Nonnegativity for hafnians of certain matrices

Abstract

We show that a complex symmetric matrix of the form $A(Y,B)=\left[\begin{array}{cc}Y& B\\ B^{\top }& \overline{Y}\end{array}\right],$ where $B$ is Hermitian positive semidefinite, has a nonnegative hafnian. These are positive scalar multiples of matrices $A(Y,B)$ that are encodable in a Gaussian boson sampler. Further, the hafnian of this matrix is non-decreasing in $B$ in the sense that $\mathrm{haf}A(Y,L)\ge \mathrm{haf}A(Y,B)$ if $L⪰B$.

Close to the published version in Linear and Multilinear Algebra