Quantum causal inference in the presence of hidden common causes: An entropic approach

Quantum causality is an emerging field of study which has the potential to greatly advance our understanding of quantum systems. In this paper, we put forth a theoretical framework for merging quantum information science and causal inference by exploiting entropic principles. For this purpose, we leverage the tradeoff between the entropy of hidden cause and the conditional mutual information of observed variables to develop a scalable algorithmic approach for inferring causality in the presence of latent confounders (common causes) in quantum systems. As an application, we consider a system of three entangled qubits and transmit the second and third qubits over separate noisy quantum channels. In this model, we validate that the first qubit is a latent confounder and the common cause of the second and third qubits. In contrast, when two entangled qubits are prepared and one of them is sent over a noisy channel, there is no common confounder. We also demonstrate that the proposed approach outperforms the results of classical causal inference for the Tubingen database when the variables are classical by exploiting quantum dependence between variables through density matrices rather than joint probability distributions. Thus, the proposed approach unifies classical and quantum causal inference in a principled way.

Figure 1

(a) Latent graph, (b) triangle graph, and (c) direct graph.

Figure 2

Two-bit symmetric channel.

Figure 3

Validation of latent graph in Model 1 (Part I) for $T=0.05$ and $\beta \in (0,1)$ via QInferGraph.

Figure 4

Validation of latent graph in Model 1 (Part I) for $T=0.01,0.005$, and $\beta \in (0,1)$ via QInferGraph.

Figure 5

Validation of latent graph in Model 1 (Part I) via classical causal inference (Algorithm 2), $T=0.001$, and $\beta \in (0,1)$.

Figure 6

Entropy of possible latent confounder $Z$: QInferGraph vs the classical InferGraph algorithm for Model 1 (Part I) with $p_1=0.1, p_2=0.2$, and mutual conditional independence threshold $T=0.05$.

Figure 7

Trade-off curve discovered by the classical LatentSearch and QLatentSearch for the pair in Model 1 (Part I) with $p_1=0.1$ and $p_2=0.2$. Each point is the output of algorithms for a different value of $\beta \in (0,1).$

Figure 8

Validation of direct graph in Model 1 (Part II) via QInferGraph and $\beta \in (0,1)$.

Figure 9

Validation of latent graph in Model 1 (Part II) via classical causal inference (Algorithm 2) and $\beta \in (0,1)$.

Figure 10

Entropy of possible latent confounder $Z$: QInferGraph vs the classical InferGraph algorithm for Model 1 (Part II) with $p=0.2$ and mutual conditional independence threshold $T=0.05$.

Figure 11

Trade-off curve discovered by the classical LatentSearch and QLatentSearch for the pair in Model 1 (Part II) with $p=0.2$. Each point is the output of algorithms for a different value of $\beta \in (0,1).$

Figure 12

Tubingen: Database with cause-effect pairs of the form (a) or (b).

Figure 13

First cause-effect pair of data from Tubingen database: altitude causes temperature.

Figure 14

Entropy of possible latent confounder $Z$: QInferGraph vs the classical InferGraph algorithm for the first cause-effect pair of Tubingen database with mutual conditional independence threshold $T=0.005$ for QInferGraph. Note that for the classical InferGraph algorithm the minimum-obtained mutual conditional independence $I(X;Y|Z)$ is 0.174265303 and there exist no $Z$ such that $I(X;Y|Z)<0.001$. Also, note that $H\left(X\right)=0.791249247$ and $H\left(Y\right)=0.989477143$, where $X$ is altitude and $Y$ is temperature.

Figure 15

Validation of latent graph in Model 1 (Part I) for $\alpha =0.2$ and $\beta \in (0,1)$ via QInferGraph and with the density matrix obtained from $0.6{\rho }_{ZXY}^{1/\sqrt{2},1/\sqrt{2}}+0.4{\rho }_{ZXY}^{0.6,0.8}$ via tracing out $Z$.

Figure 16

Validation of direct graph in Model 2 (Part II) with joint density matrix ${\rho }_{XY}=0.4\times {\rho }_{XY}^{0.6,0.8}+0.6\times {\rho }_{XY}^{1,0}$.

Figure 17

Classical approach to identify direct Ggraph for Model does not work.