## Asymptotic evaluation of the matrix permanents giving the probability amplitudes in linear bosonic networks    [PDF]

V. S. Shchesnovich
The probability amplitudes of N bosons unitarily transformed from the input $M$ modes to the output $M$ modes of a unitary linear network, given by the matrix permanents, are expressed as an average over the lattice of contingency tables with margins equal to the distributions of bosons in the input and output modes. In the limit $N\gg M$ the finite sum over the contingency tables is converted to a multidimensional integral with the integrand containing a large parameter (N) in the exponent. The integral representation allows an asymptotic estimate for the bosonic matrix permanents up to a small multiplicative error of order 1/N. The estimate depends on the solution of the scaling problem of the unitary $M\times M$-dimensional network matrix: to find the left and right diagonal matrices which scale the unitary matrix to a matrix which has specified rows and columns sums (equal to the distributions of bosons in the input and output modes). Such scaled matrices give the saddle points of the integral. In the case of simple saddle points an explicit formula giving an asymptotic estimate for the bosonic matrix permanents is given. It is compared with the exact result in the simplest case of just two-mode network (the beam-splitter) where the saddle-points are the roots of a quadratic.
View original: http://arxiv.org/abs/1304.6675