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Witnessing Wigner Negativity

Chabaud, Ulysse and Emeriau, Pierre-Emmanuel and Grosshans, Frédéric (2021) Witnessing Wigner Negativity. Quantum, 5 . Art. No. 471. ISSN 2521-327X. doi:10.22331/q-2021-06-08-471.

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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.

Item Type:Article
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URLURL TypeDescription Paper
Chabaud, Ulysse0000-0003-0135-9819
Emeriau, Pierre-Emmanuel0000-0001-5155-1783
Grosshans, Frédéric0000-0001-8170-9668
Additional Information:This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. Published: 2021-06-08. U. Chabaud acknowledges stimulating discussions with S. Gribling, T. Freiman and T. Vidick. P.-E. Emeriau acknowledges interesting discussions with A. Oustry, E. Galvão and R. Soares Barbosa. We thank J. Eisert for his valuable comments on a previous version of this work and P. Paule for providing access to the Mathematica package for implementing Zeilberger’s algorithm. U. Chabaud acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907). F. Grosshans acknowledges funding from the ANR through the ANR-17-CE24-0035 VanQuTe project.
Group:Institute for Quantum Information and Matter
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Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Agence Nationale pour la Recherche (ANR)ANR-17-CE24-0035
Record Number:CaltechAUTHORS:20210630-160717654
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109672
Deposited By: Tony Diaz
Deposited On:30 Jun 2021 18:12
Last Modified:30 Jun 2021 18:12

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