  As originally introduced, the EPR phenomenon was the ability of one party
(Alice) to steer, by her choice between two measurement settings, the quantum
system of another party (Bob) into two distinct ensembles of pure states. As
later formalized as a quantum information task, EPR-steering can be shown even
when the distinct ensembles comprise mixed states. Consider the scenario where
Alice and Bob each have a qubit and Alice performs dichotomic projective
measurements. In this case, the states in the ensembles to which she can steer
form the surface of an ellipsoid ${\cal E}$ in Bob's Bloch ball. Further, let
the steering ellipsoid ${\cal E}$ have nonzero volume. It has previously been
shown that if Alice's first measurement setting yields an ensemble comprising
two pure states, then this, plus any one other measurement setting, will
demonstrate EPR-steering. Here we consider what one can say if the ensemble
from Alice's first setting contains only one pure state $\mathsf{p}\in{\cal
E}$, occurring with probability $p_\mathsf{p}$. Using projective geometry, we
derive the necessary and sufficient condition analytically for Alice to be able
to demonstrate EPR-steering of Bob's state using this and some second setting,
when the two ensembles from these lie in a given plane. Based on this, we show
that, for a given ${\cal E}$, if $p_\mathsf{p}$ is high enough [$p_{\sf p} >
p_{\rm max}^{{\cal E}} \in [0,1)$] then any distinct second setting by Alice is
sufficient to demonstrate EPR-steering. Similarly we derive a $p_{\rm
min}^{{\cal E}}$ such that $p_\mathsf{p}>p_{\rm min}^{{\cal E}}$ is necessary
for Alice to demonstrate EPR-steering using only the first setting and some
other setting. Moreover, the expressions we derive are tight; for spherical
steering ellipsoids, the bounds coincide: $p_{\rm max}^{{\cal E}} = p_{\rm
min}^{{\cal E}}$.
