Higher rank matricial ranges and hybrid quantum error correction

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.

KEYWORDS:

• Numerical range
• matricial range
• joint higher rank matricial ranges
• quantum channel
• quantum error correction
• hybrid codes
• operator algebra

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 47L90
• 81P45
• 81P70
• 94A40
• 47A12
• 15A60
• 81P68

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

D.W.K. and B.Z. was supported by Natural Sciences and Engineering Research Council of Canada (NSERC). C.K.L. is an affiliate member of the Institute for Quantum Computing, University of Waterloo. His research was supported by USA NSF [grant number DMS 1331021] , and Simons Foundation [grant number 351047]. M.N. was supported by Mitacs and the African Institute for Mathematical Sciences.