Deterministic Joint Remote Preparation of an Arbitrary Qubit via Einstein-Podolsky-Rosen Pairs

Abstract

Joint remote state preparation is a secure and faithful method based on local operation and classical communication to transmit quantum states without the risk of full information leaking to either of the participants. In this work, we propose a new deterministic protocol for two parties to remotely prepare an arbitrary single-qubit state for a third party using two Einstein-Podolsky-Rosen pairs as the nonlocal resource. We figure out the advantages as well as the disadvantages of this new protocol in comparison with others, showing in general that the proposed protocol is superior to the existing ones. We also describe the situation when there are more than two preparers.

Appendix

In this Appendix we generalize the situation with two preparers (Alice and Bob) described in the main text to that with an arbitrary N>2 preparers (Alice, Bob 1, Bob 2, … and Bob N−1). The pre-shared quantum channel consists of N EPR pairs distributed among the N+1 participants (N preparers plus a receiver) as shown in Fig. 3.

Fig. 3

The qubits’ distribution for JRSP of the most general single-qubit state via N EPR pairs for the situation of N preparers (Alice, Bob 1, Bob 2, … and Bob N−1). Qubits are represented by dots and entangled qubits are connected by a solid line

For clarity, let us first consider N=3 in detail. The three EPR pairs of the quantum channel are $\vert \mathit{EPR} \rangle_{A_{1}B_{1}}\vert \mathit{EPR} \rangle_{A_{2}B_{2}}\vert \mathit{EPR} \rangle_{A_{3}C}=\vert Q \rangle_{A_{1}B_{1}A_{2}B_{2}A_{3}C}$, with qubits A ₁,A ₂,A ₃ hold by Alice, B ₁ by Bob 1, B ₂ by Bob 2 and C by the receiver Charlie. While Alice is allowed to know {a,b} as in the case of N=2, Bob 1 and Bob 2 now share the knowledge of φ in the following way: Bob 1 knows φ ₁ and Bob 2 knows φ ₂ where φ ₁ and φ ₂ sum up to φ.

First, Alice measures her three qubits in the basis $\{\vert u_{klm} \rangle_{A_{1}A_{2}A_{3}};k,l,m\in\{0,1\}\}$:

$$\left( \begin{array}{c} \vert u_{000}\rangle_{A_{1}A_{2}A_{3}}\\\vert u_{001} \rangle_{A_{1}A_{2}A_{3}}\\\vert u_{010} \rangle_{A_{1}A_{2}A_{3}}\\\vert u_{011} \rangle_{A_{1}A_{2}A_{3}}\\\vert u_{100} \rangle_{A_{1}A_{2}A_{3}}\\\vert u_{101} \rangle_{A_{1}A_{2}A_{3}}\\\vert u_{110} \rangle_{A_{1}A_{2}A_{3}}\\\vert u_{111} \rangle_{A_{1}A_{2}A_{3}}\end{array}\right) = \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} a & 0 & 0 &0 & 0 & 0 & 0 & b\\b & 0 & 0 & 0 & 0 & 0 & 0 & -a\\0 & a & 0 & 0 & 0 & 0 & b & 0\\0 & b & 0 & 0 & 0 & 0 & -a & 0\\0 & 0 & a & 0 & 0 & b & 0 & 0\\0 & 0 & b & 0 & 0 & -a & 0 & 0\\0 & 0 & 0 & a & b & 0 & 0 & 0\\0 & 0 & 0 & -b & a & 0 & 0 & 0\end{array}\right) \left( \begin{array} {c} \vert 000\rangle_{A_{1}A_{2}A_{3}}\\\vert 001 \rangle_{A_{1}A_{2}A_{3}}\\\vert 010 \rangle_{A_{1}A_{2}A_{3}}\\\vert 011 \rangle_{A_{1}A_{2}A_{3}}\\\vert 100 \rangle_{A_{1}A_{2}A_{3}}\\\vert 101 \rangle_{A_{1}A_{2}A_{3}}\\\vert 110 \rangle_{A_{1}A_{2}A_{3}}\\\vert 111 \rangle_{A_{1}A_{2}A_{3}}\end{array} \right) $$

