Can entanglement hide behind triangle-free graphs?

Abstract

We present an alternative approach to unveil a different kind of entanglement in bipartite quantum states whose diagonal zero patterns in suitable matrix representations admit a nice description in terms of triangle-free graphs. Upon application of a local averaging operation, the separability of such states transforms into a simple matrix positivity condition, the violation of which implies the presence of entanglement. We completely characterize the class of triangle-free graphs which allows for nontrivial entanglement detection using the above test. Moreover, we develop a recipe to construct a plethora of unique classes of positive partial transpose (PPT) entangled triangle-free states in arbitrary dimensions. Finally, we link the task of entanglement detection in general states to the well-known graph-theoretic problem of finding triangle-free-induced subgraphs in a given graph.

arXiv commentv3: Accepted version; Proof of Theorem II.3 added + Other minor changes implemented

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Cited by in corpus (6)

• Diagonal unitary and orthogonal symmetries in quantum theory
• Detecting positive quantum capacities of quantum channels
• The PPT${}^2$ conjecture holds for all Choi-type maps
• Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators
• A graphical calculus for integration over random diagonal unitary matrices
• Rank-based entanglement detection for bound entangled states