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A generalisation of the Ginibre ensemble of non-Hermitian random squarematrices is introduced. The corresponding probability measure is induced by theensemble of rectangular Gaussian matrices via a quadratisation procedure. Wederive the joint probability density of eigenvalues for such induced Ginibreensemble and study various spectral correlation functions for complex and realmatrices, and analyse universal behaviour in the limit of large dimensions. Inthis limit the eigenvalues of the induced Ginibre ensemble cover uniformly aring in the complex plane. The real induced Ginibre ensemble is shown to beuseful to describe statistical properties of evolution operators associatedwith random quantum operations, for which the dimensions of the input state andthe output state do differ.

Induced Ginibre ensemble of random matrices and quantum operations