Multiaccess quantum communication and product higher rank numerical range

TL;DR

This paper introduces the concept of product higher rank numerical range, explores its properties, and applies it to quantum error correction code construction for multiaccess channels, providing analytical tools and reverse engineering approaches.

Contribution

It is the first study of product higher rank numerical range, linking it to quantum error correction and developing methods for its analysis and application.

Findings

Defined and analyzed properties of product higher rank numerical range.

Applied the concept to construct codes for bi-unitary two-access channels.

Developed techniques for bounding and determining the shape of the range.

Abstract

In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.--D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77 (2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for studying construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi--unitary two--access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.