bibkey: debrotastacey2019luders
authors: DeBrota, John B.; Stacey, Blake C.
year: 2019
title: "Lüders channels and the existence of symmetric-informationally-complete measurements"
doi: 10.1103/PhysRevA.100.062327
url: "https://arxiv.org/abs/1907.10999v2"
claim: "The paper studies the Lüders channel of a minimal informationally complete measurement (MIC), shows that it acts as the depolarizing channel (I + rho)/(d+1) exactly when the MIC is a SIC, bounds the Haar-averaged output entropy of every Lüders MIC channel on pure inputs by log(d+1) - 2 log 2/(d+1) with equality if a SIC exists, and conjectures that equality holds only if a SIC exists (Conjecture 1)."
strata_touched:
  - D5/S3/Quantum/Measurements/LudersEntropySaturationSIC
license: citation-only
triage: anchor

DeBrota and Stacey, Lüders channels and the existence of symmetric-informationally-complete measurements

J. B. DeBrota and B. C. Stacey, Phys. Rev. A 100, 062327 (2019), DOI 10.1103/PhysRevA.100.062327; arXiv:1907.10999v2 (5 February 2020). Quotations are from the arXiv v2 TeX source lmc.tex.

A minimal informationally complete measurement in dimension $d$ is a set of $d^2$ linearly independent positive semidefinite operators $\{E_i\}$ summing to the identity; it is rank-1 if every element is proportional to a rank-one projector and equiangular if $\mathrm{tr}{E}_i{E}_j$ takes one value for $i\ne j$ and another for $i=j$, and a SIC is a rank-1 equiangular MIC (l. 80–84). The Lüders MIC channel is $\mathcal{E}(\rho )={\sum }_i\sqrt{E_i}\rho \sqrt{E_i}$ (l. 104–114).

Theorem 2 (l. 385–390): for every Lüders MIC channel, $\overline{S(\mathcal{E}(|\psi ⟩⟨\psi |))}\le \log (d+1)-\frac{2}{d+1}\log 2$, the average being over pure inputs with respect to the Haar measure, with equality achievable if a SIC exists in dimension $d$. The paper continues (l. 416–420):

Theorem \ref{avgentropybnd} would have been more forceful if the upper bound were saturated ``only if’’ a SIC exists, but we were unable to demonstrate this property, and so we leave it as a conjecture: \begin{conjecture}\label{onlyif} Equality is achievable in the statement of Theorem \ref{avgentropybnd} only if a SIC exists in dimension $d$. \end{conjecture}

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.100.062327
• Source: https://arxiv.org/abs/1907.10999v2