Mutually unbiased binary observable sets onNqubits

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 $4^N-1$ Pauli operators may be partitioned into $2^N+1\mathrm{}$ distinct subsets, each consisting of $2^N-1$ internally commuting observables. Furthermore, each such partitioning defines a unique choice of $2^N+1\mathrm{}$ 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.