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

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