Maximally Nonprojective Measurements Are Not Always Symmetric Informationally Complete

Standard quantum measurements are projective. However, the full scope of quantum measurements is represented by positive operator-valued measures (POVMs) and many of these break the limitations of projective measurements as resources in quantum information. It is therefore natural to consider how accurately an experimenter with access only to projective measurements and classical processing can simulate POVMs. The most well-known class of nonprojective measurements is called symmetric informationally complete (SIC). Such measurements are both ubiquitous in the broader scope of quantum information theory and known to be the most strongly nonprojective measurements in qubit systems. Here, we show that beyond qubit systems, the SIC property is in general not associated with the most nonprojective measurement. For this, we put forward a semidefinite programming criterion for detecting genuinely nonprojective measurements. This method allows us to determine quantitative simulability thresholds for generic POVMs and to put forward a conjecture on which qutrit and ququart measurements are most strongly nonprojective.