  The emergence of parity-time ($\mathcal{PT}$) symmetry has greatly enriched
our study of symmetry-enabled non-Hermitian physics, but the realization of
quantum $\mathcal{PT}$-symmetry faces an intrinsic issue of unavoidable
symmetry-breaking Langevin noises. Here we construct a quantum
pseudo-anti-$\mathcal{PT}$ (pseudo-$\mathcal{APT}$) symmetry in a two-mode
bosonic system without involving Langevin noises. We show that the spontaneous
pseudo-$\mathcal{APT}$ symmetry breaking leads to an exceptional point, across
which there is a transition between different types of quantum squeezing
dynamics, i.e., the squeezing factor increases exponentially (oscillates
periodically) with time in the pseudo-$\mathcal{APT}$ symmetric (broken)
region. Such dramatic changes of squeezing factors and quantum dynamics near
the exceptional point are utilized for ultra-precision quantum sensing. These
exotic quantum phenomena and sensing applications can be experimentally
observed in two physical systems: spontaneous wave mixing nonlinear optics and
atomic Bose-Einstein condensates. Our work offers a physical platform for
investigating exciting $\mathcal{APT}$ symmetry physics in the quantum realm,
paving the way for exploring fundamental quantum non-Hermitian effects and
their quantum technological applications.
