Multipartite Entanglement Detection via Correlation Minor Norm

TL;DR

The paper addresses the challenge of detecting multipartite entanglement in mixed states by extending the correlation minor norm (CMN) to multipartite systems through matricizations of the correlation tensor $\\mathcal{C}$. It derives multipartite CMN bounds for bi-separable and fully-separable states, analyzes the role of reduced-state correlations, and introduces a CMN-based global quantum discord measure, facilitating a partition-aware view of nonclassical correlations. Key contributions include tight CMN bounds that can saturate for symmetric states, a demonstration that reduced-state correlations can enhance detection, and a framework to quantify global discord via CMN with proofs and comparisons to existing criteria like the multipartite dVH bound. The work provides a practical, partition-sensitive entanglement detector and a multipartite discord formalism that may extend to continuous-variable systems, offering new insights into the structure of multipartite quantum correlations and their operational use.

Abstract

Entanglement is a uniquely quantum resource giving rise to many quantum technologies. It is therefore important to detect and characterize entangled states, but this is known to be a challenging task, especially for multipartite mixed states. The correlation minor norm (CMN) was recently suggested as a bipartite entanglement detector employing bounds on the quantum correlation matrix. In this paper we explore generalizations of the CMN to multipartite systems based on matricizations of the correlation tensor. It is shown that the CMN is able to detect and differentiate classes of multipartite entangled states. We further analyze the correlations within the reduced density matrices and show their significance for entanglement detection. Finally, we employ matricizations of the correlation tensor for introducing a measure of global quantum discord.

Figures (6)

• Figure 1: Visualization of the CMN entanglement detection scheme. The black "line" represents $\mathcal{B}$, so that states located to its right will be detected as entangled, Eq. \ref{['CMN_generic_bound']}.
• Figure 2: Illustration of the tripartite correlation tensor. The tensor slices in Eq. (\ref{['tensor_slices_matrix']}) are represented by color.
• Figure 3: Visualization of the CMN entanglement detection scheme for multipartite systems. The black "line" represents $\mathcal{B}$ where states right of the black will be detected as bi-entangled (for a specific bi-partition). The midway orange zone represents bi-separable states. Note that since there exists a bi-separable state that saturates the bounds, the black line is tangent to the bi-separable states zone.
• Figure 4: SIC-POVM for $d=2$ creates a tetrahedron. The 4 vectors are evenly spaced.
• Figure 5: Faces of the tripartite correlation tensor. A tripartite state is said to be in SFNF, if the correlations on its faces are all zero - excluding the vertex at the origin, which has a fixed nonzero value. The tensor faces are the bipartite correlation matrices of the reduces states (particle trace)

Theorems & Definitions (16)

• Definition 1
• Lemma 1
• Definition 2
• Lemma 2
• Theorem 1
• Theorem 2
• Lemma 3
• Definition 3
• Definition 4
• Theorem 3