Mutually unbiased bases with free parameters

作者：Goyeneche Dardo^*; Gomez Santiago

来源：Physical Review A, 2015, 92(6): 062325.

DOI：10.1103/PhysRevA.92.062325

We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of m real mutually unbiased bases existing in dimension N > 2 admits the introduction of (m - 1)N/2 free parameters that cannot be absorbed by a global unitary operation. As consequence, there are m = k + 1 mutually unbiased bases in every dimension N = k(2) with k(3)/2 free parameters, where k is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.

• 出版日期2015-12-14