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Nonnegativity for hafnians of certain matrices (1811.10342v2)
Published 26 Nov 2018 in quant-ph, math-ph, math.CO, math.MP, and math.NT
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 $L \succeq B$.