Private quantum codes: introduction and connection with higher rank numerical ranges

Abstract

We give a brief introduction to private quantum codes, a basic notion in quantum cryptography and key distribution. Private code states are characterized by indistinguishability of their output states under the action of a quantum channel, and we show that higher rank numerical ranges can be used to describe them. We also show how this description arises naturally via conjugate channels and the bridge between quantum error correction and cryptography.

Keywords:

• private quantum code
• quantum cryptography
• completely positive map
• density operator
• higher rank numerical range

AMS Subject Classifications:

• 15A60
• 47A12
• 81P68
• 81P94

Acknowledgments

D.W.K. was supported by NSERC Discovery Grant 400160 and NSERC Discovery Accelerator Supplement 400233. S.P. was supported by an NSERC Doctoral Scholarship.