2 citations
Publisher

Abstract

By virtue of the entangled state representation, we show that the quantum teleportated state in Bob, after Alice makes an Einstein–Podolsky–Rosen measurement and Bob makes an appropriate unitary transformation on hearing Alice's measurement result via a classical channel, is described by [Formula: see text] where F(η)≡23〈η|ρ23|η〉23, ρ23 stands for a quantum channel in 2–3 modes, |η〉23 is the bipartite entangled state, and [Formula: see text] is a displacement transform performed by Bob.

DOI
10.1142/s0217979210054208

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By virtue of the entangled state representation, we show that the quantum teleportated state in Bob, after Alice makes an Einstein–Podolsky–Rosen measurement and Bob makes an appropriate unitary transformation on hearing Alice's measurement result via a classical channel, is described by [Formula: see text] where F(η)≡23〈η|ρ23|η〉23, ρ23 stands for a quantum channel in 2–3 modes, |η〉23 is the bipartite entangled state, and [Formula: see text] is a displacement transform performed by Bob.
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.
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.
A new representation of an arbitrary and unknown N -particle state is presented at first. As an application, a scheme for teleporting an arbitrary and unknown N -particle state is proposed when N pairs of two-particle non-maximally entangled states are utilized as quantum channels. After Alice (sender) makes Bell-state measurement on her particles, Bob (recipient) introduces an auxiliary particle and carries out appropriate unitary transformation on his particle and the auxiliary particle depending on classical information from Alice. Then, von Neumann measurement that confirms whether the teleportation succeeds or not is performed by Bob on the auxiliary particle. In order to complete the teleportation, another N -1 times operations need to be performed which are similar to the above ones. It can be successfully realized with a certain probability which is determined by the product of the smaller coefficients of non-maximally entangled pairs. All possible unitary transformations are given in detail.
An unknown quantum state \ensuremath{\Vert}\ensuremath{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, ``Alice,'' and the receiver, ``Bob,'' must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state \ensuremath{\Vert}\ensuremath{\varphi}〉 which Alice destroyed.