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.
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.
In this chapter we will discuss the notion of a “dynamical correspondence”, which is applicable in both JB and JBW contexts. This generalizes the correspondence of observables and generators of one-parameter groups of automorphisms in quantum mechanics. It is closely related to Connes’ concept of orientation [36, Definition 4.11] that he used as a key property in his characterization of the natural self-dual cones associated with von Neumann algebras (i.e., the cones P ξ ♮ of Tomita—Takesaki theory). Both can be thought of as ways to specify possible Lie structures compatible with a given Jordan structure.
We present a unified framework for the study of Wishart matrices (W_p(n,Σ)), which generalize the chi-squared distribution to matrix-variate settings and model the covariance structure of multivariate Gaussian data. After recalling their defining properties - additivity under independent summation (W_1 + W_2 \sim W_p(n_1+n_2,Σ)), equivariance under linear maps (A W A^T \sim W_q(n,AΣA^T)), and their role as sample covariance matrices - we embed the positive-definite cone (S_p^+) within Monge-Ampere geometry. Here (S_p^+) acquires a Hessian manifold structure with affine-invariant metric and volume form (ω= \det(Σ)^{-(p+1)/2},dΣ), under which the Wishart density acts as a soliton of natural geometric flows. We then show that the collection of Wishart distributions forms a symmetric monoidal category (\mathcal{W}), whose objects are (W_p(n,Σ)) and whose morphisms are linear maps (A:\mathbb{R}^p\to\mathbb{R}^q). The tensor product encodes block-diagonal coupling, with braiding given by block permutation, and the axioms enforce Monge-Ampere functoriality, additivity, and convex duality via the Legendre transform. Applications to quantum error correction are discussed: Wishart laws model correlated noise, Wasserstein geodesics optimize error-mitigation cost, tensor structure captures independent error channels, and Legendre duality underpins entropy-driven decoding.
We develop a cohomological framework for entanglement entropy that unifies information theory, operator algebras, and noncommutative geometry. Entropy is a 1-cocycle in information cohomology; we generalize this to the quantum setting via Hochschild and cyclic cohomology. For an inclusion of von Neumann algebras N contained in M with a faithful normal state w admitting a cyclic and separating vector, and a conditional expectation E: M -> N (whose existence is equivalent to modular invariance of N by Takesaki's theorem), we construct the entanglement complex as the image of the transgression map tau_w on the relative cyclic complex, where A is a dense *-subalgebra of M and B = A intersect N. Our main result identifies entanglement cohomology with the kernel of the restriction map HC*(A) -> HC*(B), up to the connecting homomorphism d of the long exact sequence of the pair (A,B); when d = 0, the identification is exact. The relative modular cocycle u_t = (Dw : D(w|_N) o E)_t generates explicit entanglement classes when E is not w-invariant, with the Araki relative entropy as its infinitesimal generator. The framework accommodates Type III von Neumann algebras, where no local density matrix exists, and leads to a holographic proposal: a subspace of H^1_ent is conjecturally dual to linearized modular Hamiltonian perturbations of the bulk entanglement wedge. We verify the construction in finite-dimensional, type II_1, and type III_1 examples.
The quantum entanglement measure of concurrence is shown to be directly calculable from a Tomita- Takesaki modular operator framework constructed from the local von Neumann algebras of observables for two quantum systems. Specifically, the Tomita-Takesaki modular conjugation operator J that links two separate systems with respect to their von Neumann algebras is related to the quantum concurrence C of a pure bivariate entangled state composed from these systems. This concurrence relation provides a direct physical meaning to J as both a symmetry operator and a quantitative measure of entanglement. This procedure is then demonstrated for a supersymmetric quantum mechanical system and a real scalar field interacting with two entangled spin-21 Unruh-DeWitt qubit detectors. For the latter system, the concurrence result is shown to be consistent with some known results on the Bell-CHSH inequality for such a system.
We present a unified framework for the study of Wishart matrices WpnΣ, which generalize the chi-squared distribution to matrix-variate settings and model the covariance structure of multivariate Gaussian data. After recalling their defining properties—additivity under independent summation W1+W2∼Wpn1+n2Σ, equivariance under linear maps AWAT∼WqnAΣAT, and their role as sample covariance matrices—we embed the positive-definite cone Sp+ within Monge-Ampère geometry. Here, Sp+ acquires a Hessian manifold structure with affine-invariant metric and volume form ω=detΣ−p+12dΣ, under which the Wishart density acts as a soliton of natural geometric flows. We then show that the collection of Wishart distributions forms a symmetric monoidal category W, where objects are WpnΣ and whose morphisms are linear maps A:ℝp→ℝq. The tensor product encodes block-diagonal coupling, with braiding given by block permutation, and axioms enforcing Monge-Ampère functoriality, additivity, and convex duality via the Legendre transform. Applications to quantum error correction are discussed: Wishart laws model correlated noise, Wasserstein geodesics optimize error-mitigation cost, tensor structure captures independent error channels, and Legendre duality underpins entropy-driven decoding.
The Tomita-Takesaki modular operator for local algebras plays an important role in quantum field theory, and more recently in the study of relative entropy. However, the explicit expression of this operator, except for the case of wedges, is difficult to describe mathematically. We have obtained instead numerical results for the form of the modular Hamiltonian for a double cone in a massive scalar free field in (1+1)- and (3+1)-dimensional Minkowski space, which shows how it differs from the wedge case, in particular regarding the dependence of the modular Hamiltonian on the mass of the field.