Optimal simultaneous measurements of incompatible observables of a single photon

Heinz Dieter Dombrowski

Adetunmise C. Dada, Will McCutcheon, Erika Andersson +6 more

Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to errors in estimates. We perform the first optical implementation of the simplest possible optimal joint measurement, requiring less quantum resources than have previously often been employed. Using a heralded single-photon source, we demonstrate quantum-limited performance of the scheme on single quanta.

Willem M. de Muynck, J. M. V. A. Koelman

M. E. O. Bezerra, Francesco Albarelli, Rafał Demkowicz-Dobrzański

Abstract Quantum multiparameter metrology is hindered by incompatibility issues, such as finding a single probe state (probe incompatibility) and a single measurement (measurement incompatibility) optimal for all parameters. The simultaneous estimation of phase shift and loss in a single optical mode is a paradigmatic multiparameter metrological problem in which such tradeoffs are present. We consider two settings: single-mode or two-mode probes (with a reference lossless mode), and for each setting we consider either Gaussian states or arbitrary quantum states of light restricted only by a maximal number of photons allowed. We find numerically that, as the number of photons increases, there are quantum states of light for which probe incompatibility disappears both in the single- and two-mode scenarios. On the other hand, for Gaussian states, probe incompatibility is present in the single-mode case and may be removed only in the two-mode setting thanks to the entanglement with the reference mode. Finally, we provide strong arguments that the fundamental incompatibility aspect of the model is measurement incompatibility, which persists for all the scenarios considered, and unlike probe-incompatibility cannot be overcome even in the large photon number limit.

Hugo Ferretti, Y. Batuhan Yilmaz, Kent Bonsma-Fisher +4 more

A primary goal of quantum metrology is to understand the optimal way to measure multiple incompatible observables. Here, we report on the experimental demonstration and characterisation of a four-photon state, the “tetrahedron state”, which is the optimal state for simultaneous estimation of all parameters describing a rotation.

John Rarity