  We use the projective symmetry group analysis to classify the quantum spin
liquids on the $S=1/2$ pyrochlore magnet with a breathing anisotropy. We find
40 $\mathbb{Z}_2$ spin liquids and 16 $U(1)$ spin liquids that respect the
$F\bar{4}3m$ space group and the time reversal symmetry. As an application, we
consider the antiferromagnetic Heisenberg model, which is proposed to be the
dominant interaction in the candidate material Ba$_3$Yb$_2$Zn$_5$O$_{11}$.
Focusing on the $U(1)$ spin liquid ansatze, we find that only two of them are
physical when restricted to this model. We present an analytical solution to
the parton mean field theory for each of these two $U(1)$ spin liquids. It is
revealed that one of them has gapless, while the other one has gapped, spinon
excitations. The two $U(1)$ spin liquids are equal in energy regardless of the
degree of breathing anisotropy, and they can be differentiated by the
low-temperature heat capacity contribution from the quadratically-dispersing
gapless spinons. We further show that the latter is unaffected by fluctuations
of the $U(1)$ gauge field within the random phase approximation. Finally, we
demonstrate that a small Dzyaloshinskii-Moriya interaction lifts the degeneracy
between the two $U(1)$ spin liquids, and it eventually causes the lattice to
decouple into independent tetrahedra at strong coupling. While current model
parameters for Ba$_3$Yb$_2$Zn$_5$O$_{11}$ place it indeed in the decoupled
regime, other candidate materials may be synthesised in the near future that
realize the spin liquid states discussed in our work.
