Enhanced separability criteria based on symmetric measurements

TL;DR

The work addresses experimental entanglement detection by deriving enhanced separability criteria based on local symmetric measurements implemented via symmetric (N,M)-POVMs. It introduces a bipartite trace-norm criterion, linking measurement outcome probabilities to a matrix map $\mathcal{Q}_{a,b}$ and establishing a tight bound $\|\mathcal{Q}_{a,b}(\rho_{AB})\|_{\mathrm{tr}} \leq \sqrt{|a|^{2}+\frac{(d_A-1)(M_A^{2}x_A+d_A^{2})}{d_A M_A (M_A-1)}}\sqrt{|b|^{2}+\frac{(d_B-1)(M_B^{2}x_B+d_B^{2})}{d_B M_B (M_B-1)}}$ for separable states; corollaries for GSIC-POVMs and MUMs yield simpler forms and a multipartite generalization broadens applicability. Through isotropic and PPT-entangled-state examples, the authors demonstrate improved entanglement detection compared to prior criteria. The framework offers a practically implementable and scalable approach for bipartite and multipartite entanglement detection based on local symmetric measurements, with potential to detect genuine multipartite entanglement in future work.

Abstract

We present separability criteria based on local symmetric measurements. These experimental plausible criteria are shown to be more efficient in detecting entanglement than the current counterparts by detailed examples. Furthermore, we generalize the separability criteria from bipartite to arbitrary multipartite systems. These criteria establish a richer connection between the quantum entanglement and the probabilities of local measurement outcomes.

Figures (4)

• Figure 1: $f_1(p)$ derived from Theorem \ref{['th:1']} (solid red line). $g_1(p)$ from the Theorem $1$ of Ref.Lu2025 (dashed blue line).
• Figure 2: $f_4(q)$ 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 $\frac{1}{4} < q \leq 1$.
• Figure 3: $f_2(p)$ derived from Corollary \ref{['co:1']} (solid red line). $g_2(p)$ presented in Corollary $1$ of Ref.Lu2025 (dashed blue line).
• Figure 4: $f_3(p)$ derived from Corollary \ref{['co:2']} (solid red line). $g_3(p)$ presented in Remark $2$ of Ref.Lu2025 (dashed blue line).

Theorems & Definitions (9)

• Theorem 1
• proof
• Corollary 1
• Corollary 2
• Example 1
• Example 2
• Example 3
• Theorem 2
• proof