Mutually Unbiased Bases in 4, 8, and 16 Dimensions Generated by Means of Controlled-Phase Gates With Application to Entangled-Photon QKD Protocols

We design mutually unbiased bases (MUBs) in 4, 8, and 16 dimensions, which are compatible with quadrature phase-shift keying (QPSK) optical transmission, based on novel insights into the structure of the MUB set, viewed as the modulation of a Hadamard matrix with a set of phase masks. The new protocol is based on the generation of a pair of MUBs as tensor products of the lowest dimension (2-D) MUBs, followed by novel unbiasing transformations, which operate on pairs of MUBs and make them mutually unbiased among the pairs. We discovered that the unbiasing transformations consist of controlled-phase quantum gates acting on multiple qubits. We further conceive an abstract BB84-like higher order quantum key distribution MUB-based protocol, based on multiple entangled photons.