Optimal remote preparation of arbitrary multi-qubit real-parameter states via two-qubit entangled states

Abstract

In this paper, we present an efficient scheme for remote state preparation of arbitrary n-qubit states with real coefficients. Quantum channel is composed of n maximally two-qubit entangled states, and several appropriate mutually orthogonal bases including the real parameters of prepared states are delicately constructed without the introduction of auxiliary particles. It is noted that the successful probability is 100% by using our proposal under the condition that the parameters of prepared states are all real. Compared to general states, the probability of our protocol is improved at the cost of the information reduction in the transmitted state.

Appendix A

Suppose that quantum channel for our proposal is composed of n maximally two-qubit entangled states as follow

$$\begin{aligned} |\Psi \rangle _{jk}=\frac{1}{\sqrt{2}}(|01\rangle +|10\rangle )_{jk} \quad j=1,3,\ldots ,2n-1;~k=j+1.\nonumber \end{aligned}$$

Without loss of generality, the sender Alice and the receiver Bob have particles j and k, respectively. From Eq. (3), we can obtain that the $2^n\times 2^n$ unitary operation $U\left[ \Theta ^n_n\right]$. Thus, the whole system composed of n maximally entangled states could be given by

$$\begin{aligned} |\Psi \rangle _{12} \otimes |\Psi \rangle _{34}\cdots |\Psi \rangle _{(2n-1)2n}=\left( \frac{1}{\sqrt{2}} \right) ^n\cdot \sum ^{2^n-1}_{i=0}~~{|\Omega _i\rangle _{13\ldots 2n-1}\otimes |\Omega _i\rangle _{24\ldots 2n}}&\end{aligned}$$

here $|\Omega _i\rangle$ can be expressed as

$$\begin{aligned}{}[~|\Omega _0\rangle ,~|\Omega _1\rangle \cdots ~|\Omega _{2^n-1}\rangle ]^T=U\left[ \Theta ^n_n\right] ~[~|1\ldots 11\rangle ,~|1\ldots 10\rangle \cdots ~|0\ldots 00\rangle ]^T \end{aligned}$$

It can be found that if the measurement result is $|\Omega _i\rangle ~(i=0,1,\ldots ,2^n-1)$, particles $(2,4\ldots 2n)$ would collapse into $|\Omega _i\rangle$. Thus, the original state in Eq. (1) can be reconstructed from $|\Omega _i\rangle$ by performing some permutation operations on particles $(2,4\ldots 2n)$. The total successful probability of this scheme is equal to one. Similar to $\ \frac{1}{\sqrt{2}}(|01\rangle +|10\rangle )$ and $\frac{1}{\sqrt{2}}(|00\rangle +|10\rangle )$, others of Bell states can be used to prepare multi-qubit real-parameter states in a deterministic manner.