The effect of measurement noise on the typical quantum teleportation protocol

Quantum teleportation is an important reaseach direction of quantum communication. Quantum meaurement is one of necessary physical implementation for quantum teleportation. Thus, this paper explored the influence of measurement noise on quantum teleportation. The realization steps of quantum teleportation with measurement noise are presented. From the density operation and fidelity value, the performance of quantum teleportation is analyzed. The research results about measurement noise might be utilized for quantum communication.


INTRODUCTION
Quantum entanglement is an intriguing property of composite quantum systems [1] , and can be used to perform some quantum communication tasks, for example, quantum teleportation [2] , dense coding [3] , remote state preparation [4] , and so on. Quantum teleportation is a feasible technique for transmit quantum states from senders to remote receivers via quantum entanglement and classical communication. The first theoretical scheme of quantum teleporation was presented by Bennett et al. [2] , and this protocol is experimentally proved by Bouwmeester et al. [5] .
Due to its potential applications of quantum information, quantum teleportation has been studied [6][7][8][9][10][11] . Riebe et al [6] reported deterministic quantum-state teleportation between a pair of trapped calcium ions. Bowen et al [7] report the experimental demonstration of quantum teleportation of the quadrature amplitudes of a light field. Li et al [8] presented two different quantum channels of two three-qubit GHZ states and the six-qubit entangled state can be used for quantum teleportation of an arbitrary two-qubit. Podoshvedov et al [9] considered the implementation of a quantum teleportation protocol of an unknown qubit with an entangled hybrid state. Wei et al [10] presented an efficient protocol for probabilistic teleportation is presented with multi-parameter measurements via a non-maximally entangled state. Nevertheless, there are also some important and open subjects to be investigated in the quantum teleportation.
Quantum measurement is one of necessary physical implementation for quantum teleportation. However, quantum measurement would be influenced by human beings and instruments inevitably for quantum teleportation. Thus, this paper explored the influence of measurement noise on quantum teleportation. The detailed realization procedures of quantum teleportation with measurement noise are elaborated. Additionally, the performance of quantum teleportation is analyzed based on the density The rest of this paper are organized as follows: In Section II, the original quantum teleportation scheme would be presented. And then, the implementation procedures of quantum teleportation with measurement noise are presented in Section III. The performance of quantum teleportation will be discussed from the density operation and fidelity value in Section IV. The paper concludes with Section V.

TYPICAL QUANTUM TELEPORTATION SCHEME
The brief statement for original quantum teleportation scheme would be presented in this section. Suppose that the sender Alice want to transmit an unknown single-qubit quantum state to the receiver Bob via quantum channel. Without loss of generality, the unknown state transmitted from Alice to Bob can be shown as follow here  is a real number, and  is complex. For the teleportation of single-qubit state, the two-qubit entanglement state can be used as quantum channel According to the Bell-state measurement, the total state can be rewritten as follow In order to realize the teleportation, Alice would inform Bob of her measurement result via classical channel. Based on these results, Bob can perform corresponding unitary operations to reconstruct the original quantum state shown as Eq. (1). The special unitary transformations relative with measurement outcomes can be obtained from Table 1. These unitary transformations on Table 1 could be presented as follow Based on classical channel and quantum entanglement, we can use quantum teleportation to transmit unknown quantum states via some special unitary transformations. It is noting that quantum measurement is one of essential operations for quantum teleportation.

TELEPORTATION WITH MEASUREMENT NOISE
From the above discussion of Section II, it could be obtained that the Bell-state measurement is the first step of physical realization for original quantum teleportation. In the real system, some human beings and instruments could inevitably affect the physical implementation of quantum measurement. Therefore, the performance of quantum measurement would influence the teleportation. The concrete processes for quantum teleportation with measurement noise would be elaborated in detail. For the sake of the realization of the original teleportation scheme, the sender Alice need to perform the Bell-state measurement shown as Eq. (4). Suppose that the measurement with noise could be presented as follow After the measurement, Alice would send the measurement results to Bob with classical communication, and then Bob will perform the relative unitary gates on qubit 3. The relationship between the after-measurement state of qubit 3 and the measurement results of quit 1 and 2 could be found from

THE TELEPORTATION PERFORMANCE WITH MEASUREMENT NOISE
The density operation language provides an efficient method to describe quantum system. According to the above table, qubit 3 is on one of states   . Thus, the density operation of qubit 3 after meassurement could be defined by the equation