Tight bound on trace distance between a realistic device with partially indistinguishable bosons and the ideal Boson Sampling

Abstract: We study the closeness of an experimental unitary bosonic network with only partially indistinguishable bosons in an arbitrary mixed input state, in particular an experimental realization of the Boson Sampling, to the ideal bosonic network, where the measure of closeness of two networks is the trace distance between the output probability distributions. An upper bound on the trace distance to the ideal bosonic network is proven and also a bound on the difference between probabilities of an output configuration. Moreover, the upper bound on the trace distance is tight, provided that a physically transparent distinguishability conjecture is true. For a small distinguishability error it is shown that a realistic device with $N$ bosons is at a constant trace distance to the ideal Boson Sampling under the $O(N^{-1})$-scaling of the mismatch of internal states of bosons.