(24)

and publicly broadcasts klm, if she finds state $\vert u_{klm} \rangle_{A_{1}A_{2}A_{3}}$, projecting qubits B ₁, B ₂ and C onto an entangled state $\vert L_{klm} \rangle_{B_{1}B_{2}C}$. Expressing the quantum channel $\vert Q \rangle_{A_{1}B_{1}A_{2}B_{2}A_{3}C}$ through $\{\vert u_{klm} \rangle_{A_{1}A_{2}A_{3}}\}$,

$$\vert Q \rangle_{A_{1}B_{1}A_{2}B_{2}A_{3}C}=\frac{1}{2\sqrt{2}} \sum _{m=0}^{1}\sum_{l=0}^{1}\sum_{k=0}^{1}\vert u_{klm}\rangle_{A_{1}A_{2}A_{3}}\vert L_{klm} \rangle_{B_{1}B_{2}C}, $$

(25)

we derive $\vert L_{klm} \rangle_{B_{1}B_{2}C}$ in the form

(26)

(27)

(28)

(29)

(30)

(31)

(32)

and

$$\vert L_{111} \rangle_{B_{1}B_{2}C}=a\vert 100\rangle_{B_{1}B_{2}C}-b\vert 011 \rangle_{B_{1}B_{2}C}.$$

(33)

Next, Bob 1 and Bob 2 independently measure their qubits in a basis conditioned not only on φ ₁ and φ ₂ but also on Alice’s outcome. Concretely, when klm=000 or 010, each Bob j (j=1,2) uses a measurement basis determined by

$$\left( \begin{array}{c} \vert v_{0}\rangle_{B_{j}}\\\vert v_{1} \rangle_{B_{j}}\end{array}\right) =V^{(0)}(\varphi_{j}) \left( \begin{array}{c}\vert 0 \rangle_{B_{j}}\\\vert 1 \rangle_{B_{j}}\end{array}\right) ,$$

(34)

with V ⁽⁰⁾(φ _j ) given in (19). However, for klm=001 or 011, the basis for each Bob j is

$$\left( \begin{array}{c} \vert v_{0}\rangle_{B_{j}}\\\vert v_{1} \rangle_{Bj}\end{array}\right) =V^{(1)}(\varphi_{j}) \left( \begin{array}{c} \vert 0 \rangle_{B_{j}}\\\vert 1 \rangle_{B_{j}}\end{array} \right) ,$$

(35)

with V ⁽¹⁾(φ _j ) given in (19). In case klm=100 or 110, Bob 1 uses the basis

$$\left( \begin{array}{c} \vert v_{0}\rangle_{B_{1}}\\\vert v_{1} \rangle_{B_{1}}\end{array}\right) =V^{(0)}(\varphi_{1}) \left( \begin{array}{c} \vert 0 \rangle_{B_{1}}\\\vert 1 \rangle_{B_{1}}\end{array} \right) ,$$

(36)

but the basis for Bob 2 is

$$\left( \begin{array}{c} \vert v_{0}\rangle_{B_{2}}\\\vert v_{1} \rangle_{B_{2}}\end{array}\right) =V^{(1)}(\varphi_{2}) \left( \begin{array}{c} \vert 0 \rangle_{B_{2}}\\\vert 1 \rangle_{B_{2}}\end{array} \right) .$$

(37)

Finally, if klm=101 or 111, the bases for Bob 1 and Bob 2 are differently defined as

$$\left( \begin{array}{c} \vert v_{0}\rangle_{B_{1}}\\\vert v_{1} \rangle_{B_{1}}\end{array}\right) =V^{(1)}(\varphi_{1}) \left( \begin{array}{c} \vert 0 \rangle_{B_{1}}\\\vert 1 \rangle_{B_{1}}\end{array} \right)$$

