bibkey: haapasalopellonpaa2020extreme
authors: Erkka Haapasalo; Juha-Pekka Pellonpää
year: 2020
title: "Effective methods for constructing extreme quantum observables"
doi: 10.1088/1751-8121/ab8d52
url: https://arxiv.org/abs/1809.09935v1
claim: "An N-outcome POVM is extreme when it is not a nontrivial midpoint of two POVMs; the paper characterizes the possible rank vectors of extreme POVMs in dimensions 1 to 4 and names the rank vector (3_2, 2_6)_7 as the problematic case in dimension 7."
strata_touched:
  - D5/S3/Quantum/Measurement/ExtremePOVMRankThreeThreeTwoSix
license: citation-only
triage: anchor

Haapasalo and Pellonpää, Effective methods for constructing extreme quantum observables

E. Haapasalo and J.-P. Pellonpää, J. Phys. A: Math. Theor. 53, 245301 (2020); arXiv:1809.09935v1. Quotations are from the v1 TeX source.

An $N$-outcome POVM is a tuple of positive operators summing to the identity; it is extreme if ${\mathsf{M}}_j=\frac{1}{2}{\mathsf{A}}_j+\frac{1}{2}{\mathsf{B}}_j$ for all $j$, with POVMs $\mathsf{A},\mathsf{B}$, implies $\mathsf{A}=\mathsf{B}=\mathsf{M}$. Rank vectors list the ranks of the outcomes from highest to lowest, with rank-one outcomes omitted. Section 4 states:

In dimensions 1-4, we are able to completely characterize the possible rank vectors of extreme POVMs but in dimensions 5-7 some open questions remain.

The new problematic case now is $(3_2,2_6{)}_7$.

Verified locator

Source: https://arxiv.org/abs/1809.09935v1

• DOI: https://doi.org/10.1088/1751-8121/ab8d52 (Journal of Physics A: Mathematical and Theoretical 53(24), article 245301; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/1809.09935v1. The e-print member laatikkopaperi_v4.tex has 463 lines and md5 bcc6ad7c5a0ff663b09967a5b7450705. Locations:
  • POVMs, l. 139; extremality, l. 148–152; rank vectors, l. 220–221;
  • the low-dimensional list, l. 307–314, with the case $(3_2,2_6{)}_7$ at l. 314.