Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.
The negativity of the Wigner function is a sign of non-classicality and a necessary resource for any quantum computational speedup. Detecting this negativity for experimental quantum states is therefore crucial for the development of continuous-variable quantum technologies. However, this detection can be very difficult as it usually relies on quantum state tomography, which requires an exponential number of samples compared to the system size.
In this work, we propose an alternative, more efficient, approach which introduces specific observables equipped with threshold values, such that if the expectation value of an observable with an unknown quantum state exceeds its threshold value then the state is certified to exhibit Wigner negativity.
Our results pave the way for the characterisation of non-classical quantum states, with direct applications in quantum optics experiments. The mathematical aspects of our work also motivate further study of the set of quantum states with positive Wigner function.
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