Family of quantum communication protocols derived from structure of Gosset (E8) polytope

The 8-dimensional Gosset polytope also known as E8 root polytope has 240 vertices and can be associated either with a specific set of states of 8D quantum system expressed in some basis using only real coefficients, or as more general quantum system described by 4D complex space (ququart) . Quantum states are defined up to common multiplier and correspond to rays in projective space. Each vertex of Gosset polytope has a pair with opposite direction and for 8D real case only 120 different states can be considered. For complex case, number of states depends on orientation of polytope defined by orthogonal transformation used before transition to complex coordinates. For example, for 4D regular complex Witting polytope considered earlier for description of quantum cryptographic protocol 240 vertices corresponds to only 40 different states. Such protocol uses 40 different bases appropriate for encoding quantum states. Here is discussed yet another three configurations appropriate for similar purposes: the 8D Gosset polytope itself with 120 states and 2025 bases and two new 4D quantum communication protocols. One of them is very similar with protocol based on Witting polytope already discussed earlier and uses 120 states and 210 bases. Yet another one uses 60 states and 105 bases and does not require any orthogonal transformation before transition to complex coordinates. However, such simplification results less regular structure of the bases separated into two sets with slightly different properties with 15 and 90 bases respectively.