Causal compatibility inequalities admitting quantum violations in the triangle structure

It has long been recognized that certain quantum correlations are incompatible with a particular assumption about classical causal structure. Given a causal structure of unknown classicality, the presence of such correlations certifies the nonclassical nature of the causal structure in a device-independent fashion. In structures where all parties share a common resource, these nonclassical correlations are also known as nonlocal correlations. Any constraint satisfied by all correlations which are classically compatible with a given causal structure defines a causal compatibility criterion. Such criteria were recently derived for the triangle structure (E. Wolfe et al., arXiv:1609.00672) in the form of polynomial inequalities, begging the question of whether any of those inequalities admit violation by quantum correlations. Numerical investigation suggests that they do not, and we further conjecture that the set of correlations admitted by the classical triangle structure is equivalent to the set of correlations admitted by its quantum generalization whenever the three observable variables are binary. Our main contribution in this work, however, is the derivation of causal compatibility inequalities for the triangle structure which do admit quantum violation. This provides a robust-to-noise witness of quantum correlations in the triangle structure. We conclude by considering the possibility of quantum resources potentially qualitatively different from those known previously.

Figure 1

Bell structure consisting of two observers $A$ and $B$ together with measurement settings $S_A$ and $S_B$, respectively. The shared latent variable is labeled $\lambda$.

Figure 2

Triangle structure consisting of three observable variables $A$, $B$, and $C$ and three latent variables $X$, $Y$, and $Z$.

Figure 3

Triangle structure reimagined to mimic the Bell structure. The measurement settings $S_A$ and $S_B$ are latent nodes unlike the Bell structure (Fig. 1).

Figure 4

Fritz distribution visualized using a $4\times 4\times 4$ grid. The four outcomes of $A$, $B$, and $C$ are written in binary as a doublet of bits to illustrate that certain bits act as measurement pseudosettings.

Figure 5

Distribution ${\mathcal{N}}_{0.085}$, a noisy yet still nonclassical variant of the Fritz distribution.

Figure 6

Quantum probability distribution of the triangle structure that maximizes violation of $I_{\text{web}}$. Notice that this distribution has precisely the same possibilistic structure as the Fritz distribution, as might be expected, since $I_{\text{web}}$ was specifically generated to witness the nonclassicality of $P_{\text{F}}$.

Figure 7

Quantum probability distribution of the triangle structure that maximizes violation of $I_{\text{symmetric}}\text{web}$. Notice that this distribution has precisely the same (asymmetric) possibilistic structure as the Fritz distribution, even though $I_{\text{symmetric}}\text{web}$ itself is symmetric with respect to permutations of the variables $A$, $B$, and $C$.

Figure 8

Some inflations of the triangle structure: (a) spiral inflation, (b) wagon-wheel inflation, and (c) web inflation.

Figure 9

Some injectable sets ${\mathsf{Inj}}_{\mathcal{G}}^{}\left({\mathcal{G}}^{\prime }\right)$ of the spiral inflation ${\mathcal{G}}^{\prime }$ and their corresponding images ${\mathsf{ImInj}}_{\mathcal{G}}^{}\left({\mathcal{G}}^{\prime }\right)$ in the triangle structure $\mathcal{G}$: (a) ${\mathsf{AnSub}}_{{\mathcal{G}}^{\prime }}^{}\left(\left\{A_1\right\}\right)$, (b) ${\mathsf{AnSub}}_{\mathcal{G}}^{}\left(\left\{A\right\}\right)$, (c) ${\mathsf{AnSub}}_{{\mathcal{G}}^{\prime }}^{}\left(\{A_2,C_1\}\right)$, and (d) ${\mathsf{AnSub}}_{\mathcal{G}}^{}\left(\{A,C\}\right)$.