Quantum Uncertainty Principles for Measurements with Interventions*

Heisenberg's uncertainty principle encapsulates an iconic difference

between classical and quantum mechanics. Whereas any two observable

properties of a classical system may be probed simultaneously, the uncertainty

principle predicts the existence of incompatible observables – observables

whose measurement outcomes cannot be both known to arbitrary precision.

Yet, these observables generally represent a choice of what to measure at a

single time point – and do not represent the most powerful means of probing

unknown environments. To do that, we need interventions. Interactive

measurements permeate diverse sciences. Whether using reinforcement

learning to explore optimal strategies in competitive games or sending data

packets to probe network characteristics – intervention is critical. Can

interactive measurements also be mutually non-compatible, and if so, what are

the fundamental limits in simultaneously predicting their outcomes? Our work

addresses these questions by formulating an entropy uncertainty relation for

general interactive measurements. We then present a case study in causal

inference, illustrating a causal uncertainty relation that predicts an entropic

trade-off in learning the outcomes of measurements compatible with different

causal structures.