Higher Rank Matricial Ranges and Hybrid Quantum Error Correction

TL;DR

This paper introduces higher rank matricial ranges motivated by hybrid quantum error correction, establishing conditions for their non-emptiness and implications for hybrid code existence and advantages.

Contribution

It defines and studies higher rank matricial ranges in the context of hybrid quantum error correction, linking operator properties to code existence.

Findings

Bounds on Hilbert space dimension for non-empty matricial ranges

Conditions guaranteeing hybrid code existence

Examples illustrating hybrid code advantages

Abstract

We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty, and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.