  The $\Xi^0$ asymmetry parameters are measured using entangled quantum
$\Xi^0$-$\bar{\Xi}^0$ pairs from a sample of $(448.1 \pm 2.9) \times 10^6$
$\psi(3686)$ events collected with the BESIII detector at BEPCII. The relative
phase between the transition amplitudes of the $\Xi^0 \bar{\Xi}^0$ helicity
states is measured to be $\Delta \Phi = -0.050 \pm 0.150 \pm 0.020$~rad, which
implies that there is no obvious polarization at the current level of
statistics. The decay parameters of the $\Xi^0$ hyperon $(\alpha_{\Xi^0},
\alpha_{\bar{\Xi}^0}, \phi_{\Xi^0}, \phi_{\bar{\Xi}^0})$ and the angular
distribution parameter $(\alpha_{\psi(3686)})$ and $\Delta \Phi$ are measured
simultaneously for the first time. In addition, the $CP$ asymmetry observables
are determined to be $A^{\Xi^0}_{CP} = (\alpha_{\Xi^0} +
\alpha_{\bar{\Xi}^0})/(\alpha_{\Xi^0} -
  \alpha_{\bar{\Xi}^0})$ $= -0.007$ $\pm$ 0.082 $\pm$ 0.025 and $\Delta
\phi^{\Xi^0}_{CP} = (\phi_{\Xi^0} + \phi_{\bar{\Xi}^0})/2$ $= -0.079$ $\pm$
0.082 $\pm$ 0.010 rad, which are consistent with $CP$ conservation.
