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