Mutually unbiased binary observable sets on<i>N</i>qubits

Abstract

The Pauli operators (tensor products of Pauli matrices) provide a complete basis of operators on the Hilbert space of N qubits. We prove that the set of ${4}^{N}\ensuremath{-}1$ Pauli operators may be partitioned into ${2}^{N}+1$ distinct subsets, each consisting of ${2}^{N}\ensuremath{-}1$ internally commuting observables. Furthermore, each such partitioning defines a unique choice of ${2}^{N}+1$ mutually unbiased basis sets in the N-qubit Hilbert space. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.