  We present the Y-formalism for the non-minimal pure spinor quantization of
superstrings. In the framework of this formalism we compute, at the quantum
level, the explicit form of the compound operators involved in the construction
of the $b$ ghost, their normal-ordering contributions and the relevant
relations among them. We use these results to construct the quantum-mechanical
$b$ ghost in the non-minimal pure spinor formalism. Moreover we show that this
non-minimal $b$ ghost is cohomologically equivalent to the non-covariant $b$
ghost.
