Approximate orthogonality of permutation operators, with application to quantum information

TL;DR

This paper investigates the approximate orthogonality of permutation operators in quantum systems and demonstrates their applications in quantum information, random matrix theory, and quantum circuit complexity.

Contribution

It provides a collective measure of approximate orthogonality for permutation matrices and applies this to various problems in quantum information and random matrix theory.

Findings

Permutation matrices are approximately orthogonal when the dimension is large.

Random maximally entangled states resemble fully random states.

Product tests cannot be efficiently performed using LOCC without many copies.

Abstract

Consider the $n!$ different unitary matrices that permute $n$ $d$-dimensional quantum systems. If $d\geq n$ then they are linearly independent. This paper discusses a sense in which they are approximately orthogonal (with respect to the 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 these matrices, but here we show a more collective statement, quantified in terms of the operator norm distance of the Gram matrix to the identity matrix. This simple point has several applications in quantum information and random matrix theory: (1) showing that random maximally entangled states resemble fully random states, (2) showing that Boson sampling output probabilities resemble those from Gaussian matrices, (3) improving the Eggeling-Werner scheme for multipartite data hiding, (4) proving that the product test of Harrow-Montanaro cannot be performed using LOCC without a large number of copies of the state to be tested, (5) proving that the purity of a quantum state also cannot be efficiently tested using LOCC, and (6, published separately) helping prove that poly-size random quantum circuits are poly-designs.