Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Detecting genuine multipartite entanglement using moments of positive maps

Saheli Mukherjee, Bivas Mallick, Sahil Gopalkrishna Naik, Ananda G. Maity, A. S. Majumdar

TL;DR

The paper introduces a practical framework for detecting genuine multipartite entanglement by using truncated moments of positive maps, derived from Lindblad dynamics, and evaluated via Hankel-matrix criteria on transposition-based maps. It avoids full state tomography by enabling experimental estimation of moments through shadow tomography and local operator measurements, and demonstrates the approach with tripartite and quadripartite examples, including GHZ and W states and noisy variants. Key contributions include the formulation of a GME-moment criterion, explicit tripartite and quadripartite demonstrations, and an experimental realization proposal for measuring the moments. Collectively, the work offers a scalable, information-efficient route to certify GME in complex quantum systems and suggests avenues for applying similar moment-based techniques to other positive maps.

Abstract

Genuine multipartite entanglement (GME) represents the strongest form of entanglement in multipartite systems, providing significant advantages in various quantum information processing tasks. In this work, we propose an experimentally feasible scheme for detecting GME, based on the truncated moments of positive maps. Our method avoids the need for full state tomography, making it scalable for larger systems. We provide illustrative examples of both pure and mixed states to demonstrate the efficacy of our formalism in detecting inequivalent classes of tripartite genuine entanglement. We further demonstrate the detection of quadripartite genuine entanglement, underscoring the effectiveness of our method in identifying entanglement beyond the tripartite case. Finally, we present a proposal for realising these moments in real experiments.

Detecting genuine multipartite entanglement using moments of positive maps

TL;DR

The paper introduces a practical framework for detecting genuine multipartite entanglement by using truncated moments of positive maps, derived from Lindblad dynamics, and evaluated via Hankel-matrix criteria on transposition-based maps. It avoids full state tomography by enabling experimental estimation of moments through shadow tomography and local operator measurements, and demonstrates the approach with tripartite and quadripartite examples, including GHZ and W states and noisy variants. Key contributions include the formulation of a GME-moment criterion, explicit tripartite and quadripartite demonstrations, and an experimental realization proposal for measuring the moments. Collectively, the work offers a scalable, information-efficient route to certify GME in complex quantum systems and suggests avenues for applying similar moment-based techniques to other positive maps.

Abstract

Genuine multipartite entanglement (GME) represents the strongest form of entanglement in multipartite systems, providing significant advantages in various quantum information processing tasks. In this work, we propose an experimentally feasible scheme for detecting GME, based on the truncated moments of positive maps. Our method avoids the need for full state tomography, making it scalable for larger systems. We provide illustrative examples of both pure and mixed states to demonstrate the efficacy of our formalism in detecting inequivalent classes of tripartite genuine entanglement. We further demonstrate the detection of quadripartite genuine entanglement, underscoring the effectiveness of our method in identifying entanglement beyond the tripartite case. Finally, we present a proposal for realising these moments in real experiments.

Paper Structure

This paper contains 19 sections, 1 theorem, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Positive maps in the realm of bipartite entanglement detection
  4. Generation of positive maps from Lindblad structure
  5. Genuine multipartite entanglement detection
  6. Partial transpose moment criterion
  7. Moment-based genuine multipartite entanglement detection
  8. Examples for $N=3$
  9. Example for $N=4$
  10. Proposal for experimental realization of the moments
  11. Conclusions
  12. Acknowledgements
  13. Genuine entanglement detection using moments of reduction map
  14. Detection of $GHZ$ state using the moments of reduction map:
  15. Detection of noisy $GHZ$ state using moments of reduction map:
  16. ...and 4 more sections

Key Result

Theorem 1

If a state $\rho_{2-\text{sep}} \in \mathcal{D}(\mathbb{C}^d \otimes \mathbb{C}^d \otimes ....\otimes \mathbb{C}^d )$ is biseparable, where $[H_{l}(\mathbf{s^{(\mathcal{T})}})]_{ij} = s_{i+j+1}^{(\mathcal{T})}$ for $i,j \in \{0,1,...,l\}$, $l \in \mathbb{N}$ and $\mathbf{s^{(\mathcal{T})}}=(s_1^{(\mathcal{T})}, s_2^{(\mathcal{T})},...,s_n^{(\mathcal{T})})$ is defined in Eq. moments.

Figures (6)

  • Figure 1: Detection of genuine entanglement of tripartite noisy $GHZ$ state using moments of modified transposition map.
  • Figure 2: Detection of genuine entanglement of tripartite noisy $W$ state using moments of transposition map.
  • Figure 3: Detection of genuine entanglement of a convex mixture of $GHZ$ and $W$ state using moments of modified transposition map.
  • Figure 4: Detection of genuine entanglement of $4$-qubit noisy $GHZ$ state using moments of modified transposition map.
  • Figure 5: Detecting genuine entanglement of tripartite noisy $GHZ$ state using the moments of reduction map
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7