Quantum Logical Information Theory

Abstract

The transition to density matrices in QM is facilitated by reformulating the ‘classical’ (i.e., non-quantum) results about logical entropy using density matrices. Then the transition to the quantum version of logical entropy is made using the semi-algorithmic procedure of “linearization.” Given a concept applied to sets, apply that concept to the basis set of a vector space and whatever it linearly generates gives the corresponding vector space concept. Classically, the logical entropy $h\left ( f^{-1}\right )$ of the inverse-image partition of $f:U\rightarrow \mathbb {R}$ with point probabilities on U is the application of the product probability p × p measure on U × U to the ditset $\operatorname {dit}\left ( f^{-1}\right )$. That is linearized in quantum logical information theory with qudit spaces replacing the ditsets. Another set of classical definitions start with two probability distributions $p=\left ( p_{1},\ldots ,p_{n}\right )$ and $q=\left ( q_{1},\ldots ,q_{n}\right )$ over the same index set. Since probability distributions are supplied by density matrices, the corresponding quantum logical notions are developed starting with two density matrices ρ and τ. Finally, we show how to make the generalization of Hamming distance and that it in fact ends up being the same as the existing quantum notion of the Hilbert-Schmidt distance between two density matrices—which is the quantum version of the Euclidean distance squared d(p||q) between two probability distributions in ‘classical’ logical information theory.