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arXiv:quant-ph/0104012 · Physical Review A

Mutually unbiased binary observable sets onNqubits

Jay Lawrence, Časlav Brukner, Anton Zeilinger

Published Feb 27, 2002• 158 citations
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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.

DOI
10.1103/physreva.65.032320

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One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.
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