  We study the generation of planar quantum squeezed (PQS) states by quantum
non-demolition (QND) measurement of a cold ensemble of $^{87}$Rb atoms. Precise
calibration of the QND measurement allows us to infer the conditional
covariance matrix describing the $F_y$ and $F_z$ components of the PQS,
revealing the dual squeezing characteristic of PQS. PQS states have been
proposed for single-shot phase estimation without prior knowledge of the likely
values of the phase. We show that for an arbitrary phase, the generated PQS
gives a metrological advantage of at least 3.1 dB relative to classical states.
The PQS also beats traditional squeezed states generated with the same QND
resources, except for a narrow range of phase values. Using spin squeezing
inequalities, we show that spin-spin entanglement is responsible for the
metrological advantage.
