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Symmetric Laplacians, quantum density matrices and their Von Neumann entropy
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Simmons, David, Coon, Justin and Datta, Animesh (2017) Symmetric Laplacians, quantum density matrices and their Von Neumann entropy. Linear Algebra and Its Applications, 532 . pp. 534-549. ISSN 0024-3795.
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Official URL: https://doi.org/10.1016/j.laa.2017.06.038
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 R\'enyi-p entropy. Specifically, we show that
(1) the complete graph achieves maximum entropy,
(2) the 2-regular graph: a) achieves minimum R\'enyi-2 entropy
among all k-regular graphs, b) is within log4/3 of the minimum R\'enyi-2 entropy and log42‾√/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.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QC Physics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Physics | |||||||||
| Library of Congress Subject Headings (LCSH): | Laplacian matrices, Hilbert space | |||||||||
| Journal or Publication Title: | Linear Algebra and Its Applications | |||||||||
| Publisher: | Elsevier Inc | |||||||||
| ISSN: | 0024-3795 | |||||||||
| Official Date: | 1 November 2017 | |||||||||
| Dates: |
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| Volume: | 532 | |||||||||
| Page Range: | pp. 534-549 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Date of first compliant deposit: | 17 July 2017 | |||||||||
| Date of first compliant Open Access: | 14 July 2018 | |||||||||
| RIOXX Funder/Project Grant: |
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