  We study the problem of designing optical receivers to discriminate between
multiple coherent states using coherent processing receivers---i.e., one that
uses arbitrary coherent feedback control and quantum-noise-limited direct
detection---which was shown by Dolinar to achieve the minimum error probability
in discriminating any two coherent states. We first derive and re-interpret
Dolinar's binary-hypothesis minimum-probability-of-error receiver as the one
that optimizes the information efficiency at each time instant, based on
recursive Bayesian updates within the receiver. Using this viewpoint, we
propose a natural generalization of Dolinar's receiver design to discriminate
$M$ coherent states each of which could now be a codeword, i.e., a sequence of
$N$ coherent states each drawn from a modulation alphabet. We analyze the
channel capacity of the pure-loss optical channel with a general
coherent-processing receiver in the low-photon number regime and compare it
with the capacity achievable with direct detection and the Holevo limit
(achieving the latter would require a quantum joint-detection receiver). We
show compelling evidence that despite the optimal performance of Dolinar's
receiver for the binary coherent-state hypothesis test (either in error
probability or mutual information), the asymptotic communication rate
achievable by such a coherent-processing receiver is only as good as direct
detection. This suggests that in the infinitely-long codeword limit, all
potential benefits of coherent processing at the receiver can be obtained by
designing a good code and direct detection, with no feedback within the
receiver.
