Note on von Neumann and Rényi entropies of a Graph

Dairyko, Michael; Hogben, Leslie; Lin, Jephian C. -H.; Lockhart, Joshua; Roberson, David A.; Severini, Simone; Young, Michael

doi:10.48550/arxiv.1609.00420

Cited by 1 publication

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 1. Theorem 4 contrasts sharply with the result that was established in [7] for combinatorial Laplacians. In particular, that work demonstrated that the star graph's combinatorial Laplacian achieves minimum Rényi-2 entropy and almost always achieves minimum Von Neumann entropy.…”

Section: Extremal Entropies Of Symmetric Graph Laplacianscontrasting

confidence: 57%

“…We then proceed to study the entropic properties of this state: we show that the complete graph achieves maximum Von Neumann entropy, while the 2-regular graph is within log 4 √ 2/3 of the smallest Von Neumann entropic values among all connected graphs. This contrasts with work in [7], where it was conjectured that the star graph minimizes the Von Neumann entropy. Here we show that the star graph (asymptotically) maximizes the Von Neumann entropy.…”

Section: Introductionmentioning

confidence: 60%

“…Specifically, in that work it was demonstrated that the Von Neumann entropy should be interpreted as the number of entangled bits that are represented by the graph's quantum state. Further extensions were made in [7], where it was shown that, as the number of vertices grows, almost all connected graphs have a Von Neumann entropy that is greater than the star graph.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symmetric Laplacians, quantum density matrices and their Von-Neumann entropy

Simmons¹,

Coon²,

Datta³

2017

Linear Algebra and its Applications

View full text Add to dashboard Cite

Abstract. We show that the (normalized) symmetric Laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite Hilbert space (one part corresponding to the vertices, the other corresponding to the edges). This suggests an interpretation of the symmetric Laplacian's Von Neumann entropy as a measure of bipartite entanglement present between the two parts of the state. We then study extreme values for a connected graph's generalized Rényi-p entropy. Specifically, we show that(1) the complete graph achieves maximum entropy, (2) the 2-regular graph: (a) achieves minimum Rényi-2 entropy among all k-regular graphs, (b) is within log 4/3 of the minimum Rényi-2 entropy and log 4 √ 2/3 of the minimum Von Neumann entropy among all connected graphs, (c) achieves a Von Neumann entropy less than the star graph. Point (2) contrasts sharply with similar work applied to (normalized) combinatorial Laplacians, where it has been shown that the star graph almost always achieves minimum Von Neumann entropy. In this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound.

show abstract

Section: Extremal Entropies Of Symmetric Graph Laplacianscontrasting

confidence: 57%

Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation