Partial distinguishability and photon counting probabilities in linear multiport devices

TL;DR

This paper generalizes photon counting probabilities in multiport devices for arbitrary quantum states and partial distinguishability, providing formulas useful for analyzing Boson Sampling experiments and their error bounds.

Contribution

It introduces a unified framework for calculating photon counting probabilities with arbitrary quantum states and partial distinguishability in multiport devices.

Findings

Probability formulas involve matrix permanents and Husimi functions.

Reduces to known formulas for Fock states and distinguishable photons.

Provides tools for analyzing experimental Boson Sampling errors.

Abstract

Probabilities of photon counts at the output of a multiport optical device are generalised for optical sources of arbitrary quantum states in partially distinguishable optical modes. For the single-mode photon sources, the generating function for the probabilities is a linear combination of the matrix permanents of positive semi-definite Hermitian matrices, where each Hermitian matrix is a Hadamard product of a submatrix of the multiport matrix and a Hermitian matrix describing partial distinguishability. For the multi-mode sources the generating function is given by an integral of the Husimi functions of the sources. When each photon source outputs exactly a Fock state, the obtained expression reduces to the probability formula derived for partially distinguishable photons, \textit{Physical Review A \textbf{91}, 013844 (2015)}. The derived probability formula can be useful in analysing experiments with partially distinguishable sources and error bounds of experimental Boson Sampling devices.