Remote state preparation

Remote state preparation uses a shared entangled state, local quantum measurements and classical communication to prepare a state known to the sender at the receiver. The sender can tailor the measurement to the desired state, so restricted state families can require less communication than quantum teleportation. A real great-circle family of qubit states admits one-bit remote preparation of real qubit states.

Prepare independent candidate blocks by one-bit heralding of a remote state preparation block, but send only the first successful block's index. Success eventually occurs with probability one; a prefix code for a rare geometric success uses at most $n+e/(e-1)$ expected bits per block. The expected singlet state consumption is $n{2}^n$. Alternatively, limiting the number of trials to $n{2}^n$ gives failure probability at most $e^{-n}$ and a fixed message length $n+{\log }_{2}n+O(1)$. Both versions approach one classical bit per remotely prepared qubit in their respective expected-length or high-success senses.

// Target: quantum-theory.bigb

For a target pure state $|\psi ⟩$ of a qubit, choose its orthogonal complement so thatMeasuring the sender's half of this singlet state in that basis leaves the receiver in $|\psi ⟩$ when the sender obtains $|{\psi }^{\perp }⟩$. The outcome has probability one half. One classical success flag heralds the exact preparation without conveying the target's classical description.

// Target: quantum-theory.bigb

Apply heralded remote preparation of an arbitrary qubit independently to $n$ shared singlet states. The sender reports one bit indicating whether every local outcome was successful. A positive flag certifies that the receiver's block is the desired product of $n$ pure states; its probability is $2^{-n}$. A negative flag need not identify which states failed, since the whole block can be discarded.

// Target: algebra.bigb

Alice holds the first two qubits and Bob the third of a three-vertex graph-state wire. Alice measures at $\alpha$ and $(-1{)}^r\beta$, with outcomes $r,s$, and sends these two bits. Bob applies $Z^r{X}^s$, the inverse of the known Pauli frame $X^s{Z}^r$, obtaining $U(\beta )U(\alpha )|+⟩$ up to global phase. Bob need not know the angles. Every branch succeeds, with branch probability $1/4$, and no postselection is needed. Alice's measurements and Bob's corrections constitute local operations and classical communication on the previously shared entangled resource.

For $|{\alpha }_{\theta }⟩=\cos \theta |0⟩+\sin \theta |1⟩$, Alice measures her half of a Bell state in the real basis $\{|{\alpha }_{\theta }⟩,|{\beta }_{\theta }⟩\}$, with $|{\beta }_{\theta }⟩=-\sin \theta |0⟩+\cos \theta |1⟩$. Bob receives one outcome bit and applies $I$ or $iY$. Since $iY|{\beta }_{\theta }⟩=|{\alpha }_{\theta }⟩$ for every $\theta$, the preparation is exact and the correction is independent of the unknown parameter.