The geometry of Bloch space in the context of quantum random access codes

We study the communication protocol known as a quantum random access code (QRAC) which encodes n classical bits into m qubits (m< n) with a probability of recovering any of the initial n bits of at least p>12. Such a code is denoted by (n, m, p)-QRAC. If cooperation is allowed through a shared random string, we call it a QRAC with shared randomness. We prove that for any (n, m, p)-QRAC with shared randomness the parameter p is upper bounded by 12+122m-1n. For m= 2 , this gives a new bound of p≤12+12n confirming a conjecture by Imamichi and Raymond (AQIS’18). Our bound implies that the previously known analytical constructions of (3,2,12+16)- , (4,2,12+122)- and (6,2,12+123)-QRACs are optimal. To obtain our bound, we investigate the geometry of quantum states in the Bloch vector representation and make use of a geometric interpretation of the fact that any two quantum states have a nonnegative overlap.