Entanglement swapping for generalized nonlocal correlations

We consider an analog of entanglement-swapping for a set of black boxes with the most general nonlocal correlations consistent with relativity (including correlations which are stronger than any attainable in quantum theory). In an attempt to incorporate this phenomenon, we consider expanding the space of objects to include not only correlated boxes, but “couplers,” which are an analog for boxes of measurements with entangled eigenstates in quantum theory. Surprisingly, we find that no couplers exist for two binary-input–binary-output boxes, and hence that there is no analog of entanglement swapping for such boxes.

Figure 1

Two correlated boxes held by Alice and Bob, in the state $P(ab\mid xy)$. The dashed line between the two boxes represents that they are nonlocally correlated.

Figure 2

Diagram showing the two PR states held by Alice and Bob, and by Bob and Charlie, and a circuit applied by Bob to his two boxes.

Figure 3

In order to generate nonlocal correlations between Alice and Charlie, Bob attaches a coupler to the input and output channels of his two boxes. The coupler generates a one-bit output $b^{\prime }$.

Figure 4

Alice and Charlie bring their boxes together and apply a coupler between then, while Bob applies his coupler as normal. If Alice and Charlie applies their coupler first and obtain $a^{\prime }=0$, Bob’s two boxes will collapse into a standard PR state. In this case, Bob’s coupler will output the same result as if it had been applied directly to a PR state.

Figure 5

Diagram showing how to generate $\chi (b,b_1{b}_2{y}_1{y}_2)$ for the standard potential couplers $(\alpha =\beta =\gamma =\delta =ϵ=0)$ for each of the five classes defined in Table using wiring and logic gates (AND and XOR). In each circuit, the box on the left is the box with input $y_1$.