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Consider the $n!$ different unitary matrices that permute $n$ $d$-dimensionalquantum systems. If $d\geq n$ then they are linearly independent. This paperdiscusses a sense in which they are approximately orthogonal (with respect tothe Hilbert-Schmidt inner product) if $d\gg n^2$, or, in a different sense, if$d\gg n$. Previous work had shown pairwise approximate orthogonality of thesematrices, but here we show a more collective statement, quantified in terms ofthe operator norm distance of the Gram matrix to the identity matrix.  This simple point has several applications in quantum information and randommatrix theory: (1) showing that random maximally entangled states resemblefully random states, (2) showing that Boson sampling output probabilitiesresemble those from Gaussian matrices, (3) improving the Eggeling-Werner schemefor multipartite data hiding, (4) proving that the product test ofHarrow-Montanaro cannot be performed using LOCC without a large number ofcopies of the state to be tested, (5) proving that the purity of a quantumstate also cannot be efficiently tested using LOCC, and (6, publishedseparately) helping prove that poly-size random quantum circuits arepoly-designs.

Approximate orthogonality of permutation operators, with application to quantum information