Research problems on numerical ranges in quantum computing

Abstract

We describe some instances in quantum information processing where numerical range techniques arise. We focus on two basic settings: higher-rank numerical ranges and their relevance in theoretical quantum error correction, and the classical numerical range and its use for comparing quantum information processing operations. We present the basic theory, discuss examples and formulate open problems.

Keywords:

• numerical range
• higher-rank numerical range
• quantum information processing
• quantum error correction
• gate fidelity

AMS Subject Classifications:

• 15A60
• 15A90
• 47N50
• 81P68

Acknowledgements

D.W. Kribs is grateful for several interesting conversations with participants of WONRA08, and for helpful conversations with M.B. Ruskai. D.W. Kribs was partially supported by NSERC grant 400160, by NSERC Discovery Accelerator Supplement 400233, and by Ontario Early Researcher Award 048142. A. Pasieka was partially supported by an Ontario Graduate Scholarship. M.P. da Silva and C. Ryan were partially supported by NSERC, M. Laforest was partially supported by NSERC and FQRNT.