  Light-field microscopy (LFM) is a 3D microscopy technique whereby volumetric
information of a sample is gained by simultaneously capturing both the position
and momentum (angular) information of light illuminating a scene. Conventional
LFM designs generally require a trade-off between position and momentum
resolution, requiring one to sacrifice resolving power for increased depth of
field (DOF) or vice versa. In this work, we demonstrate a LFM design that does
not require this trade-off by utilizing the inherent correlations between
spatial-temporal entangled photon pairs. Here, one photon from the pair is used
to illuminate a sample from which the position information of the photon is
captured directly by a camera. By virtue of the strong momentum
anti-correlation between the two photons, the momentum information of the
illumination photon can then be inferred by measuring the angle of its
entangled partner on a different camera. By using a combination of ray-tracing
and a Gerchberg-Saxton type algorithm for the light field reconstruction, we
demonstrate that a resolving power of 5 $\mu$m can be maintained with a DOF of
$\sim500$ $\mu$m, approximately 3 times of the latest LFM designs or $>100$
time that of a conventional microscope. In the extreme, at a resolving power of
100 $\mu$m, it is possible to achieve near infinite DOF.
