Fig. 1: Remote state preparation (RSP) in a quantum network.

a Two distant nodes, Alice and Bob, first use quantum repeaters (QRs) to share Einstein–Podolsky–Rosen (EPR) entangled pairs for long-distance communication. b To prepare Bob’s remote state: \(\left\vert {{{\bf{s}}}}\right\rangle =U\left\vert {{{{\bf{s}}}}}_{0}\right\rangle\), as shown on the equatorial of the Bloch sphere³⁶, Alice performs the operation U^† and then measures her qubit in the basis \(\{\left\vert {{{{\bf{s}}}}}_{0}\right\rangle ,\left\vert {{{{\bf{s}}}}}_{0}^{\perp }\right\rangle \}\). Here, the states \({\left\vert 0\right\rangle }_{1}\), \({\left\vert 0\right\rangle }_{2}\), and \({\left\vert 0\right\rangle }_{3}\) on the Bloch sphere denote the eigenvectors of the Pauli-X, Pauli-Y, and Pauli-Z observables, respectively, with the eigenvalue 1. If Alice obtains the measurement outcome \(\left\vert {{{{\bf{s}}}}}_{0}^{\perp }\right\rangle\) (\(\left\vert {{{{\bf{s}}}}}_{0}\right\rangle\)), she sends Bob the message 1 ( − 1) via a classical channel to show that his correction unitary operator, \(\hat{\pi }\), is unnecessary (necessary). Here, the identity operator I represents doing nothing. The RSP process can be considered a quantum operation, Eq. (3), transforming the initial state \(\left\vert {{{{\bf{s}}}}}_{0}\right\rangle\) to the final state \(\left\vert {{{\bf{s}}}}\right\rangle\), corresponding to the rotation from the Bloch vector \({\overrightarrow{{{{\bf{s}}}}}}_{0}\) to the final one, \(\overrightarrow{{{{\bf{r}}}}}\). c With an extended quantum channel or quantum network (\({{{{\mathscr{E}}}}}_{{{{\rm{QN}}}}}\)) consisting of teleportation and (or) QR, Bob can send the prepared state \(\left\vert {{{\bf{s}}}}\right\rangle\) to a third distant end node, Charlie.