Witnessing Wigner Negativity

Negativity of the Wigner function is arguably one most striking non-classical features quantum states. Beyond its fundamental relevance, it also a necessary resource for speedup with continuous variables. As technologies emerge, need to identify and characterize resources which provide an advantage over existing classical becomes more pressing. Here we derive witnesses negativity single mode multimode states, based on fidelities Fock can be reliably measured using standard detection setups. They possess threshold expectation value indicating whether state has negative function. Moreover, amount violation provides operational quantification negativity. We phrase problem finding values our as infinite-dimensional linear optimisation. By relaxing restricting corresponding programs, two hierarchies semidefinite numerical sequences increasingly tighter upper lower bounds values. further show that both converge value. form complete family – each detected by at least witness thus providing reliable method experimentally witnessing states from few measurements. From foundational perspective, findings insights set positive functions still lacks proper characterisation.