Quantifying quantum causal influences

Causal influences are at the core of any empirical science, the reason why its quantification is of paramount relevance for the mathematical theory of causality and applications. Quantum correlations, however, challenge our notion of cause and effect, implying that tools and concepts developed over the years having in mind a classical world have to be reevaluated in the presence of quantum effects. Here, we propose the quantum version of the most common causality quantifier, the average causal effect, measuring how much a target quantum system is changed by interventions on its presumed cause. Not only does it offer an innate manner to quantify causation in two-qubit gates but also in alternative quantum computation models such as the measurement-based version, suggesting that causality can be used as a proxy for optimizing quantum algorithms. Considering quantum teleportation, we show that any pure entangled state offers an advantage in terms of causal effects as compared to separable states. This broadness of different uses showcases that, just as in the classical case, the quantification of causal influence has foundational and applied consequences and can lead to a yet totally unexplored tool for quantum information science.

Figure 1

The setup to calculate the causal influence of quantum gates. Here the influence is measured from the entry $|a\rangle$ to the output of $|b\rangle$ (red lines). We perform a Haar-random average over $|b\rangle$ and a partial trace over the first output qubit.

Figure 2

${\mathcal{A}}_Q$ as a function of the concurrence. The green, red, blue, and orange curves correspond to families of states $|H_2^{\epsilon }\rangle , {\rho }_{\mathrm{iso}}, |G_2^{\epsilon }\rangle$, and $|F_2^{\epsilon }\rangle$, respectively, as defined in the main text. The black horizontal line delimits the shaded region and corresponds to ${\mathcal{A}}_Q^{\mathrm{sep}}=2/\pi$. Points correspond to 10000 uniformly sampled pure states.

Figure 3

${\mathcal{A}}_Q$ as a function of the concurrence. The green and red curves correspond to families of states $|F_2^{\epsilon }\rangle$ and ${\rho }_{\mathrm{iso}}$, respectively. The blue curve corresponds to either $|H_2^{\epsilon }\rangle$ or $|G_2^{\epsilon }\rangle$. The black horizontal line delimits the shaded region and corresponds to ${\mathcal{A}}_Q\left({\rho }_{\text{sep}}\right)=1/2$. Orange points correspond to 10000 uniformly-sampled pure states, whereas purple points correspond to 10000 randomly sampled mixed states.