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Multipartite entanglement based on realignment moments

Hui Zhao, Shu-Ying Zhuang, Naihuan Jing, Mei-Ming Zhang

TL;DR

The paper addresses entanglement detection in high-dimensional bipartite and multipartite quantum systems using realignment moments. It introduces parameterized tests based on realignment moments $T_k$ for bipartite states and defines multipartite moments $T_k^R$, along with new criteria $V_1(a)$, $V_2(u)$, and $V_3(v)$ that yield sufficient conditions for entanglement under various partitions. The approach can detect bound entangled states as well as NPT entangled states, and, in the multipartite case, extends the method to arbitrary dimensions with improved detection demonstrated on concrete examples such as GHZ-W mixtures and four-qubit states. Overall, the work broadens the toolkit for entanglement certification by providing tunable, moment-based criteria that outperform some existing methods in both bipartite and multipartite scenarios.

Abstract

Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.

Multipartite entanglement based on realignment moments

TL;DR

The paper addresses entanglement detection in high-dimensional bipartite and multipartite quantum systems using realignment moments. It introduces parameterized tests based on realignment moments for bipartite states and defines multipartite moments , along with new criteria , , and that yield sufficient conditions for entanglement under various partitions. The approach can detect bound entangled states as well as NPT entangled states, and, in the multipartite case, extends the method to arbitrary dimensions with improved detection demonstrated on concrete examples such as GHZ-W mixtures and four-qubit states. Overall, the work broadens the toolkit for entanglement certification by providing tunable, moment-based criteria that outperform some existing methods in both bipartite and multipartite scenarios.

Abstract

Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.

Paper Structure

This paper contains 4 sections, 3 theorems, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Bipartite entanglement criterion with parameter based on moments
  3. Multipartite entanglement criterion based on realignment moments
  4. Conclusion

Key Result

Theorem 1

Suppose $\rho$ is a separable bipartite state, let $\Delta=(T_{1}^{2}-T_{1})^2-2(T_{1}^{2}-T_{2})T_1^2$ and then we have that (i) if $\Delta\leq0$, then $V_1(a)\leq1$ for $a>0$; (ii)if $\Delta>0$, then $V_1(a)\leq1$ for $0<a\leq\frac{T_1-T_1^2-\sqrt{\Delta}}{T_1^2-T_2}$ or $a\geq\frac{T_1-T_1^2+\sqrt{\Delta}}{T_1^2-T_2}$.

Figures (4)

  • Figure 1: $V_1$ from our Theorem \ref{['thm1']} when $a=2$ (solid red cure), the x-axis represents the state parameter $d$.
  • Figure 2: $V_1$ from our result when $a=0.2$ (solid red cure), criterion from Theorem $1$ in ref17 (dashed blue curve), criterion in ref13 (dash-dot green curve), the state is entangled when the value (dashed blue curve, dash-dot green curve) greater than zero.
  • Figure 3: $V_2$ from our Theorem \ref{['thm2']} for the state $\rho_{GHZ,W}$ when $l=2$ and $u=5$ (solid red line), the x-axis represents the state parameter $q$.
  • Figure 4: $V_3$ from our Theorem \ref{['thm3']} for the state $\rho$ when $l=2$ and $v=0.01$ (solid red line), method from Theorem $4$ in ref26 (dashed blue curve), the state is entangled when the value (dashed blue curve) greater than zero.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof