Witnessing Wigner Negativity

Ulysse Chabaud1,2, Pierre-Emmanuel Emeriau3, and Frédéric Grosshans3

1Institute for Quantum Information and Matter, Caltech
2Université de Paris, IRIF, CNRS, France
3Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

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Updated version: The authors have uploaded version v6 of this work to the arXiv which may contain updates or corrections not contained in the published version v4. The authors left the following comment on the arXiv:
29 pages + 40 pages of appendices, 6 figures, 4 tables. v2: Added link to the set of Wigner positive states. v3: Extension to the multimode setting. v4: Accepted to Quantum; fixed typos. v5: Mistake in Eq.(57) in v4 and revision accordingly. v6: more details on programs p.6


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.

Continuous-variable quantum information uses information encoded in the continuous degrees of freedom of quantum systems and is a promising candidate for quantum computing. In continuous variables, quantum states may be represented equivalently in phase space via their Wigner function.

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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