Abstract passive interferometers with applications to conservative logic

Interferometers with two complex inputs and two complex outputs can be viewed as state machines over a 4-D hyperspace, where the relationship between outputs and inputs are transition functions. In the simplest case, there is full symmetry between inputs and between outputs. The nonzero inputs, the origin and the outputs define a 2-D plane in the 4-D space. The interferometer rotates the input vectors by π/4 in the hyperspace. Cascading the symmetric units defines a simple Hamiltonian path of length 8 in a 2-D subspace. Breaking symmetry by introducing a relative phase shift in the input leads to more complex Hamiltonian paths that can require all four dimensions to describe. The special symmetry breaking caused by a π2 phase shift in one of the input arms leads to outputs which, upon detection, are recognizable as Boolean and other simple operations on properly encrypted and interpreted inputs!