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

TL;DR

This paper establishes new bounds on quantum random access codes using geometric analysis of quantum states in Bloch space, confirming optimality of certain known constructions and extending understanding of quantum encoding limits.

Contribution

It derives a universal upper bound on the success probability of QRACs with shared randomness based on Bloch space geometry, and confirms the optimality of specific known QRACs.

Findings

Bound p ≤ 1/2 + 1/√(2n) for 2-qubit QRACs.

Proved optimality of certain existing QRAC constructions.

Connected quantum state geometry to QRAC success probabilities.

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.