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"Enough" Wigner negativity implies genuine multipartite entanglement

Lin Htoo Zaw, Jiajie Guo, Qiongyi He, Matteo Fadel, Shuheng Liu

TL;DR

This work reveals two rigorous links between Wigner negativity and genuine multipartite entanglement (GME) in multimode continuous-variable systems. It shows that either a large enough negativity volume along a carefully chosen two-dimensional phase-space slice or persistent negativity of the centre-of-mass Wigner function after Gaussian smoothing certifies GME, with the latter also bounding the trace distance to non-GME states. A nonclassicality-depth condition on the centre-of-mass mode provides a practical sufficiency for generating GME via vacuum interference with a maximally mixing multimode interferometer, complementing known necessary conditions. Furthermore, the authors derive experimentally friendly GME criteria based on measuring a finite region of the Wigner function or a finite set of characteristic-function points, enabling implementation in cQED, cQAD, and trapped-ion platforms where quadrature measurements are not readily available.

Abstract

Wigner negativity and genuine multipartite entanglement (GME) are key nonclassical resources that enable computational advantages and broader quantum-information tasks. In this work, we prove two theorems for multimode continuous-variable systems that relate these nonclassical resources. Both theorems show that "enough" Wigner negativity -- either a large-enough Wigner negativity volume along a suitably-chosen two-dimensional slice, or a large-enough nonclassicality depth of the centre-of-mass of a system -- certifies the presence of GME. Moreover, violations of the latter inequality provide lower bounds of the trace distance to the set of non-GME states. Our results also provide sufficient conditions for generating GME by interfering a state with the vacuum through a multiport interferometer, complementing long-known necessary conditions. Beyond these fundamental connections, our methods have practical advantages for systems with native phase-space measurements: they require only measuring the Wigner function over a finite region, or measuring a finite number of characteristic function points. Such measurements are frequently performed with readouts common in circuit/cavity quantum electrodynamic systems, trapped ions and atoms, and circuit quantum acoustodynamic systems. As such, our GME criteria are readily implementable in these platforms.

"Enough" Wigner negativity implies genuine multipartite entanglement

TL;DR

This work reveals two rigorous links between Wigner negativity and genuine multipartite entanglement (GME) in multimode continuous-variable systems. It shows that either a large enough negativity volume along a carefully chosen two-dimensional phase-space slice or persistent negativity of the centre-of-mass Wigner function after Gaussian smoothing certifies GME, with the latter also bounding the trace distance to non-GME states. A nonclassicality-depth condition on the centre-of-mass mode provides a practical sufficiency for generating GME via vacuum interference with a maximally mixing multimode interferometer, complementing known necessary conditions. Furthermore, the authors derive experimentally friendly GME criteria based on measuring a finite region of the Wigner function or a finite set of characteristic-function points, enabling implementation in cQED, cQAD, and trapped-ion platforms where quadrature measurements are not readily available.

Abstract

Wigner negativity and genuine multipartite entanglement (GME) are key nonclassical resources that enable computational advantages and broader quantum-information tasks. In this work, we prove two theorems for multimode continuous-variable systems that relate these nonclassical resources. Both theorems show that "enough" Wigner negativity -- either a large-enough Wigner negativity volume along a suitably-chosen two-dimensional slice, or a large-enough nonclassicality depth of the centre-of-mass of a system -- certifies the presence of GME. Moreover, violations of the latter inequality provide lower bounds of the trace distance to the set of non-GME states. Our results also provide sufficient conditions for generating GME by interfering a state with the vacuum through a multiport interferometer, complementing long-known necessary conditions. Beyond these fundamental connections, our methods have practical advantages for systems with native phase-space measurements: they require only measuring the Wigner function over a finite region, or measuring a finite number of characteristic function points. Such measurements are frequently performed with readouts common in circuit/cavity quantum electrodynamic systems, trapped ions and atoms, and circuit quantum acoustodynamic systems. As such, our GME criteria are readily implementable in these platforms.

Paper Structure

This paper contains 5 sections, 8 theorems, 68 equations, 2 figures.

Table of Contents

  1. Proof of Theorem \ref{['thm:GME-2D-slice']}
  2. Proof of Theorem \ref{['thm:GMN-reduced-state']}
  3. Proof of Corollary \ref{['col:nonclassicality-depth-reduced-state']}
  4. Proof of Corollary \ref{['col:GME-Wigner-witness']}
  5. Proof of Corollary \ref{['col:GME-characteristic-witness']}

Key Result

Theorem 1

Choose some coefficients $\vec{y},\vec{z}\in\mathbb{C}^M$ such that $\vec{y}\circ\vec{y}^* - \vec{z}\circ\vec{z}^* = \vec{1}$, where $[\mathbf{A}\circ\mathbf{B}]_{m,n} = [\mathbf{A}]_{m,n}[\mathbf{B}]_{m,n}$ is the elementwise product and $\vec{1} = (1,1,\dots,1)$ is a vector of ones. This specifies Then, $\mathcal{N}_{2D}(\rho) > \mathcal{N}_{2D}^{\operatorname{GME}}(\rho)$ implies that $\rho$ is

Figures (2)

  • Figure 1: An illustration of our results that relate negativities in particular regions of the Wigner function to the presence of GME. The central figure is a three-dimensional slice of the Wigner function of an exemplary tripartite state specified in the End Matter. \ref{['thm:GME-2D-slice']} states that a large-enough negativity volume along a suitably-chosen two-dimensional slice implies GME, while \ref{['thm:GMN-reduced-state']} states that the persistence of negativities in the centre-of-mass Wigner function after a suitable smoothing process also implies GME.
  • Figure 2: (a) The Wigner function $W_{\ket{\operatorname{W}_{3}}}(\vec{\alpha})$ of the tripartite W state along the slice $\alpha_1=\alpha_2=\alpha_3$. The GME of this state can be certified using \ref{['col:GME-Wigner-witness']} by just integrating $W_{\ket{\operatorname{W}_{3}}}(\vec{\alpha})$ along this two-dimensional slice over the region $0 \leq \abs{\alpha_m} \lesssim r$ for any $r \gtrsim 0.7$. (b) The characteristic function $\chi_{\ket{\operatorname{W}_{3}}}(\vec{\xi})$ of $\ket{\operatorname{W}_{3}}$ along $\xi_1=\xi_2=\xi_3$. Its GME is certified using \ref{['col:GME-characteristic-witness']} by just measuring $\chi_{\ket{\operatorname{W}_{3}}}(\vec{\xi})$ at 10 of the 19 points marked out as crosses, with the other 9 values obtained from the symmetry $\chi_{\ket{\operatorname{W}_{3}}}(-\vec{\xi}) = \chi_{\ket{\operatorname{W}_{3}}}^*(\vec{\xi})$.

Theorems & Definitions (15)

  • Theorem 1: Enough Wigner negativity volume along a two-dimensional slice implies GME
  • Theorem 2: Negativity of the smoothed Wigner function of the centre-of-mass implies GME
  • Corollary 1: Enough nonclassicality depth of the centre-of-mass implies GME
  • Corollary 2: GME criterion with Wigner function measurements over a finite region
  • Corollary 3: GME criterion with characteristic function measurements over finite points
  • Lemma 1
  • proof
  • proof
  • Lemma 2: Simplified and rephrased from Theorem 2 of Ref. ZawCertifiable2024
  • Lemma 3
  • ...and 5 more