Entanglement in Mutually Unbiased Bases

Higher-dimensional Hilbert spaces are still not fully explored. One issue concerns mutually unbiased bases (MUBs). For primes [1] and their powers (e.g. [2]), full sets of MUBs are known. The question of existence of all MUBs in composite dimensions is still open. We show that for all full sets of MUBs of a given dimension a certain entanglement measure of the bases is constant. This fact could be an argument either for or against the existence of full sets of MUBs in some dimensions and tells us that almost all MUBs are maximally entangled for high-dimensional composite systems, whereas this is not the case for prime dimensions. We present a new construction of MUBs in squared prime dimensions. We use only one entangling operation, which simplifies possible experiments. The construction gives only product states and maximally entangled states. \\[4pt] [1] I. D. Ivanovi\'c, J. Phys. A 14, 3241 (1981). \\[0pt] [2] W. K. Wootters and B. D. Fields, Ann. Phys. (N.Y.) 191, 363 (1989).