  We show that a complex symmetric matrix of the form $A(Y,B) =
\begin{bmatrix}Y & B\\ B^\top & \overline{Y} \end{bmatrix},$ 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 $L \succeq B$.
