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

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.

arXiv comment36 pages, 6 figures

References in corpus (12)

• Quantum correlations with no causal order
• Informational derivation of Quantum Theory
• Experimental Superposition of Orders of Quantum Gates
• Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
• Quantum causal modelling
• Perfect discrimination of no-signalling channels via quantum superposition of causal structures
• Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication
• Quantum Steering Ellipsoids
• Inferring causal structure: a quantum advantage
• Qubit Entanglement Breaking Channels
• Quantum steering ellipsoids, extremal physical states and monogamy
• A universal set of qubit quantum channels

Cited by in corpus (6)

• Causal classification of spatiotemporal quantum correlations
• Positive Maps From Irreducibly Covariant Operators
• Quantum observation scheme universally identifying causalities from correlations
• The Min-Entropy of Classical-Quantum Combs for Measurement-Based Applications
• Quantifying Causal Influence in Quantum Mechanics
• Quantum causal models via QBism