  In the cuprates, the lightly-doped region is of major interest because
superconductivity, antiferromagnetism, and the pseudogap state
\cite{Timusk,Lee,Anderson} come together near a critical doping value $x_c$.
These states are deeply influenced by phase fluctuations \cite{Emery} which
lead to a vortex-liquid state that surrounds the superconducting region
\cite{WangPRB01,WangPRB06}. However, many questions
\cite{Doniach,Fisher,FisherLee,Tesanovic,Sachdev} related to the nature of the
transition and vortex-liquid state at very low tempera- tures $T$ remain open
because the diamagnetic signal is difficult to resolve in this region. Here, we
report torque magnetometry results on $\rm La_{2-x}Sr_xCuO_4$ (LSCO) which show
that superconductivity is lost at $x_c$ by quantum phase fluctuations. We find
that, in a magnetic field $H$, the vortex solid-to-liquid transition occurs at
field $H_m$ much lower than the depairing field $H_{c2}$. The vortex liquid
exists in the large field interval $H_m \ll H_{c2}$, even in the limit $T\to$0.
The resulting phase diagram reveals the large fraction of the $x$-$H$ plane
occupied by the quantum vortex liquid.
