Cyclic Einstein-Podolsky-Rosen steering

Einstein-Podolsky-Rosen (EPR) steering is a form of quantum correlation that exhibits a fundamental asymmetry in the properties of quantum systems. Given two observers, Alice and Bob, there are known to exist bipartite entangled states which are one-way steerable in the sense that Alice can steer Bob's state, but Bob cannot steer Alice's state. Here, we generalize this phenomenon to three parties and find a cyclic property of tripartite EPR steering. In particular, we identify a three-qubit state whose reduced bipartite states are one-way steerable for arbitrary projective measurements. Moreover, the three-qubit state has a cyclic steering property in the sense that by arranging the system in a triangular configuration the neighboring parties can steer each other's states only in the same (e.g., clockwise) direction. That is, Alice can steer Bob's state, Bob can steer Charlie's state, and Charlie can steer Alice's state, but not the other way around.

Figure 1

Setup for cyclic EPR steering. The figure depicts a three-qubit state with its two-party reduced states shown as green rectangles. The yellow arrows indicate that steering takes place in the clockwise direction. That is, steering occurs from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. However, steering is not possible in the other (anticlockwise) direction, even if the untrusted party can perform arbitrary projective measurements.

Figure 2

Plot for the Bloch vectors of Alice's and Bob's measurements in EPR-steering scenario 1. Each measurement axis is defined by a pair of circles. The red circles correspond to Alice's Bloch vectors $\pm {\vec{a}}_x$, and the blue circles correspond to Bob's Bloch vectors $\pm {\vec{b}}_x$. Note that ${\vec{a}}_6$ is the zero-length vector corresponding to a degenerate measurement.