We present a case study of quantum-like probabilities that are motivated by quantum cognition. We introduce quantum-like evolution that is l2 norm preserving but in which the matrix does not need to be unitary. We show how to map any 2 × 2 stochastic matrix to an l2 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.

中文翻译：

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类量子进化原理

我们提出了一个由量子认知激发的类量子概率的案例研究。我们引入了类量子进化，即 l2范数保留，但矩阵不需要是单一的。我们展示了如何将任何 2 × 2 随机矩阵映射到一个 l2范数保持平衡相位矩阵，将长度为 1 的实向量映射为长度为 1 的复向量。类量子进化可以模拟一个开放系统的概率分布，其中算子不是单一的而是规范保持的。这种行为在量子认知中得到了研究。通过张量积可以构建更高维的平衡相位矩阵。类量子演化可以通过将输入向量的相位编码为平衡相位矩阵的相位、马尔可夫链或可导致固定、周期性或混沌行为导致奇怪振荡的替代演化来模拟酉开路演化。