Resource theory of interactive quantum instruments

Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a nontrivial way, instruments must have the capability to interact coherently with the state that they measure. Here we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and noninteractive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterizes the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.