TY - JOUR

T1 - The geometry of Bloch space in the context of quantum random access codes

AU - Mančinska, Laura

AU - Storgaard, Sigurd A.L.

N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Bloch vector representation

KW - Geometry of Bloch space

KW - Optimality of success probability

KW - Quantum random access codes

UR - http://www.scopus.com/inward/record.url?scp=85127570872&partnerID=8YFLogxK

U2 - 10.1007/s11128-022-03470-4

DO - 10.1007/s11128-022-03470-4

M3 - Journal article

AN - SCOPUS:85127570872

VL - 21

SP - 1

EP - 16

JO - Quantum Information Processing

JF - Quantum Information Processing

SN - 1570-0755

IS - 4

M1 - 143

ER -