  We prove new one shot achievability results for measurement compression of
quantum instruments with side information at the receiver. Unlike previous one
shot results for this problem, our one shot bounds are nearly optimal and do
not need catalytic randomness. In fact, we state a more general problem called
centralised multi link measurement compression with quantum side information
and provide one shot achievability results for it. As a simple corollary, we
obtain one shot measurement compression results for quantum instruments with
side information that we mentioned earlier. All our one shot results lead to
the standard results for this problem in the asymptotic iid setting. We prove
our achievability bounds by first proving a novel sequential classical quantum
multipartite covering lemma, which should be of independent interest.
