Extremal steering assemblages

Nonlocal correlations between a fully characterized quantum system and an untrusted black box device are described by an assemblage of conditional quantum states. These assemblages form a convex set, whose extremal points are relevant in many operational contexts. We give necessary and sufficient conditions for an assemblage to be extremal using linear independence conditions and an algorithm to decompose a generic assemblage into extremal points. A matlab implementation of this algorithm is also provided.

Figure 1

The semi-device-independent scenario: One party is treated as a black box, in which classical inputs can be chosen, with the device then giving a corresponding classical output. The other party holds a completely controllable quantum system. During operation the two systems are prevented from communication.

Figure 2

For any nonextremal point of a convex set, we can decompose it along a perturbation. By contrast, any attempt to do this for an extremal point will take us outside the set.

Figure 3

An illustration of how repeated perturbations lead to a decomposition of any point into extremals. Note how different choices in the direction of each perturbation can lead to completely different decompositions.