Although I've voted to close because there is a trivial answer, based on your interest in analogies between quantum and statistical physics I think the following may be of interest to you. Since a comment isn't appropriate here I've CW'd this.

Let $u$ be a generic unitary matrix, so that $\sum_j u_{ij}\bar u_{kj} = \delta_{ik}$. If we set $v_{ij} := |u_{ij}|^2$, then it is easy to show that $v$ is a doubly stochastic matrix (though not all doubly stochastic matrices are of this form [1]). Indeed such a matrix is called a unitary-stochastic transition [2] or unistochastic [1] matrix. When one starts with a unitary matrix that is the propagator representing a time evolution operator associated to some Hamiltonian acting on a finite-dimensional Hilbert space, then taking the squared norms yields the associated transition matrix.

[1] See appendix A of Pakonski, P. et al. “Classical 1D maps, quantum graphs and ensembles of unitary matrices”. J. Phys. A 34, 9303 (2001).

[2] Marshall, A. W., and Olkin, I. Inequalities: Theory of Majorization and Its Applications. Academic Press (1979). Cited in [1] and in Tanner, G. “Unitary-stochastic matrix ensembles and spectral statistics”. J. Phys. A 34, 8485 (2001).