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

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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.

TidsskriftQuantum Information Processing
Udgave nummer4
Sider (fra-til)1-16
StatusUdgivet - 2022

Bibliografisk note

Funding Information:
This paper is based on S. Storgaard’s bachelor’s thesis. L. Mančinska acknowledges support by Villum Fonden via the QMATH Centre of Excellence (Grant No. 10059) and Villum Young Investigator grant (No. 37532).

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

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