Nonseparability and steerability of two-qubit states from the geometry of steering outcomes

When two qubits, $A$ and $B$, are in an appropriate state, Alice can remotely steer Bob's system $B$ into different ensembles by making different measurements on $A$. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on $A$ (made by Alice) and an ensemble of $B$ (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of $B$ corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the four-dimensional geometry of Alice's steering outcomes completely determines both the nonseparability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into nonseparable and steerable classes is equivalent to geometrically classifying certain four-dimensional skewed double cones.

Figure 1

Three-dimensional representation of Alice's steering outcomes ${\mathcal{M}}_A^{\prime }$ inside Bob's positive light cone ${\mathcal{M}}_B^{+}$ together with their projective projections on the Bob's Bloch hyperplane.

Figure 2

Two-dimensional cross sections of the boundaries of the polyhedral box based on the uniform ansatz (red solid lines), the steering outcomes of the Werner state at $p=\frac{1}{2}$ (green dashed lines), and the steering outcomes of the modified Werner state at $p=0.4, q\approx 0.75$ (blue dotted lines). The outermost black lines present the cross section of the forward light cone at the origin.