Sinkhorn normal form for unitary matrices

TL;DR

This paper proves a conjecture that any unitary matrix can be decomposed into diagonal unitaries and a matrix with equal row and column sums, using symplectic topology, with applications in linear optics.

Contribution

It establishes a non-constructive decomposition of unitary matrices analogous to Sinkhorn's theorem, linking symplectic topology and linear optics.

Findings

Decomposition of any unitary into diagonal unitaries and a matrix with uniform sums

Application to linear optics arrays with phase shifters and Fourier transforms

Provides a new perspective on unitary matrix factorization

Abstract

Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into two diagonal unitaries and one whose row- and column sums are equal to one. The proof is non-constructive and based on a reformulation in terms of symplectic topology. As a corollary, we obtain a decomposition of unitary matrices into an interlaced product of unitary diagonal matrices and discrete Fourier transformations. This provides a new decomposition of linear optics arrays into phase shifters and canonical multiports described by Fourier transformations.