Observable-projected ensembles

Measurements in many-body quantum systems can generate non-trivial phenomena, such as preparation of long-range entangled states, dynamical phase transitions, or measurement-altered criticality. Here, we introduce a new measurement scheme that produces an ensemble of mixed states in a subsystem, obtained by measuring a local Hermitian observable on part of its complement. We refer to this as the $\textit{observable-projected ensemble}$. Unlike standard projected ensembles-where pure states are generated by projective measurements on the complement-our approach involves projective partial measurements of specific observables. This setup has two main advantages: theoretically, it is amenable to analytical computations, especially within conformal field theories. Experimentally, it requires only a linear number of measurements, rather than an exponential one, to probe the properties of the ensemble. As a first step in exploring the observable-projected ensemble, we investigate its entanglement properties in conformal field theory and perform a detailed analysis of the free compact boson.