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Tensor free probability theory: asymptotic tensor freeness and central limit theorem

Ion Nechita, Sang-Jun Park

TL;DR

This work generalizes Voiculescu’s asymptotic freeness to random matrices with tensor product structures by introducing an $r$-partite tensor probability space and tensor free cumulants. It develops a full combinatorial framework—tensor trace invariants, tensor free cumulants, and tensor freeness—and proves that independent locally unitarily invariant ensembles, as well as certain non-independent models (partial transposes, tensor embeddings), become asymptotically tensor free. A tensor central limit theorem is established, yielding limiting laws as linear combinations of tensor semicircular elements and connecting to known tensor-GUE limits; in addition, a route to recover classical CLT via tensor freeness is provided. The results offer a unified approach to large-scale random tensors and multi-matrix models, with implications for quantum information and tensor-structured data analysis. Overall, the paper broadens the toolbox for analyzing high-dimensional tensor ensembles and their asymptotic joint distributions under tensor invariance constraints.

Abstract

Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this work, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the \emph{tensor distribution} limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of freeness, which we term \emph{tensor freeness}. It can be defined via the vanishing of mixed \emph{tensor free cumulants}, allowing the joint tensor distribution of tensor free elements to be described in terms of that of individual elements. We present several applications of these results in the context of random matrices with a tensor product structure, such as partial transpositions of (local) unitarily invariant random matrices and tensor embeddings of random matrices. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables.

Tensor free probability theory: asymptotic tensor freeness and central limit theorem

TL;DR

This work generalizes Voiculescu’s asymptotic freeness to random matrices with tensor product structures by introducing an -partite tensor probability space and tensor free cumulants. It develops a full combinatorial framework—tensor trace invariants, tensor free cumulants, and tensor freeness—and proves that independent locally unitarily invariant ensembles, as well as certain non-independent models (partial transposes, tensor embeddings), become asymptotically tensor free. A tensor central limit theorem is established, yielding limiting laws as linear combinations of tensor semicircular elements and connecting to known tensor-GUE limits; in addition, a route to recover classical CLT via tensor freeness is provided. The results offer a unified approach to large-scale random tensors and multi-matrix models, with implications for quantum information and tensor-structured data analysis. Overall, the paper broadens the toolbox for analyzing high-dimensional tensor ensembles and their asymptotic joint distributions under tensor invariance constraints.

Abstract

Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this work, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the \emph{tensor distribution} limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of freeness, which we term \emph{tensor freeness}. It can be defined via the vanishing of mixed \emph{tensor free cumulants}, allowing the joint tensor distribution of tensor free elements to be described in terms of that of individual elements. We present several applications of these results in the context of random matrices with a tensor product structure, such as partial transpositions of (local) unitarily invariant random matrices and tensor embeddings of random matrices. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables.

Paper Structure

This paper contains 19 sections, 65 theorems, 263 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on random matrix theory and free probability
  3. Permutations and partitions
  4. Moments and free cumulants associated to permutations
  5. Unitary and orthogonal Weingarten calculus
  6. Unitary and orthogonal invariant random matrices
  7. Algebraic tensor probability spaces
  8. Tensor free cumulants
  9. Tensor free independence
  10. Moment conditions for tensor freeness
  11. Classical independence from tensor freeness
  12. Asymptotic tensor freeness of globally invariant random matrices
  13. Asymptotic tensor freeness of independent locally invariant random matrices
  14. Asymptotic tensor freeness of non-independent random matrices
  15. Partial transposes of random matrices
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X_N^{(1)},\ldots, X_N^{(L)}$ be independent families of $N^r\times N^r$ random matrices, and let us identify $\mathcal{M}_{N^r}(\mathbb{C})\cong \mathcal{M}_{N}(\mathbb{C})^{\otimes r}$.

Figures (14)

  • Figure 1: Factoring of $\pi = (1\, 2)(-2 \, 3)(-3\, 4)(-1 \, -4)$ as $\pi = \textcolor{blue}{\varepsilon} \textcolor{red}{\sigma\delta\sigma^{-1}}\textcolor{blue}{\varepsilon}$, with $\textcolor{blue}{\varepsilon} = (-1 \, 1)$ and $\textcolor{red}{\sigma\delta\sigma^{-1}} = \prod_{i=1}^4 (-i\;\sigma(i))$.
  • Figure 2: Diagram for the matrix product $(U \otimes U) X (U \otimes U)^*$.
  • Figure 3: Two removal diagrams obtained from \ref{['fig:E-UU-X-UUstar']}, using different pairings $\pi,\rho$. On the left panel, we have $(\textcolor{red}{\pi}, \textcolor{blue}{\rho}) = (\textcolor{red}{(1\,2)(-1\,-2)}, \textcolor{blue}{(1\,-2)(-1\,2)})$. On the right panel, we have $(\textcolor{red}{\pi}, \textcolor{blue}{\rho}) = (\textcolor{red}{(1\,-1)(2\,-2)}, \textcolor{blue}{(1\,2)(-1\,-2)})$.
  • Figure 4: The trace invariant from \ref{['eq:orthogonal-trace-invariant']}. In the top row, the invariant is constructed from the paring $(1,2)(-1,-2)(-3,-5)(5,4)(-4,3) \in \mathcal{P}_2(\pm 5)$. The bottom diagram is obtained from the top diagram by moving the boxes and represents the invariant in terms of the matrices and their transposes.
  • Figure 5: The diagram for the trace invariant $\Tr_{(12)(3), (1)(23)}(X_1,X_2,X_3)$. The summation is over the six indices $a_{1,2,3},b_{1,2,3}$. The first permutation $\alpha_1=(12)(3)$ connects the top legs of the tensors, while the second permutation $\alpha_2=(1)(23)$ connects the bottom legs. On each level $s=1,2$, the input (leg on the RHS) of the $i$-th box is connected to the output (leg on the LHS) of the $\alpha_s(i)$-th box.
  • ...and 9 more figures

Theorems & Definitions (135)

  • Theorem 1.1: \ref{['thm-indepTranspose']}, Asymptotic tensor freeness of (local) unitary invariant random matrices
  • Theorem 1.2: \ref{['thm-TensorCLT', 'thm-CLTBipartite']}, Central limit theorems for tensor free elements
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4: MP13
  • Lemma 2.5: MP19
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • ...and 125 more