  We present a theory of a hybrid quantum liquid state, $\textit{quantum
spin-dipole liquid}$ (QSDL), in a hydrogen-bonded electron system, by combining
a quantum proton ice and Anderson's resonating valence bond spin liquid theory,
motivated by the recent experimental discovery of a quantum spin liquid with
proton fluctuations in $\kappa$-H$_3$(Cat-EDT-TTF)$_2$ (a.k.a. H-Cat). In our
theory, an electron spin liquid and a proton dipole liquid are realized
simultaneously in the ground state called $\textit{quantum valence bond ice}$.
In this state, neither of them can be established independently of the other.
Analytical and numerical calculations reveal that this state has a large
entanglement entropy between spins and dipoles, which is far beyond the (crude)
Born-Oppenheimer approximation. We also examine the stability of QSDL with
respect to perturbations and discuss implications for experiments in H-Cat and
its deuterated analog D-Cat.
