Efficient Characterization of Phase Relations in Linear Optical Devices

Linear optical networks play a paramount role in current implementations of quantum information protocols using photons. A paradigmatic example of their potential is given by boson sampling experiments, which recently have reached the quantum advantage regime [1], [2]. Such linear devices are essentially <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> -mode interferometers, whose operation is entirely captured by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N\times N$</tex> unitary matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$U$</tex> with complex entries <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$u_{j i}=\left\vert u_{j i}\right\vert e^{\theta_{j i}}$</tex>. This matrix characterizes, in a classical framework, the transformation of the phasors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon_{j}^{(in)}$</tex> of the input field across the different modes into the phasors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon_{j}^{(out)}$</tex> of the output fields according to the linear relation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon_i^{({out })}=\sum_i u_{j i} \varepsilon_i^{({in })}$</tex>).