Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IntechOpen eBooks

Non-Gaussian Entanglement and Wigner Function

Mustapha Ziane, M. El Baz

Published Oct 30, 2019• 1 citation
Publisher

Abstract

A measure of non-Gaussian entanglement in continuous variable (CV) systems based on the volume of the negative part of the Wigner function is proposed. We analyze comparatively this quantity with a numerical evaluation of the negativity of the partial transpose (NPT) considering a system of Bell states formed in the coherent state basis (quasi-Bell states).

DOI
10.5772/intechopen.86426

Related papers

The negativity of partial transpose versus the negativity of Wigner function

Mustapha Ziane, Fatima-Zahra Siyouri, M. El Baz +1 more

We investigate the multipartite entanglement in the phase space using the negativity of Wigner function (NWF) and in the Hilbert space using the negativity of partial transpose (NPT). We analyze comparatively these quantities and the different types of entanglements that are present in two major classes — GHZ and [Formula: see text] — made of coherent states. We show that the negativity of Wigner function can be used as a good measure of genuine entanglement in multipartite systems. However, the negativity of partial transpose is a good quantifier for only the bipartite entanglement in tripartite systems.

International Journal of Geometric Methods in Modern Physics•Apr 3, 2019•3 citations

On the negativity of the Wigner function as a measure of entanglement under quantum polarization converter devices

Mustapha Ziane, M. El Baz

We study the behaviour of the Negativity of Wigner Function (NWF) as a measure of entanglement in non-Gaussian states under quantum polarisation converter devices. We analyze comparatively this quantity with other measures of entanglement in a system prepared in a superposition of two-mode coherent states. We show that the (WF) can be identified as a quantifier of non-Gaussian entanglement.

arXiv:2101.09182•Jan 22, 2021•0 citations

The role of non-Gaussian entanglement of continuous variable quantum states in quatum technologies

Michal Matulík

We present an effort to find a measure of non-Gaussianity of non-Gaussian states. We compute the difference between the logarithmic negativities of a non-Gaussian state and its Gaussified form as a non-Gaussian entanglement resource. We use the latter to assess the resource content of non-Gaussianity of experimentally relevant state - two- mode squeezed vacuum with a subtracted photon. We also used symplectic analysis to discard the Gaussian entanglement. Therefore the measure is reduced to the logarithmic negativity of a non-Gaussian state only. We compare the measure with the Wigner distribution and its negativity and discuss the results. 1

•Jan 1, 2023•0 citations

Non-Gaussian Quantum States and Where to Find Them

Mattia Walschaers

Gaussian states have played an important role in the physics of continuous-variable quantum systems.They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description.Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements.In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states.We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems.We then use this framework in various ways to explore the structure of the state space.We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank.For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations.To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state.Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states.On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations.Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage.At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.

arXiv:2104.12596•PRX Quantum•Sep 28, 2021•311 citations

Exploring the possibility of a complex-valued non-Gaussianity measure for quantum states of light

Andrew J. Pizzimenti, Prajit Dhara, Zacharie Van Herstraeten +2 more

We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.

arXiv:2303.00880•APL Quantum•Aug 27, 2024•3 citations

Entanglement and Wigner Function Negativity of Multimode Non-Gaussian States

Mattia Walschaers, Claude Fabre, Valentina Parigi +1 more

Non-Gaussian operations are essential to exploit the quantum advantages in optical continuous variable quantum information protocols. We focus on mode-selective photon addition and subtraction as experimentally promising processes to create multimode non-Gaussian states. Our approach is based on correlation functions, as is common in quantum statistical mechanics and condensed matter physics, mixed with quantum optics tools. We formulate an analytical expression of the Wigner function after the subtraction or addition of a single photon, for arbitrarily many modes. It is used to demonstrate entanglement properties specific to non-Gaussian states and also leads to a practical and elegant condition for Wigner function negativity. Finally, we analyze the potential of photon addition and subtraction for an experimentally generated multimode Gaussian state.

arXiv:1707.02285•Physical Review Letters•Oct 31, 2017•94 citations
A measure of non-Gaussian entanglement in continuous variable (CV) systems based on the volume of the negative part of the Wigner function is proposed. We analyze comparatively this quantity with a numerical evaluation of the negativity of the partial transpose (NPT) considering a system of Bell states formed in the coherent state basis (quasi-Bell states).
We investigate the multipartite entanglement in the phase space using the negativity of Wigner function (NWF) and in the Hilbert space using the negativity of partial transpose (NPT). We analyze comparatively these quantities and the different types of entanglements that are present in two major classes — GHZ and [Formula: see text] — made of coherent states. We show that the negativity of Wigner function can be used as a good measure of genuine entanglement in multipartite systems. However, the negativity of partial transpose is a good quantifier for only the bipartite entanglement in tripartite systems.
We present an effort to find a measure of non-Gaussianity of non-Gaussian states. We compute the difference between the logarithmic negativities of a non-Gaussian state and its Gaussified form as a non-Gaussian entanglement resource. We use the latter to assess the resource content of non-Gaussianity of experimentally relevant state - two- mode squeezed vacuum with a subtracted photon. We also used symplectic analysis to discard the Gaussian entanglement. Therefore the measure is reduced to the logarithmic negativity of a non-Gaussian state only. We compare the measure with the Wigner distribution and its negativity and discuss the results. 1
Gaussian states have played an important role in the physics of continuous-variable quantum systems.They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description.Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements.In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states.We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems.We then use this framework in various ways to explore the structure of the state space.We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank.For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations.To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state.Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states.On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations.Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage.At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.
We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.
Non-Gaussian operations are essential to exploit the quantum advantages in optical continuous variable quantum information protocols. We focus on mode-selective photon addition and subtraction as experimentally promising processes to create multimode non-Gaussian states. Our approach is based on correlation functions, as is common in quantum statistical mechanics and condensed matter physics, mixed with quantum optics tools. We formulate an analytical expression of the Wigner function after the subtraction or addition of a single photon, for arbitrarily many modes. It is used to demonstrate entanglement properties specific to non-Gaussian states and also leads to a practical and elegant condition for Wigner function negativity. Finally, we analyze the potential of photon addition and subtraction for an experimentally generated multimode Gaussian state.