bibkey: devos2014scalingunitary
authors: Alexis De Vos; Stijn De Baerdemacker
year: 2014
title: "Scaling a unitary matrix"
doi: 10.1142/S1230161214500139
url: https://arxiv.org/abs/1401.7883v2
claim: "The paper studies the scaling of a unitary matrix by diagonal unitaries to a matrix with all line sums equal to one, introduces the landscape Ψ(λ, ρ) = n² − |sum(B)|² with B_jk = e^(i(λ_j+ρ_k)) A_jk, and conjectures that all its stationary points are maxima, saddle points or global minima, i.e. that no local minima exist, and that the global minima satisfy Ψ = 0."
strata_touched:
  - D5/S3/Quantum/Matrix/UnitaryScalingLandscapeRefutation
license: citation-only
triage: anchor

Scaling a unitary matrix

Alexis De Vos and Stijn De Baerdemacker, arXiv:1401.7883v2 (math-ph), 2015-02-06; published in Open Syst. Inf. Dyn. 21 (2014) 1450013. Quotations are from the v2 TeX source.

The landscape (Appendix B):

Given a unitary matrix~$A$, finding the scaled matrix $B$ is equivalent to solving the (non-linear) eqn [ n^2 - |\som(B)|^2 = 0 \ . ] As \begin{equation} B_{jk} = e^{i(\lambda_j + \rho_k)}, A_{jk}\ , \label{B} \end{equation} we introduce a $2n$-dimensional landscape $\Psi$, given by [ \Psi(\lambda_1, \lambda_2, …, \lambda_n, \rho_1, \rho_2, …, \rho_n) = n^2-|\som(B)|^2 \ . ] We have to find the minimum of this function, i.e.\ the point $\Psi =0$.

The conjecture (Appendix B):

We conjecture that all these stationary points are either maxima or saddle points or global minima. In other words: we conjecture that no local minima exist. Moreover, we conjecture that the global minima satisfy~$\Psi =0$.

The same conjecture in Section 2:

\item or $0<\Psi <n^2$. \end{itemize} If the first case occurs, then the stationary point is a global maximum; if the second case occurs, then the stationary point is a global minimum. We conjecture that the third class consists of saddle points. In other words:

Verified locator

• URL: https://arxiv.org/abs/1401.7883v2 (v2, 2015-02-06; source sinkhorn-update.tex, md5 a2350283fac809546c7d40886d8ec1f1): the landscape (l. 882–896), the conjecture (l. 938–940) and its statement in Section 2 (l. 237–242).
• DOI: 10.1142/S1230161214500139 (Open Syst. Inf. Dyn. 21, 1450013). The published text was not read.