Abstract
We study quantum causal inference in a setup proposed by Ried et al (2015 Nat. Phys. 11 414) in which a common cause scenario can be mixed with a cause–effect scenario, and for which it was found that quantum mechanics can bring an advantage in distinguishing the two scenarios: whereas in classical statistics, interventions such as randomized trials are needed, a quantum observational scheme can be enough to detect the causal structure if the common cause results from a maximally entangled state. We analyze this setup in terms of the geometry of unital positive but not completely positive qubit-maps, arising from the mixture of qubit channels and steering maps. We find the range of mixing parameters that can generate given correlations, and prove a quantum advantage in a more general setup, allowing arbitrary unital channels and initial states with fully mixed reduced states. This is achieved by establishing new bounds on signed singular values of sums of matrices. Based on the geometry, we quantify and identify the origin of the quantum advantage depending on the observed correlations, and discuss how additional constraints can lead to a unique solution of the problem.
Highlights
Imagine a scenario where two experimenters, Alice and Bob, sit in two distinct laboratories
We showed how the geometry of the set of signed singular values (SSV) of correlation matrices representing positive maps of the density operator ρA → ρB determines the possibility to reconstruct the causal structure linking ρA and ρB
We showed that there are more cases than previously known for which a complete solution of the causal inference problem can be found without additional constraints, namely all correlations created by maps whose signed singular values of the correlation matrix lie on the edges of the cube of positive maps C defined in (14)
Summary
Imagine a scenario where two experimenters, Alice and Bob, sit in two distinct laboratories. Some time later Bob obtains a coin and he checks whether it shows heads or tails This experiment is repeated many times (ideally: infinitely many times) and after this they meet and analyze their joint outcomes. Assuming their joint probability distribution entails correlations, there must be some underlying causal mechanism which causally connects their coins [1] This could be an unobserved confounder (acting as a common-cause), and they measured two distinct coins influenced by the confounder. The task of Alice and Bob is to determine the underlying causal structure, i.e. to distinguish the two scenarios This would be rather easy if Alice could prepare her coin after the observation by her choice and check whether this influences the joint probability (so-called “interventionist scheme”).
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