  Multi-photon emissions constitute a fundamental source of noise in quantum
repeaters and other quantum communication protocols when probabilistic photon
sources are employed. In this paper, it is shown that by alternating the Bell
state measurement (BSM) basis in concatenated entanglement swapping links one
can automatically identify and discard many errors from stimulated multi-photon
emissions. The proposed protocol is shown to completely eliminate the dominant
quadratic growth of multi-photon errors with the length of the repeater chain.
Furthermore, it is shown that the protocol can be employed in
satellite-assisted entanglement distribution links to enable links which are
more robust in the presence of imbalanced channel losses. The analysis
introduces a convenient calculus based on Clifford algebra for modeling
concatenated entanglement swapping links with multi-photon emissions. In
particular, we present a compact expression for the fidelity of the Bell state
produced by a repeater chain of arbitrary length including noise from
double-pair emissions.
