  The electric dipole moment (EDM) of elementary particles, arising from
flavor-diagonal $CP$ violation, serves as a powerful probe for new physics
beyond the Standard Model and holds the potential to provide novel insights in
unraveling the puzzle of the matter-dominated Universe. Hyperon EDM is a
largely unexplored territory. In this paper, we present a comprehensive angular
analysis that focuses on entangled hyperon-antihyperon pairs in $J/\psi$ decays
for the indirect extraction of hyperon EDM. The statistical sensitivities are
investigated for BESIII and the proposed Super Tau-Charm Facility (STCF).
Leveraging the statistics from the BESIII experiment, the estimated sensitivity
for $\Lambda$ EDM can reach an impressive level of $10^{-19}$ $e$ cm, achieving
a 3-orders-of-magnitude improvement over the only existing measurement in a
fixed-target experiment at Fermilab with similar statistics. The estimated
sensitivities for the $\Sigma^+$, $\Xi^-$, and $\Xi^0$ hyperons at the same
level of $10^{-19}$ $e$ cm will mark the first-ever achievement and the latter
two will be the first exploration of hyperons with two strange valence quarks.
The EDM measurements for hyperons conducted at the BESIII experiment will be a
significant milestone and serve as a litmus test for new physics such as
supersymmetry and the left-right symmetrical model. Furthermore, at the STCF
experiment, the sensitivity of hyperon EDM measurements can be further enhanced
by 2 orders of magnitude. Additionally, this angular analysis enables the
determination of $CP$ violation in hyperon decays, the effective weak mixing
angle, and beam polarization.
