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

• URL: http://arxiv.org/abs/2106.00155v2
• Date: Thu, 24 Feb 2022 13:06:18 GMT
• Title: The geometry of Bloch space in the context of quantum random access codes
• Authors: Laura Man\v{c}inska and Sigurd A. L. Storgaard
• Abstract summary: We study the communication protocol known as a Quantum Random Access Code (QRAC) We prove that for any $(n,m,p)$-QRAC with shared randomness the parameter $p$ is upper bounded by $ tfrac12+tfrac12sqrttfrac2m-1n$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $p>\tfrac{1}{2}$. 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 $ \tfrac{1}{2}+\tfrac{1}{2}\sqrt{\tfrac{2^{m-1}}{n}}$. For $m=2$ this gives a new bound of $p\le \tfrac{1}{2}+\tfrac{1}{\sqrt{2n}}$ confirming a conjecture by Imamichi and Raymond (AQIS'18). Our bound implies that the previously known analytical constructions of $(3,2,\tfrac{1}{2}+\tfrac{1}{\sqrt{6}})$- , $(4,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{2}})$- and $(6,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{3}})$-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.

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