Principles of Quantum-like Evolution

We present a case study of quantum-like probabilities that are motivated by quantum cognition. We introduce quantum-like evolution that is l(2) norm preserving but in which the matrix does not need to be unitary. We show how to map any 2 x 2 stochastic matrix to an l(2) norm preserving balanced phase matrix that maps real vectors of length one into complex vectors of length one. Quantum-like evolution can simulate a probability distribution of open system in which the operator is not unitary but norm preserving. Such a kind of behaviour is studied in quantum cognition. By tensor product higher dimensional balanced phase matrices can be built. Quantum-like evolution can simulate either unitary open one by coding the phase of input vector into the phase of a balanced phase matrix, a Markov chain or an alternative evolution that can lead to fixed, periodic or chaotic behaviour resulting in strange oscillations.