An important object appearing in the framework of the Tomita–Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the connection between von Neumann algebras and quantum field theory, von Neumann algebras have become increasingly important for (higher) category theory and topology. We show explicitly how an example of a class of cones discovered by Connes–Araki–Haagerup (CAH), invariant under the modular automorphism group, are related to Wishart laws and information geometry. Given its relation to 2D quantum field theory this highlights new relations between (quantum) information geometry and quantum geometry. Additionally, our results have applications to qubits. The cones and the Wishart laws defined on them carry interesting monoidal structures, revealing the possibility of using a more abstract formalism of monoidal categories. Such a language is interesting as it opens a deeper possibility for generalizations of results.