bibkey: godsil2013averagemixing authors: Chris Godsil year: 2013 title: “Average mixing of continuous quantum walks” doi: 10.1016/j.jcta.2013.05.006 url: https://arxiv.org/abs/1103.2578v3 claim: “Section 11, Question 1 asks whether the discriminant D of the adjacency minimal polynomial always makes D times the average mixing matrix integral. Lemma 1.1 gives the spectral Schur-square formula; Lemma 3.1 proves the D squared bound.” strata_touched:

• D5/S3/Quantum/Dynamics/AverageMixingDiscriminantIntegrality license: citation-only triage: anchor

Average mixing of continuous quantum walks

Chris Godsil, arXiv:1103.2578v3; Journal of Combinatorial Theory, Series A 120 (2013), 1649–1662, doi:10.1016/j.jcta.2013.05.006. Page numbers below refer to the arXiv v3 PDF.

Section 11, Question 1 (page 20) asks:

Is it true that if $D$ is the discriminant of the minimal polynomial of $X$, then $D{\hat{M}}_X$ is an integer matrix?

The minimal polynomial of the graph means the minimal polynomial of its adjacency matrix over the rationals. Section 1 (page 2) specifies the spectral projections:

where ${\theta }_r$ runs over the distinct eigenvalues of $A$ and $E_r$ is the matrix representing orthogonal projection onto the eigenspace belonging to ${\theta }_r$.

Lemma 1.1 (page 3) states:

If $A={\sum }_r{\theta }_r{E}_r$ is the spectral decomposition of $S=A(X)$, then ${\hat{M}}_X={\sum }_r{E}_r^{\circ 2}$.

Here $\circ$ is the Schur (entrywise) product and $E_r^{\circ 2}=E_r\circ E_r$. This is the spectral expression used by the avgMixing definition in D5/S3/Quantum/Dynamics/AverageMixingTraceMaximum.lean.

Lemma 3.1 (page 5) states:

If $D$ is the discriminant of the minimal polynomial of $A$, then $D^2{\hat{M}}_X$ is an integer matrix.

Question 1 asks for the stronger factor $D$. Questions 2 and 3 of section 11 concern an algorithm over the rationals and the graphs whose average mixing matrix is a linear combination of $I$ and $J$.

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DOI: https://doi.org/10.1016/j.jcta.2013.05.006. Source version: https://arxiv.org/abs/1103.2578v3. The quoted statements occur on PDF pages 3 (Lemma 1.1), 5 (Lemma 3.1) and 20 (section 11, Question 1).