Finite-Dimensional Type I von Neumann Algebras in PyTorch: A GPU-Accelerated Framework for Random Block-Diagonal Operators

• URL: http://arxiv.org/abs/2606.15882v1
• Date: Sun, 14 Jun 2026 16:06:26 GMT
• Title: Finite-Dimensional Type I von Neumann Algebras in PyTorch: A GPU-Accelerated Framework for Random Block-Diagonal Operators
• Abstract summary: We present texttttorch_vn_algebra, an open-source Python library built on PyTorch for numerical experiments with finite-dimensional Type I von Neumann algebras.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present \texttt{torch\_vn\_algebra}, an open-source Python library built on PyTorch for numerical experiments with finite-dimensional Type I von Neumann algebras (direct sums of matrix algebras). The library provides: $\bullet$ a compact batched tensor representation $(B,C,k_{\max},k_{\max})$ that handles both Monte Carlo samples and multiple direct summands; $\bullet$ lazy evaluation of operators to avoid unnecessary memory allocation; $\bullet$ generation of random operators with arbitrary eigenvalue distributions (user-provided samplers) and various unitary ensembles (Haar, $\mathrm{SU}(n)$, COE, CSE, diagonal phases); $\bullet$ functional calculus via SVD (absolute value, square root, inverse, entropy) and a hybrid method for extreme eigenvalues (exact diagonalisation for $k_{\max}\le256$, otherwise power iteration); $\bullet$ three trace functionals (blunt, normalised subspace trace, and the von Neumann tracial state); $\bullet$ GPU-accelerated batched linear algebra for moderate-scale Monte Carlo studies (e.g., $2\times10^4$ samples of $100\times100$ operators). The library is validated against analytical expectations (Haar moments, trace properties). Performance benchmarks on a Tesla P100 GPU are presented and discussed. Limitations and future work are outlined. The code is open-source.

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