Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering

In a recent work [Phys. Rev. Lett. 98, 140402 (2007)] we defined “steering,” a type of quantum nonlocality that is logically distinct from both nonseparability and Bell nonlocality. In the bipartite setting, it hinges on the question of whether Alice can affect Bob’s state at a distance through her choice of measurement. More precisely and operationally, it hinges on the question of whether Alice, with classical communication, can convince Bob that they share an entangled state under the circumstances that Bob trusts nothing that Alice says. We argue that if she can, then this demonstrates the nonlocal effect first identified in the famous Einstein-Podolsky-Rosen paper [Phys. Rev. 47, 777 (1935)] as a universal effect for pure entangled states. This ability of Alice to remotely prepare Bob’s state was subsequently called steering by Schrödinger, whose terminology we adopt. The phenomenon of steering has been largely overlooked, and prior to our work had not even been given a rigorous definition that is applicable to mixed states as well as pure states. Armed with our rigorous definition, we proved that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. In this work we expand on these results and provide further examples of steerable states. We also elaborate on the connection with the original EPR paradox.

Figure 1

(Color online) Operational definitions for classes of entangled states. Bell-nonlocal states (a) can be defined via a three-party task (involving Alice [A], Bob [B], and Charlie [C]). Steerable states (b) may be defined using a two-party task. Defining an entangled state (c) essentially requires only one party. In all cases shading indicates the skeptical party, dotted arrows indicate two-way communication, and solid arrows indicate trust and two-way communication.

Figure 2

(Color online) Boundaries between classes of entangled states for Werner (a) and isotropic (b) states $W_d^{\eta }$, inept states $W_{ϵ}^{\eta }$ (c), and two-mode symmetric Gaussian states $W_{\overline{n}}^{\eta }$ (d). The bottom (blue) line is ${\eta }_{\mathrm{ent}}$, above which states are entangled. The next (red) line is ${\eta }_{\text{steer}}$, above which states are steerable. In cases (c) and (d) the down arrows indicate that we have only an upper bound on ${\eta }_{\text{steer}}$. The top (green) line with down arrows is an upper bound on ${\eta }_{\text{Bell}}$, above which states are Bell nonlocal. The up arrows in cases (a) and (b) are lower bounds on ${\eta }_{\text{Bell}}$ for $d=2$. This lower bound establishes that the classes are strictly distinct. In cases (a) and (b), dots join values at finite $d$ with those at $d=\infty$. The separate point in (c) is explained at the end of Sec. 4C.