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Quantum Separability Criteria Based on Symmetric Measurements

Yu Lu, Wen Zhou, Meng Su, Hong-Xing Wu, Shao-Ming Fei, Zhi-Xi Wang

TL;DR

Addressing the challenge of entanglement detection, the paper develops a separability criterion based on local symmetric measurements mapped through the $ (N,M) $ POVM framework. It defines a probability matrix $ ext{P}( ho)$ and a trace-norm bound $ig\| ext{M}_{\mu, u}^{l}( ho_{AB})ig\|_{tr}$ that separates separable from entangled states, with explicit bounds depending on the POVM parameters and their local implementations. Corollaries for GSICPOVMs and MUMs, together with detailed examples (including isotropic and PPT-entangled states), demonstrate improved entanglement detection over several existing criteria. The framework naturally generalizes to multipartite systems, enabling entanglement verification without full tomography and offering pathways to tighter concurrence bounds and extensions to generalized symmetric measurements.

Abstract

We propose experimentally feasible separability criteria for bipartite systems based on local symmetric measurements. Through detailed examples, we demonstrate that our criteria can detect entanglement more effectively compared to existing counterparts. Furthermore,we demonstrate the potential for our results to be generalized to general multipartite systems.

Quantum Separability Criteria Based on Symmetric Measurements

TL;DR

Addressing the challenge of entanglement detection, the paper develops a separability criterion based on local symmetric measurements mapped through the POVM framework. It defines a probability matrix and a trace-norm bound that separates separable from entangled states, with explicit bounds depending on the POVM parameters and their local implementations. Corollaries for GSICPOVMs and MUMs, together with detailed examples (including isotropic and PPT-entangled states), demonstrate improved entanglement detection over several existing criteria. The framework naturally generalizes to multipartite systems, enabling entanglement verification without full tomography and offering pathways to tighter concurrence bounds and extensions to generalized symmetric measurements.

Abstract

We propose experimentally feasible separability criteria for bipartite systems based on local symmetric measurements. Through detailed examples, we demonstrate that our criteria can detect entanglement more effectively compared to existing counterparts. Furthermore,we demonstrate the potential for our results to be generalized to general multipartite systems.

Paper Structure

This paper contains 5 sections, 5 theorems, 57 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Detecting entanglement of bipartite states
  4. DETECTION OF MULTIPARTITE ENTANGLEMENT
  5. CONCLUSIONS

Key Result

Theorem 1

If a bipartite state $\rho_{AB}\in H_{A}\otimes H_{B}$ is separable, then where $x_A$ ($x_B$) is the parameter in the $(N,M)$ POVMs (Eq:1) on subsystem $H_A$ ($H_B$); $\|G\|_{\mathrm{tr}}=\mathrm{tr}\left(\sqrt{G^{\dagger} G}\right)$ is the trace norm of a matrix $G$.

Figures (6)

  • Figure 1: $f_1(p)$  derived from Theorem \ref{['th:1']} (solid red line),from which  $\rho_p$  is entangled for   $0.670 093 \leq p \leq 1$. $g_1(p)$  presented in Theorem $1$ of Ref.tang2023enhancing (dashed blue line), from which  $\rho_p$  is entangled for  $0.882 182 \leq p \leq 1$.
  • Figure 2: The orange region above the blue zero plane indicates that entanglement of $\rho_p$ for $0.8822 \leq p \leq 1$ is detected by Theorem $1$ in Ref.shen2018improved.
  • Figure 3: $f_2(p)$  derived from Corollary \ref{['co:1']} (solid red line), from which  $\rho_p$  is entangled for $0.837 933 \leq p \leq 1$. $g_2(p)$  presented in Remark $2$ of Ref.tang2023enhancing (dashed blue line), from which  $\rho_p$  is entangled for  $0.882 577 \leq p \leq 1$.
  • Figure 4: $f_3(p)$  derived from Corollary \ref{['co:2']} (solid red line), from which  $\rho_p$  is entangled for  $0.728 219 \leq p \leq 1$. $g_3(p)$  presented in Remark $3$ of Ref.tang2023enhancing (dashed blue line), from which  $\rho_p$  is entangled for  $0.882 178 \leq p \leq 1$.
  • Figure 5: $f_4(q)$  in Theorem \ref{['th:1']} (solid red line),  $f_5(q)$  in Corollary \ref{['co:1']} (dashed blue line), and  $f_6(q)$  in Corollary \ref{['co:2']} (dash-dotted orange line). It can be seen that  $\rho_{\mathrm{iso}}$  is entangled for  $\frac{1}{4} < q \leq 1$.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 2
  • proof
  • ...and 1 more