Randomness-free Detection of Non-projective Measurements: Qubits & beyond

Abstract: Non-projective measurements play a crucial role in various information-processing protocols. In this work, we propose an operational task to identify measurements that are neither projective nor classical post-processing of data obtained from projective measurements. Our setup involves space-like separated parties with access to a shared state with bounded local dimensions. Specifically, in the case of qubits, we focus on a bipartite scenario with different sets of target correlations. While some of these correlations can be obtained through non-projective measurements on a shared two-qubit state, it is impossible to generate these correlations using {\it projective simulable} measurements on bipartite qubit states, or equivalently, by using one bit of shared randomness and local post-processing. For certain target correlations, we show that detecting qubit non-projective measurements is robust under {\it arbitrary} depolarising noise, except in the limiting case. We extend this task for qutrits and demonstrate that some correlations achievable via local non-projective measurements cannot be reproduced by both parties performing the same qutrit {\it projective simulable} measurements on their pre-shared state. We provide numerical evidence for the robustness of this scheme under {\it arbitrary} depolarising noise. For a more generic consideration (bipartite and tripartite scenario), we provide numerical evidence for a projective-simulable bound on the reward function for our task. We also show a violation of this bound by using qutrit POVMs. From a foundational perspective, we extend the notion of non-projective measurements to general probabilistic theories (GPTs) and use a randomness-free test to demonstrate that a class of GPTs, called {\it square-bits or box-world} are unphysical.