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 ( m12 . 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+12√(2^m-1n) . For m=2 , this gives a new bound of p≤12+1√(2n) confirming a conjecture by Imamichi and Raymond (AQIS’18). Our bound implies that the previously known analytical constructions of (3,2,12+1√(6)) - , (4,2,12+12√(2)) - and (6,2,12+12√(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 nonnegative overlap.