Focusing in Arthurs-Kelly-type joint measurements with correlated probes

Joint approximate measurement schemes of position and momentum provide us with a means of inferring pieces of complementary information if we allow for the irreducible noise required by quantum theory. One such scheme is given by the Arthurs-Kelly model, where information about a system is extracted via indirect probe measurements, assuming separable uncorrelated probes. Here, following Di Lorenzo [Phys. Rev. Lett. 110, 120403 (2013)], we extend this model to both entangled and classically correlated probes, achieving full generality. We show that correlated probes can produce more precise joint measurement outcomes than the same probes can achieve if applied alone to realize a position or momentum measurement. This phenomenon of focusing may be useful where one tries to optimize measurements with limited physical resources. Contrary to Di Lorenzo's claim, we find that there are no violations of Heisenberg's error-disturbance relation in these generalized Arthurs-Kelly models. This is simply due to the fact that, as we show, the measured observable of the system under consideration is covariant under phase space translations and as such is known to obey a tight joint measurement error relation.