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Communications in Theoretical Physics

Trace Distance of Remote State Preparation Through Noisy Channels

Ai-Xi Chen, Jiahua Li, Jin Li-Xia

Published Apr 1, 2005• 12 citations
Publisher

Abstract

In this paper, we study remote state preparation (RSP) by w state through noisy channels. The trace distance is used to describe how close the original state is to the output state. Studies show the trace distance is a function of decohenrence rates and angles of the state to be prepared. At the same time, we investigate the influence of different types of noises on the trace distance and find the various types of noises have different degrees of influence on the trace distance for a definite qubit. We also study changes of the trace distance against polar and azimuthal angles.

DOI
10.1088/0253-6102/43/4/014

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In this paper, we study remote state preparation (RSP) by w state through noisy channels. The trace distance is used to describe how close the original state is to the output state. Studies show the trace distance is a function of decohenrence rates and angles of the state to be prepared. At the same time, we investigate the influence of different types of noises on the trace distance and find the various types of noises have different degrees of influence on the trace distance for a definite qubit. We also study changes of the trace distance against polar and azimuthal angles.
The influence of noise on the remote preparation of a qubit by an entangled pair is investigated. In Bloch sphere representation, we use the trace distance to describe how close the final state is to the original state to be prepared. Our studies include two cases. Firstly, we consider the sender and the receiver sharing a mixture of two Bell states as quantum channel in remote state preparation (RSP) and calculate the trace distance as a function of mixture ratio and angle of a state to be prepared. Secondly, the system–environment interaction is taken into account by including stochastic fluctuating terms in the system Hamiltonian. Solving the Bloch equations, we obtain the evolution density matrix of the system. We then apply this stochastic model to study the effect of noise on the trace distance of RSP.
We investigate the influence of quantum noises on the processes of remote preparation of a qubit by using an Einstein-Podolsky-Rosen （EPR） state and a bipartite partially entangled state as the quantum channel respectively. By solving the master equation in the Lindblad form, we obtain the time evolution of the quantum channels. Then we use the trace distance to describe how close the output state and the initial state are in various noisy conditions. Our results show that the influence of the noise acting on z direction is the weakest, and the influence of the noises acting simultaneously on x, y, and z directions is the strongest.
Using a quantum channel consisting of a GHZ state exposed to noisy environment, we investigate how to remotely prepare an entangled state and a qubit state, respectively. By solving the master equation in the Lindblad form, the influence of the various types of noises on the GHZ state is first discussed. Then we use the fidelity to describe how close the remotely prepared state and the initial state are. Our results show that the fidelity is a function of the decoherence rates and the angles of the initial state. It is found that for each of the two RSP schemes, the influence of the noise acting simultaneously in x, y, and z directions on the average fidelity is the strongest while the influence of the noise acting in x or z direction on the average fidelity is relatively weaker.
In this paper, the theoretical investigation of remote preparation of an entangled state is studied in nonideal conditions. Our studies include two parts. In the first part, we consider the remote state preparation (RSP) of an entangled state through two equally noisy quantum channel states, namely, a mixture of Bell states. Studies show there is a particular mixed-state channel for which all pure entangled states remain entangled after this inexact RSP. In the second part, we suppose that noises which quantum channels suffer from can be expressed as the Lindblad operators. The master equation of the system can be expressed in the Lindblad form. Through solving the master equation, we calculate the fidelity as a function of decoherence rates and parameters of the state to be prepared. For a given entangled state, we investigate the influence of different types of noises on the fidelity.
In this paper,we propose a scheme in which one can realize remote state preparation(RSP) of an entangled state through entanglement swapping.In ideal conditions where quantum systems about RSP do not suffer unwanted interactions with the outside environment,two two-particle entangled states are used as quantum channels to remotely prepare an entangled state.Furthermore,we investigate RSP of an entangled state through noisy channels.Studies show the fidelity of the process of RSP is different under influences of different types of noises.We also investigate the influence of decoherence rate on remote state preparation for a deterministic state.
We investigate the influence of a noisy environment on the remote preparation of the multi-qubit equatorial state, and specifically deduce the final states and fidelities of the remote preparation of the three-qubit and four-qubit equatorial states under diverse types of noisy environments, namely, amplitude damping, bit flip, phase damping, phase flip, bit-phase flip, depolarization, and non-Markov environments. The results show that when the decoherence factors of the front six noises are equal, the influence degrees of phase damped noise, bit flip noise, phase flip noise, and bit-phase flip noise are similar, while the information loss caused by the amplitude damped noise and depolarizing noise is less. In particular, the bit flip noise and depolarizing noise may have more complex effects on the remote state preparation (RSP) schemes depending on the phase information of the target states, even for the ideal cases where the fidelity values are always 1 for specific phase relations. In the non-Markov environment, owing to the back and forth of information between the environment and systems, fidelities exhibit oscillating behavior and the minimum value may stay greater than zero for a long evolutionary time. These results are expected to have potential applications for understanding and avoiding the influence of noise on remote quantum communication and quantum networks.