Abstract
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.
GENERAL SCIENTIFIC SUMMARY Introduction and background. Quantum complementarity, as expressed by the Heisenberg uncertainty relation, for example, states that two observables are maximally complementary if total knowledge about the result of one of them precludes any knowledge about the result of the other. The eigenbases of the corresponding measurement operators are said to be mutually unbiased. Their number remains unknown for d-level quantum systems, where d is not a power of a prime. From a practical perspective, such bases find applications in quantum communication scenarios, e.g. quantum cryptography, and therefore their implementation and physical features are of vital interest, and not only their number.
Main results. We study properties of entanglement, which Einstein called 'spooky action at a distance', present in maximally complementary bases. We show that for a bipartite system with d = a b levels any (hypothetical) set of d+1 mutually unbiased bases contains a fixed quantity of entanglement, being solely a function of a and b. In the case of b = a, with a being a prime, we provide an explicit construction of such sets which uses only one entangling gate applied a suitable number of times to appropriate product states.
Wider implications. The requirement of a fixed overall quantity of entanglement can be regarded as an important hint towards settling the question of what the number of maximally complementary observables is. Also, the fact that by necessity entanglement for such very large systems becomes the rule rather than an exception is expected to be important in many quantum information protocols.