Compounds of symmetric informationally complete measurements and their application in quantum key distribution

TL;DR

This paper introduces a new complex structure called SIC-compound, constructed from symmetric informationally complete measurements, and demonstrates its application in enhancing quantum key distribution security.

Contribution

The paper constructs an explicit example of SIC-compounds in four dimensions and explores their relation to mutually unbiased bases for quantum state discrimination.

Findings

Explicit construction of SIC-compound in 4D

SIC-compounds enable secure quantum key distribution with high error tolerance

Relation established between SIC-compounds and mutually unbiased bases

Abstract

Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is defined to be a collection of $d^3$ vectors in $d$-dimensional Hilbert space that can be partitioned in two different ways: into $d$ SICs and into $d^2$ orthonormal bases. While a priori their existence may appear unlikely when $d>2$, we surprisingly answer it in the positive through an explicit construction for $d=4$. Remarkably this SIC-compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC-compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalisation of the six-state protocol.