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arXiv:2608.22604

Wishart Matrices and Quantum Geometry: Foundations and Applications in Quantum Information

Noémie C. Combe

Published Aug 23, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2608.22604

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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 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.
With this follow-up paper, we continue developing a mathematical framework based on information geometry for representing physical objects. The long-term goal is to lay down informational foundations for physics, especially quantum physics. We assume that we can now model information sources as univariate normal probability distributions N (μ, σ0), as before, but with a constant σ0 not necessarily equal to 1. Then, we also relaxed the independence condition when modeling m sources of information. Now, we model m sources with a multivariate normal probability distribution Nm(μ,Σ0) with a constant variance–covariance matrix Σ0 not necessarily diagonal, i.e., with covariance values different to 0, which leads to the concept of modes rather than sources. Invoking Schrödinger’s equation, we can still break the information into m quantum harmonic oscillators, one for each mode, and with energy levels independent of the values of σ0, altogether leading to the concept of “intrinsic”. Similarly, as in our previous work with the estimator’s variance, we found that the expectation of the quadratic Mahalanobis distance to the sample mean equals the energy levels of the quantum harmonic oscillator, being the minimum quadratic Mahalanobis distance at the minimum energy level of the oscillator and reaching the “intrinsic” Cramér–Rao lower bound at the lowest energy level. Also, we demonstrate that the global probability density function of the collective mode of a set of m quantum harmonic oscillators at the lowest energy level still equals the posterior probability distribution calculated using Bayes’ theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. While these new assumptions certainly add complexity to the mathematical framework, the results proven are invariant under transformations, leading to the concept of “intrinsic” information-theoretic models, which are essential for developing physics.
Random quantum states are useful in various areas of quantum information science. Distributions of random quantum states using Gaussian distributions have been used in various scenarios in quantum information science. One of this is the distribution of random quantum states derived using the Wishart distibution usually used in statistics. This distribution of random quantum states using the Wishart distribution has recently been named as the quantum Wishart distribution \cite{Han}. The quantum Wishart distribution has been found for non-central distribution with a general covariance matrix and zero mean matrix. Here, we find out the closed form expression for the distribution of random quantum states pertaining to non-central Wishart distribution with any general rank one mean matrix and a general covariance matrix for arbitrary dimensions in both real and complex Hilbert space. We term this as the non-zero mean quantum Wishart distribution. We find out the method for the desired placement of its peak position in the real and complex Hilbert space for arbitrary dimensions. We also show an application of this via a fast and efficient algorithm for the random sampling of quantum states, mainly for qubits where the target distribution is a well behaved arbitrary probability distribution function occurring in the context of quantum state estimation experimental data .
Polar duality is a well-known concept from convex geometry and analysis. In the present paper, we study two symplectically covariant versions of polar duality keeping in mind their applications to quantum mechanics. The first variant makes use of the symplectic form on phase space and allows a precise study of the covariance matrix of a density operator. The latter is a fundamental object in quantum information theory., The second variant is a symplectically covariant version of the usual polar duality highlighting the role played by Lagrangian planes. It allows us to define the notion of "geometric quantum states" with are in bijection with generalized Gaussians.
N C K SC K SC Recent advances at the intersection of quantum information geometry and machine-learning approaches to the renormalisation group suggest a fertile reinterpretation of the Similarity Renormalization Group (SRG). The continuous unitary transformations that drive a Hamiltonian toward band-diagonal form may be viewed as geodesics on a manifold of quantum states (or of density operators) equipped with a monotone metric, or equivalently as successive layers of a deep network that progressively disentangle degrees of freedom. This independent study develops that correspondence. It reviews the information-geometric structures relevant to quantum field theories and many-body Hamiltonians, examines neural-network renormalisation-group (NN-RG) architectures, and maps the SRG flow—originally clarified in the pedagogical setting of Szpigel & Perry (2000)—onto both geometric and learning-theoretic languages. The resulting perspective yields analytic insights into the optimality of particular generators and practical algorithms for approximate Hamiltonian diagonalisation that combine SRG ideas with modern differentiable programming. Error-scaling diagnostics originating in the two-dimensional laboratory are re-expressed as statements about geodesic deviation and about generalisation error in the associated learning problem. Also available on Academia.edu: https://www.academia.edu/attachments/133749950/download_file?s=portfolio
The von Neumann entropy S( D ^ ) generates in the space of quantum density matrices D ^ the Riemannian metric ds2 = −d2S( D ^ ), which is physically founded and which characterises the amount of quantum information lost by mixing D ^ and D ^ + d D ^ . A rich geometric structure is thereby implemented in quantum mechanics. It includes a canonical mapping between the spaces of states and of observables, which involves the Legendre transform of S( D ^ ). The Kubo scalar product is recovered within the space of observables. Applications are given to equilibrium and non equilibrium quantum statistical mechanics. There the formalism is specialised to the relevant space of observables and to the associated reduced states issued from the maximum entropy criterion, which result from the exact states through an orthogonal projection. Von Neumann’s entropy specialises into a relevant entropy. Comparison is made with other metrics. The Riemannian properties of the metric ds2 = −d2S( D ^ ) are derived. The curvature arises from the non-Abelian nature of quantum mechanics; its general expression and its explicit form for q-bits are given, as well as geodesics.