Towards Quantifying Complexity with Quantum Mechanics

While we have intuitive notions of structure and complexity, the formalization of this intuition is non-trivial. The statistical complexity is a popular candidate. It is based on the idea that the complexity of a process can be quantified by the complexity of its simplest mathematical model - the model that requires the least past information for optimal future prediction. Here we review how such models, known as $\epsilon$-machines can be further simplified through quantum logic, and explore the resulting consequences for understanding complexity. In particular, we propose a new measure of complexity based on quantum $\epsilon$-machines. We apply this to a simple system undergoing constant thermalization. The resulting quantum measure of complexity aligns more closely with our intuition of how complexity should behave.