Entanglement in mutually unbiased bases

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  • 10 February 2011
Abstract
One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon is most strongly manifested when observables are related to mutually unbiased bases. In this article 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, independently 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 experimentally-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we show basic properties of product mutually unbiased bases and their implications for entanglement of states when building the remaining bases. The constructions are illustrated with explicit examples in low dimensions. We believe that properties of entanglement in mutually unbiased bases might be one of the ingredients to settle the question of existence of the complete sets. We also expect that they will be relevant in applications of bases in the experimental realization of quantum protocols in higher dimensional Hilbert spaces.

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