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Physical Review A

Efficient detection of nonclassicality using moments of the Wigner function

Bivas Mallick, Sudip Chakrabarty, Saheli Mukherjee, Ananda G. Maity, A. S. Majumdar

Published Mar 4, 2025• 9 citations
Publisher

Abstract

States with a negative Wigner function, a significant subclass of nonclassical states, serve as a valuable resource for various quantum information processing tasks. Here we provide a criterion for detecting such quantum states characterized by a negative Wigner function. Our method relies on evaluating moments of the Wigner function, which involves computing simple functionals and can be implemented in a real experiment without the need for full state tomography or Wigner function reconstruction. We provide explicit examples to support our detection scheme. Further, we propose an experimental method utilizing the mode swap operator to realize these moments in a real experiment.

DOI
10.1103/physreva.111.032406

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States with a negative Wigner function, a significant subclass of nonclassical states, serve as a valuable resource for various quantum information processing tasks. Here we provide a criterion for detecting such quantum states characterized by a negative Wigner function. Our method relies on evaluating moments of the Wigner function, which involves computing simple functionals and can be implemented in a real experiment without the need for full state tomography or Wigner function reconstruction. We provide explicit examples to support our detection scheme. Further, we propose an experimental method utilizing the mode swap operator to realize these moments in a real experiment.
States with negative Wigner functions constitute a fundamental nonclassical resource underlying quantum advantage. However, their experimental certification typically relies on reconstructing the full phase-space distribution, resulting in a prohibitive measurement overhead for arbitrary quantum states. In this letter, we overcome this limitation by introducing Wigner moments, a family of global phase-space quantities that admit an exact multicopy realization as parity expectation values, and are therefore directly measurable from only a modest number of state copies. This operational correspondence enables systematic hierarchies of detection criteria together with experimentally accessible lower bounds on the logarithmic Wigner negativity, and constitutes a genuine measure of nonclassicality. Numerical benchmarking of our protocol reveals a substantial reduction in copy budget relative to conventional Wigner tomography, establishing Wigner moments as an efficient framework for certifying continuous-variable quantum resources such as genuine multipartite entanglement.
States with negative Wigner function, a significant subclass of non-classical states, serve as a valuable resource for various quantum information processing tasks. Here, we provide a criterion for detecting such quantum states exhibiting negative Wigner function. Our method relies on evaluating moments of the Wigner function which involves computing simple functionals and can be implemented in a real experiment without the need for full state tomography or Wigner function reconstruction. We then provide explicit examples to support our detection scheme. Further, we propose an experimental method utilizing the continuous variable SWAP operator to realize these moments in a real experiment.
Complex numbers, intrinsic to the formulation of quantum theory, play a pivotal role in enabling advantages across a broad range of quantum information-processing tasks. Despite their fundamental importance, practical and scalable criteria for detecting quantum imaginarity remain relatively underexplored, particularly methods that enable its identification with reduced experimental overhead. In this work, we propose a realistic and experimentally feasible method to detect quantum imaginarity using moment-based approach. Our framework relies on experimentally accessible moments of the Kirkwood-Dirac quasiprobability distribution, enabling scalable detection in many-body and high-dimensional systems without requiring full state tomography. We then present an illustrative example to support our detection scheme. Finally, we present an interferometric scheme for measuring these moments, paving the way for experimental implementation of our detection protocol.
The Kirkwood-Dirac (KD) distribution has recently emerged as a powerful quasiprobability framework with wide-ranging applications in quantum information processing tasks. In this work, we introduce an experimentally motivated criterion for detecting nonclassical signatures of the KD distribution using its statistical moments and demonstrate its effectiveness through explicit examples. We further show that this approach extends naturally to identifying other quantum resources, such as quantum coherence and nonclassical extractable work -- that are intrinsically connected to the KD distribution. Our criteria involves the evaluation of simple functionals, making it well-suited for efficient experimental implementation.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
Higher-dimensional entanglement is a valuable resource for several quantum information processing tasks, and is often characterized by the Schmidt number and specific classes of entangled states beyond qubit-qubit and qubit-qutrit systems. We propose a criterion to detect higher-dimensional entanglement, focusing on determining the Schmidt number of quantum states and identifying significant classes of positive partial transposition and negative partial transposition entangled states. Our approach relies on evaluating moments of generalized positive maps which can be efficiently simulated in real experiments without the requirement of full-state tomography. We demonstrate the effectiveness of our detection scheme through various illustrative examples. As a direct application, we explore the implications of our moment-based detection schemes in identifying useful quantum channels such as non-Schmidt-number breaking channels and non-entanglement breaking channels. Finally, we present an operational implication of our proposed moment criterion through its manifestation in channel discrimination tasks.