Combinatorial structures in quantum correlation: A new perspective

TL;DR

The work introduces A_α-graph states, a novel class of quantum states built from graph adjacency and degree matrices with a tunable α, and analyzes their physical validity and entanglement properties through PPT and p_3-PPT criteria. By expressing PPT via graph parameters such as degree sequences and Frobenius norms, the authors provide graph-theoretic entanglement tests that are experimentally accessible via moments of the partial transpose. They derive explicit inequalities and demonstrate them on concrete graphs, connecting combinatorial structure to quantum correlations. The framework enables targeted entanglement detection for graph-derived states and suggests future directions for multipartite extensions and NISQ applications.

Abstract

Graph-theoretic structures play a central role in the description and analysis of quantum systems. In this work, we introduce a new class of quantum states, called $A_α$-graph states, which are constructed from either unweighted or weighted graphs by taking the normalised convex combination of the degree matrix $D$ and the adjacency matrix $A_G$ of a graph $G$. The constructed states are different from the standard graph states arising from stabiliser formalism. Our approach is also different from the approach used by Braunstein et al. This class of states depend on a tunable mixing parameter $α\in (0,1]$. We first establish the conditions under which the associated operator $ρ_α^{A_G}$ is positive semidefinite and hence represents a valid quantum state. We then derive a positive partial transposition (PPT) condition for $A_α$-graph states in terms of graph parameters. This PPT condition involves only the Frobenius norm of the adjacency matrix of the graph, the degrees of the vertices and the total number of vertices. For simple graphs, we obtain the range of the parameter $α$ for which the $A_α$-graph states represent a class of entangled states. We then develop a graph-theoretic formulation of a moments-based entanglement detection criterion, focusing on the recently proposed $p_3$-PPT criterion, which relies on the second and third moments of the partial transposition. Since the estimation of these moments is experimentally accessible via randomised measurements, swap operations, and machine-learning-based protocols, our approach provides a physically relevant framework for detecting entanglement in structured quantum states derived from graphs. This work bridges graph theory and moments-based entanglement detection, offering a new perspective on the role of combinatorial structures in quantum correlations.

Figures (10)

• Figure 1: (a) Graph $G_1$ and (b) its partial transpose ${G_1}^{T_B}$.
• Figure 2: Graph $G_2$ and its partial transpose graph
• Figure 3: $G_1$
• Figure 4: $G_2$
• Figure 5: $G_3$

Theorems & Definitions (20)

• Definition 1
• Example 1
• Example 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 5
• Corollary 6
• Example 3