Quantum random access code ($n⟶m\text{ qubits}$)

An encoding stores $n$ classical bits in $m$ qubits so that a receiver who later chooses a bit index can recover that bit with a specified success probability. The sender cannot tailor the encoding to the unknown requested index. This task permits approximate recovery of any one bit rather than simultaneous recovery of all bits, and must specify whether additional communication, shared randomness, or entanglement is allowed.

Encode signs $s_j=(-1{)}^{b_j}$ by the unit Bloch vector $(s_1,s_2,s_3)/\sqrt{3}$. Perform the requested Pauli measurement and interpret a positive outcome as zero. Every input and index has success $p_{\ast }$. To prove optimality, write each binary-measurement difference as $D_j=t_{j}I+v_j\cdot \sigma$, where positivity gives $|v_j|\le 1-|t_j|$. Averaging over all eight inputs cancels the $t_j$ terms. For any encoding vectors of norm at most one, the average success is at most $1/2+(1/48){\sum }_s|{\sum }_j{s}_j{v}_j|$. The Cauchy-Schwarz inequality bounds the sum by $8\sqrt{{\sum }_j|v_j{|}^2}\le 8\sqrt{3}$, proving the optimum even for mixed encodings and general binary measurements. The constant-success construction also maximizes worst-case success.