Joint higher rank numerical range of Pauli group

Abstract

For a noisy quantum channel, a quantum error correction code of dimension k exists if and only if the joint rank- numerical range associated with the error operators of the channel is non-empty. In this paper, the joint rank- numerical range of an -tuple of elements in the -qubit Pauli group are discussed. Let be a Hermitian triple in the -qubit Pauli group. We state a constructive method to obtain the joint rank- numerical range of . Moreover, we show that .

Keywords:

• joint higher rank numerical range
• Pauli group
• quantum error correction
• tensor product

AMS Subject Classifications:

• 15A60
• 81P45
• 15A69
• 15A90