Abstract Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g. tensor eigenvalue/vector distributions, is interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of 0D quantum field theories. In this paper, using this method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb {R})$ , $O(N,\mathbb {C})$ , and $U(N,\mathbb {C})$ , respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.Abstract Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g. tensor eigenvalue/vector distributions, is interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of 0D quantum field theories. In this paper, using this method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb {R})$, $O(N,\mathbb {C})$, and $U(N,\mathbb {C})$, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.

arXiv:2408.01030Progress of Theoretical and Experimental PhysicsAug 31, 20240 citations

Elements of Classical Computation

Springer eBooksFeb 2, 20070 citations

Revisiting the Page curve and its moments. A combinatorial approach

Gero von Gersdorff

We revisit the calculation of the von Neumann (or "entanglement") entropy of a subsystem of a pure quantum state, under the assumption that the latter is drawn at random from a uniform distribution on the full Hilbert space. We derive simple and closed expressions for all power moments, from which the moments of the entropy can be computed by simple differentiation. Our approach (different from the usual one based on random matrix theory and Laguerre polynomials) makes use of Schur-Weyl duality and the character theory of the symmetric group $S_N$ . The paper is self-contained, providing all the necessary mathematical background.We revisit the calculation of the von Neumann (or "entanglement") entropy of a subsystem of a pure quantum state, under the assumption that the latter is drawn at random from a uniform distribution on the full Hilbert space. We derive simple and closed expressions for all power moments, from which the moments of the entropy can be computed by simple differentiation. Our approach (different from the usual one based on random matrix theory and Laguerre polynomials) makes use of Schur-Weyl duality and the character theory of the symmetric group $S_N$ . The paper is self-contained, providing all the necessary mathematical background.

quant-pharXiv:2606.30941Jul 1, 20260 citations

TEQF: A Mathematical Framework for Adaptive Perturbation Operators, Normalized Transforms, and Decision-State Dynamics

Amil Abraham Thomas

Description TEQF: A Mathematical Framework for Adaptive Perturbation Operators, Normalized Transforms, and Decision-State Dynamics This research monograph presents the mathematical foundations, operator formulations, computational investigations, and proposed extensions of the TEQF (Temporal-Equivariant Quantum Field) framework. The research investigates adaptive perturbation operators acting on mathematical state spaces, with particular emphasis on Hilbert-space formulations, normalized transformations, information pathways, and state-dependent decision mechanisms. The monograph examines: Operator theory: Mathematical definitions, domains, codomains, perturbation mappings, and composition rules. Hilbert-space analysis: Linearity, boundedness, norm preservation, isometries, and associated mathematical conditions. Adaptive decision architecture: State-dependent pathway selection, decision landscapes, and perturbation-based state transitions. Normalized transforms: Mathematical relationships between perturbation operators and subsequent state transformations. Computational validation: Reproducible numerical experiments, randomized testing, mathematical consistency checks, and illustrative counterexamples. Physical interpretation: Exploratory connections to gravitational modelling, wave-state interactions, temporal dynamics, and other potential applications, without asserting experimental confirmation. The accompanying research package includes the consolidated mathematical monograph, historical source documents, original research sketches, supporting diagrams, Python validation scripts, and numerical test results. The computational experiments examine an illustrative adaptive-operator implementation rather than establish the empirical validity of a new physical theory. The work distinguishes established mathematical principles, proposed theoretical constructions, conditional proofs, and unresolved research questions. Research status: Independent mathematical research and computational investigation. The TEQF framework remains under development and has not been independently validated as a physical theory or production-ready computational system. Author: Amil Abraham ThomasResearch affiliation: AIDEN — Independent Mathematical Research and Systems ArchitectureYear: 2026 Keywords: TEQF, Adaptive Operators, Operator Theory, Hilbert Spaces, Perturbation Theory, Nonlinear Operators, Mathematical Transforms, Decision-State Dynamics, Computational Mathematics, Numerical Validation, Mathematical Physics, Applied Mathematics.Description TEQF: A Mathematical Framework for Adaptive Perturbation Operators, Normalized Transforms, and Decision-State Dynamics This research monograph presents the mathematical foundations, operator formulations, computational investigations, and proposed extensions of the TEQF (Temporal-Equivariant Quantum Field) framework. The research investigates adaptive perturbation operators acting on mathematical state spaces, with particular emphasis on Hilbert-space formulations, normalized transformations, information pathways, and state-dependent decision mechanisms. The monograph examines: Operator theory: Mathematical definitions, domains, codomains, perturbation mappings, and composition rules. Hilbert-space analysis: Linearity, boundedness, norm preservation, isometries, and associated mathematical conditions. Adaptive decision architecture: State-dependent pathway selection, decision landscapes, and perturbation-based state transitions. Normalized transforms: Mathematical relationships between perturbation operators and subsequent state transformations. Computational validation: Reproducible numerical experiments, randomized testing, mathematical consistency checks, and illustrative counterexamples. Physical interpretation: Exploratory connections to gravitational modelling, wave-state interactions, temporal dynamics, and other potential applications, without asserting experimental confirmation. The accompanying research package includes the consolidated mathematical monograph, historical source documents, original research sketches, supporting diagrams, Python validation scripts, and numerical test results. The computational experiments examine an illustrative adaptive-operator implementation rather than establish the empirical validity of a new physical theory. The work distinguishes established mathematical principles, proposed theoretical constructions, conditional proofs, and unresolved research questions. Research status: Independent mathematical research and computational investigation. The TEQF framework remains under development and has not been independently validated as a physical theory or production-ready computational system. Author: Amil Abraham ThomasResearch affiliation: AIDEN — Independent Mathematical Research and Systems ArchitectureYear: 2026 Keywords: TEQF, Adaptive Operators, Operator Theory, Hilbert Spaces, Perturbation Theory, Nonlinear Operators, Mathematical Transforms, Decision-State Dynamics, Computational Mathematics, Numerical Validation, Mathematical Physics, Applied Mathematics.

Oct 20, 20160 citations

O(N,\mathbb {R})

O(N,\mathbb {C})

, and

U(N,\mathbb {C})

, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.

S_N

. The paper is self-contained, providing all the necessary mathematical background.