Analyzing causal structures using Tsallis entropies

Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realizable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional) independence between classical random variables and apply these to causal structures. We find that the number of independent constraints needed to characterize the causal structure is prohibitively high such that the computations required for the standard entropy vector method cannot be employed even for small causal structures. Instead, without solving the whole problem, we find new Tsallis entropic constraints for the triangle causal structure by generalizing known Shannon constraints. Our results reveal mathematical properties of classical and quantum Tsallis entropies and highlight difficulties of using Tsallis entropies for analyzing causal structures.

Figure 1

Some causal structures: Observed nodes are circled and the uncircled ones correspond to unobserved nodes. (a) The bipartite Bell causal structure. The nodes $A$ and $B$ represent the random variables corresponding to Alice's and Bob's choices of input while $X$ and $Y$ represent the random variables corresponding to their outputs. $\mathrm{\Lambda }$ here is the only potentially unobserved node and is the common cause of $X$ and $Y$. (b) The triangle causal structure. Here, the three observed nodes $X$, $Y$, and $Z$ have unobserved, pairwise common causes $A$, $B$, and $C$, but no joint common cause.

Figure 2

A circuit decomposition of the channel $\mathcal{E}:\mathcal{S}\left({\mathcal{H}}_{\mathbf{C}}\right)\to \mathcal{S}({\mathcal{H}}_{\mathbf{A}}\otimes {\mathcal{H}}_{\mathbf{B}})$ when $\mathbf{C}$ is a complete common cause of $\mathbf{A}$ and $\mathbf{B}$: If the map $\mathcal{E}$ from the system $C$ to the systems $A$ and $B$ can be decomposed as shown here, then $C$ is a complete common cause of $A$ and $B$ ([47]). We build up our result step by step considering the channels given by ${\mathcal{E}}_{\text{i}}$ (unitary), ${\mathcal{E}}_{\text{ii}}$ (unitary followed by local isometries), and ${\mathcal{E}}_{\text{iii}}=\mathcal{E}$.