Mutually unbiased phase states, phase uncertainties, and Gauss sums

Abstract

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/ 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters.Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspects.2

Hjelmslev geometry of mutually unbiased bases

“…This concept plays a key role in a search for a rigorous formulation of quantum complementarity and lends itself to numerous applications in quantum information theory. It is a well-known fact (see, e.g., [1]- [9] and references therein) that H q supports at most q + 1 pairwise mutually unbiased bases (MUBs) and various algebraic geometrical constructions of such q + 1, or complete, sets of MUBs have been found when q = p r , with p being a prime and r a positive integer. In our recent paper [10] we have demonstrated that the bases of such a set can be viewed as points of a proper conic (or, more generally, of an oval) in a projective plane of order q.…”

Practical implementation of mutually unbiased bases using quantum circuits

“…Different explicit constructions of MUBs in prime power dimensions have been suggested in a number of recent papers [49][50][51][52][53][54][55]. We follow here the approach established in Refs.…”

Optimal reconstruction of the states in qutrit systems

“…As shown in Ref [44], the only MUB structure for a two-qutrit system is (4,6), where 4 is the number of the separable bases and 6 is the number of the bipartite entangled bases. However in three-qutrit case, there are five sets of MUBs with different structures, namely {(0, 12, 16), (1,9,18), (2,6,20), (3,3,22), (4,0,24)} [21,44]. It is easy to see that the (0, 12, 16) set of MUBs has the minimum physical complexity.…”