On the power of one pure steered state for EPR-steering with a pair of qubits

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, provided they are pure enough and different enough. 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: $p_{\rm max}^{{\cal E}} = p_{\rm min}^{{\cal E}}$.
Comment: 30 pages, 9 figures