Error-free interconversion of nonlocal boxes

Understanding the structure of nonlocal correlations is important in many fields ranging from fundamental questions of physics to device-independent cryptography. We present a protocol that can convert extremal two-party–two-input nonlocal no-signaling boxes of any type into any other extremal two-party–two-input nonlocal no-signaling box perfectly. Our results are exact, and even though the number of required boxes cannot be determined in advance, their expected number is finite. Our protocol is adaptive and demonstrates for the first time the usefulness of using no-signaling boxes in different causal orders by the parties.

Figure 1

Nontrivial output pairs of the parties and their interdependence when applying Protocol 4 on two boxes with $d=2$. Pairs of numbers represent output pairs $a_1{a}_2$ or $b_1{b}_2$ when $x=y=1$ (highlighted in bold in Table 1). If there is a nonzero probability for a given pair of outputs, then an arrow is drawn from one pair to the other in the sense that Alice's output stands at the base, while Bob's output stands at the point of the arrow. The colors correspond to the colored entries of Table 1. Due to the nature of the $d$-boxes, the correlations are unique, i.e., $a_1{a}_2$ determines $b_1{b}_2$ and vice versa. Therefore, there is only one arrow starting and ending at every point.

Figure 2

Nontrivial output pairs of the parties and their interdependence when applying Protocol 4 on two boxes with $d=5$. Pairs of numbers represent output pairs $a_1{a}_2$ or $b_1{b}_2$. If there is a correlation between a given pair of outputs, then an arrow is drawn from one pair to the other in the sense that Alice's output stands at the base, while Bob's output stands at the point of the arrow. The four possible six-cycles are highlighted in red, green, blue, and black.

Figure 3

Nontrivial output pairs of the parties and their interdependence when applying the modified version of Protocol 4 in the case of $d=5$.

Figure 4

Nontrivial output pairs of the parties and their interdependence when applying the second modified version of Protocol 4 in the case of $d=5$ and $s=4$.