A Free Object in Quantum Information Theory

We consider three examples of affine monoids. The first stems from information theory and provides a natural model of image distortion, as well as a higher-dimensional analogue of a binary symmetric channel. The second, from physics, describes the process of teleporting quantum information with a given entangled state. The third is purely a mathematical construction, the free affine monoid over the Klein four group. We prove that all three of these objects are isomorphic.