An SU(2n)-valued nonlinear Fourier transform

We define a nonlinear Fourier transform which maps sequences of contractive $n\times n$ matrices to $SU(2n)$-valued functions on the circle $\mathbb{T}$. We characterize the image of finitely supported sequences and square-summable sequences on the half-line, and construct an inverse for $SU(2n)$-valued functions whose diagonal $n\times n$ blocks are outer matrix functions. As an application, we relate this nonlinear Fourier transform with quantum signal processing over $U(2n)$ and multivariate quantum signal processing.