/complex_probabilities

Visualization of complex valued L1 norm probabilites that arise from stochastic matrix transitions. An example outside of quantum mechanics where complex probability spaces are found.

Primary LanguageRust

Theory

Oscillations occur when there are negative eigenvalues in the stochastic transition matrix. The continous extension of each transformation then results in complex valued probabilites. For a stocastic matrix, there is always one eigenvalue of 1, and all others fall between [-1, 1]. For a two state system, a negative eigenvalue results when P(A|A) + P(B|B) < 1, where P(X|X) is the self transition probabilities of each state.

Example for a two state system

Stochastic Matrix
  ┌         ┐
  │ 0.5 0.9 │
  │ 0.5 0.1 │
  └         ┘


Eigenvalues
  ┌      ┐
  │    1 │
  │ -0.4 │
  └      ┘


Eigenvectors
  ┌                     ┐
  │  0.6428571428571428 │
  │ 0.35714285714285715 │
  └                     ┘


  ┌    ┐
  │  1 │
  │ -1 │
  └    ┘

Probability State Trajectory

The red line represents the space of real probabilities where P(A) + P(B) = 1. The green line represents the asymptotic eigenvector corresponding to eigenvalue 1.

Usage

First install rust and cargo. cargo run (-- --help) Outputs to plot.svg.