The $\mathfrak S_k$-circular limit of random tensor flattenings

TL;DR

This paper investigates the asymptotic joint distribution of large random tensor flattenings in quantum information, revealing convergence to an operator-valued circular system and describing the law of large random density matrices.

Contribution

It introduces a new free probability framework for tensor flattenings and characterizes their limit distribution in the context of quantum states.

Findings

Convergence to an operator-valued circular system with permutation group covariance.

Description of the law of large random density matrices for bosonic quantum states.

Provides a free probability approach to tensor flattenings in quantum information.

Abstract

The tensor flattenings appear naturally in quantum information when one produces a density matrix by partially tracing the degrees of freedom of a pure quantum state. In this paper, we study the joint $^*$-distribution of the flattenings of large random tensors under mild assumptions, in the sense of free probability theory. We show the convergence toward an operator-valued circular system with amalgamation on permutation group algebras for which we describe the covariance structure. As an application we describe the law of large random density matrix of bosonic quantum states.