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arXiv · 1811.10342

Nonnegativity for hafnians of certain matrices

Kamil Bradler·Shmuel Friedland·Robert Israel
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Abstract

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 $L \succeq B$.

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Kamil Bradler, Shmuel Friedland, Robert Israel. 2018-11-26. Nonnegativity for hafnians of certain matrices. https://doi.org/10.1080/03081087.2021.1892022

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