Necessary conditions for steerability of two qubits from consideration of local operations

Einstein-Podolsky-Rosen (EPR) steering refers to the ability of one observer to convince a distant observer that they share entanglement by making local measurements. Determining which states allow a demonstration of EPR steering remains an open problem in general. Here, we outline and demonstrate a method of analytically constructing new classes of two-qubit states which are nonsteerable by arbitrary projective measurements, from consideration of local operations performed by the steering party on states known to be nonsteerable.

Figure 1

Hierarchy of quantum correlations. The motivation of this paper is to investigate the boundary of nonsteerability for two-qubit states when the steering party can make projective measurements.

Figure 2

Sections of the state space of the displaced Werner state for which LHS models are known. The white region contains separable states, gray region entangled states, and those in the black region are not quantum states. Nonsteerable regions are below the orange [Eq. (7)] and blue lines [Eq. (22)]. The border constructed by performing CPTP maps on Alice's side of nonsteerable T states does not improve upon known results here.

Figure 3

Borders of regions guaranteed to be nonsteerable for two types of displaced $T$ states in the section of the $T^{\prime }=\text{diag}[t_1^{\prime },t_2^{\prime },t_3^{\prime }]$ space where all $t_i^{\prime }\mathrm{s}$ are negative. The white region denotes separable states, the gray region entangled states, and the black region nonquantum states. Orange curves coincide with equality in Eq. (7), blue curves are the convex hull of states obtained by applying extremal maps on Alice's part of a T state satisfying Eq. (6). States to the top-right of these curves are nonsteerable. (a) Nonsteerability borders for two-qubit states where $\mathit{a}=0.1{\widehat{\mathit{e}}}_{\mathit{z}}$ and $t_1^{\prime }=t_2^{\prime }$, given by Eq. (25). The methods presented in this paper uncover interesting regions of nonsteerable states, away from $t_1^{\prime }=t_2^{\prime }\approx t_3^{\prime }$. (b) Nonsteerability borders for $(\mathit{a}=0.2{\widehat{\mathit{e}}}_{\mathit{z}},T^{\prime }=\text{diag}[t_1^{\prime },t_2^{\prime }/2,t_3^{\prime }])$ states. By applying maps on Alice's qubit prior to steering we can prove a set of nonsteerable states which is a superset of those guaranteed by Eq. (7).