(38)

and

$$\left( \begin{array} {c} \vert v_{0}\rangle_{B_{2}}\\\vert v_{1} \rangle_{B_{2}}\end{array}\right) =V^{(0)}(\varphi_{2}) \left( \begin{array}{c} \vert 0 \rangle_{B_{2}}\\\vert 1 \rangle_{B_{2}}\end{array} \right) ,$$

(39)

respectively. Then the states $\vert L_{klm} \rangle_{B_{1}B_{2}C}$ in (25) can be, up to an unimportant global phase factor, decomposed in terms of $\{\vert v_{n} \rangle_{B_{1}},\vert v_{s} \rangle_{B_{2}}\}$ as follows

$$\vert L_{klm} \rangle_{B_{1}B_{2}C}=\frac{1}{2}\sum _{s=0}^{1}\sum _{n=0}^{1}\vert v_{n}\rangle_{B_{1}}\vert v_{s} \rangle_{B_{2}}R_{klmns}^{+}\vert \Psi \rangle_{C}, $$

(40)

with R _klmns either the identity operator or a Pauli one. This implies that after Alice, Bob 1 and Bob 2 announce their measurement outcomes, Charlie will be able to convert the qubit C to be in the desired state |Ψ〉 _C by applying R _klmns on C. The reconstruction operators R _klmns that Charlie needs in the last step are shown in Table 2. Because each of the 2⁵=32 possible outcomes klmns is associated with a reconstruction operator R _klmns , the total success probability is obviously 1.

Table 2 The reconstruction operator R _klmns , conditioned on the measurement outcomes klm, n and s of Alice, Bob 1 and Bob 2, respectively. I is the identity operator, X={{0,1},{1,0}} the bit-flip operator and Z={{1,0},{0,−1}} the phase-flip one

As inferred from the cases of N=2 and N=3, for any N>3 the quantum channel should consist of N EPR pairs, $\vert Q \rangle_{A_{1}B_{1}A_{2}B_{2}\ldots A_{N-1}B_{N-1}A_{N}C}=\vert \mathit{EPR} \rangle_{A_{1}B_{1}}\vert \mathit{EPR}\rangle_{A_{2}B_{2}}\ldots \vert \mathit{EPR}\rangle_{A_{N-1}B_{N-1}}\vert \mathit{EPR} \rangle_{A_{N}C}$ (see Fig. 3). The information is split in such a way that Alice still knows {a,b}, but Bob j (j=1,2,…,N−1) just knows φ _j in such a way that the constraint $\sum_{j=1}^{N-1}\varphi_{j}=\varphi$ is satisfied. The basis for Alice to measure N qubits A ₁, A ₂,…,A _N is spanned by 2^N orthonormal states $\{(-1)^{i_{1}}a\vert i_{1}i_{2}\ldots i_{N} \rangle_{A_{1}A_{2}\ldots A_{N}}+b\vert \overline {i_{1}}\overline{i_{2}}\ldots \overline{i_{N}} \rangle_{A_{1}A_{2}\ldots A_{N}}; i_{n}\in\{0,1\}; \overline{i_{n}}=1-i_{n}\}$. As for Bob j, each of them independently measures qubit B _j in the basis $V^{(0)}(\varphi_{j}).\{\vert 0\rangle_{B_{j}},\vert 1 \rangle_{B_{j}}\}$ or $V^{(1)}(\varphi_{j}).\{\vert 0 \rangle_{B_{j}},\vert 1\rangle_{B_{j}}\}$, depending on Alice’s outcome. Such generalized JRSP protocol works deterministically since for each of the 2^2N−1 possible measurement outcomes of the N preparers there exists a corresponding reconstruction operator for the receiver to obtain the target state |Ψ〉 _C .