LAUSR

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 > 1 2 . 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 1 2 + 1 2 √ 2 m−1 n . Form = 2 this gives a new bound of p ≤ 1 2 + 1 √ 2n confirming a conjecture by Imamichi and Raymond (AQIS’18). Our bound implies that the previously known analytical constructions of (3,2, 1 2 + 1 √ 6 ), (4,2, 1 2 + 1 2 √ 2 )and (6,2, 1 2 + 1 2 √ 3 )-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 non-negative overlap.