Quantum modelling of causal influence and knowledge of causal influence

Develop a rigorous framework for modelling causal influence and knowledge of causal influence in quantum theory, including specifying the appropriate quantum objects that play the roles of deterministic functions and probability distributions over functions so that resource theories analogous to RTCaus and RTKnowCaus can be formulated.

Background

The paper develops two classical resource theories: RTCaus, which treats deterministic functions as resources to quantify causal influence, and RTKnowCaus, which treats probability distributions over functions as resources to quantify knowledge of causal influence. Extending these ideas to the quantum domain requires identifying the correct quantum counterparts of these classical objects.

The authors emphasize that completely positive trace-preserving (CPTP) maps are quantum analogues of stochastic maps rather than of distributions over functions, so existing frameworks such as quantum Shannon theory and resource theories based solely on CPTP maps do not capture knowledge of causal influence. They also note that superpositions of causal orders and other quantum features will likely require distinct formal treatment from the classical setting, and suggest that categorical methods may offer a useful direction.