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

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 $p>\frac{1}{2}$. 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 $\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2^{m-1}}{n}}$. For $m=2$ this gives a new bound of $p\le \frac{1}{2}+\frac{1}{\sqrt{2n}}$ confirming a conjecture by Imamichi and Raymond (AQIS'18). Our bound implies that the previously known analytical constructions of $(3,2,\frac{1}{2}+\frac{1}{\sqrt{6}})$- , $(4,2,\frac{1}{2}+\frac{1}{2\sqrt{2}})$- and $(6,2,\frac{1}{2}+\frac{1}{2\sqrt{3}})$-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.

arXiv comment14 pages, 2 figures

References in corpus (7)

• Bounding the set of finite dimensional quantum correlations
• Brief History of Quantum Cryptography: A Personal Perspective
• (4,1)-Quantum Random Access Coding Does Not Exist
• Geometry of the set of mixed quantum states: An apophatic approach
• Improved Classical and Quantum Random Access Codes
• Unbounded-error One-way Classical and Quantum Communication Complexity
• A general bound for the dimension of quantum behaviours in the prepare-and-measure scenario

Cited by in corpus (2)

• Quantum Advantages in (n,d)->1 Random Access Codes
• Recursive Quantum Relaxation for Combinatorial Optimization Problems