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

TL;DR

This paper establishes a tight upper bound on how close a realistic bosonic device with partially indistinguishable bosons can be to the ideal Boson Sampling, using trace distance as a measure, with implications for experimental fidelity.

Contribution

The paper provides the first tight bound on the trace distance between realistic and ideal Boson Sampling devices, considering partial indistinguishability of bosons.

Findings

Upper bound on trace distance proven

Bound is tight under a conjecture

Device with small distinguishability error is close to ideal in trace distance

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.