Two-qubit causal structures and the geometry of positive qubit-maps

TL;DR

This paper explores quantum causal inference involving mixed cause-effect and common-cause scenarios, demonstrating how quantum mechanics can outperform classical methods in identifying causal structures through the geometry of qubit-maps.

Contribution

It introduces a geometric framework for analyzing unital positive qubit-maps, establishing bounds and conditions for quantum advantage in causal inference.

Findings

Quantum advantage in causal discrimination with unital channels.

Derived bounds on singular values for sums of matrices.

Identified conditions for unique causal structure solutions.

Abstract

We study quantum causal inference in a set-up proposed by Ried et al. [Nat. Phys. 11, 414 (2015)] 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.