bibkey: vomende2022strictpositivity
authors: Frederik vom Ende
year: 2022
title: "Strict positivity and D-majorization"
doi: 10.1080/03081087.2020.1860887
url: https://arxiv.org/abs/2004.05613v4
claim: "For a positive definite D, a matrix A is D-majorized by B when some quantum channel maps B to A and fixes D. The paper relates strict positivity of channels to this order. It shows that D is the unique minimal element of its trace hyperplane, and that the pure basis state for a minimal entry of d is maximal among positive semidefinite matrices of that trace. In the vector case the set M_D(A) of matrices D-majorized by A is a polytope; in the matrix case it has infinitely many extreme points, and the paper asks whether finitely many remain after identifying extreme points related by unitaries commuting with D."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/DMajorizationExtremeClassInfinitude
license: citation-only
triage: anchor

vom Ende, Strict positivity and D-majorization

F. vom Ende, arXiv:2004.05613v4 [quant-ph]; Linear and Multilinear Algebra 70(19) (2022), 4023–4048. Quotations are from the v4 TeX source, with the source’s macros \unitvector and \conj expanded.

A channel is a linear, completely positive, trace-preserving map, and $Q(n)$ denotes the channels on ${\mathbb{C}}^{n\times n}$. Definition defi_matrix_D_maj:

Let $D\in {\mathbb{C}}^{n\times n}$ positive definite and $A,B\in {\mathbb{C}}^{n\times n}$. Now $A$ is said to be \textit{$D$-majorized} by $B$, denoted by $A{\prec }_{D}B$, if there exists $T\in Q(n)$ such that $T(B)=A$ and $T(D)=D$.

Equation eq:def_MD defines $M_D(X)$ as the set of all matrices $D$-majorized by $X$.

Theorem d_matrix_minmax (ii), for $d\in {\mathbb{R}}_{++}^n$ and ${\mathfrak{h}}_d^{+}$ the positive semidefinite matrices of trace $e^{T}d$:

$(e^{T}d)|e_k⟩⟨e_k|$ is maximal in ${\mathfrak{h}}_d^{+}$ with respect to ${\prec }_D$ where $k$ is chosen such that $d_k$ is minimal in $d$. It is the unique maximal element in ${\mathfrak{h}}_d^{+}$ with respect to ${\prec }_D$ if and only if $d_k$ is the unique minimal element of $d$.

The question in the Conclusion and Outlook:

In the matrix case, however, the number of extreme points is infinite […] a straightforward argument shows that if $X$ is extremal in $M_D(A)$ then so is $U^{\ast }XU$ for all unitaries $U$ which satisfy $[U,D]=0$. A way of bridging this gap could be to define the unitary equivalence relation $$\sim :=\{(X,UX{U}^{\ast })|X\in M_D(A),U\in {\mathbb{C}}^{n\times n}\text{ unitary with }[U,D]=0\}$$ for arbitrary $d\in {\mathbb{R}}_{++}^n$ and $A\in {\mathbb{C}}^{n\times n}$, and then look at the equivalence class of an extreme point $X$ of $M_D(A)$ under $\sim$. Whether one ends up with finitely many extreme points after factoring out the unitary equivalents for now remains open.

Verified locator

Source: https://arxiv.org/abs/2004.05613v4

• DOI: https://doi.org/10.1080/03081087.2020.1860887 (Linear and Multilinear Algebra 70(19), 4023–4048; published online 17 December 2020; title, author, journal, volume, issue and pages from Crossref).
• URL: https://arxiv.org/abs/2004.05613v4. The e-print is a gzipped tar archive; its member strictly_positive_qc_v2.tex has 1257 lines and md5 b8b3abce192b148593dafd02809eaccf. Locations:
  • Section ch:prelim, l. 99; $Q(n)$ and $Q_X(n)$, l. 140–146;
  • Section sec:matrix_d_maj, l. 398; Definition defi_matrix_D_maj, l. 466–469;
  • Theorem d_matrix_minmax, l. 874–885, proof to l. 921;
  • Eq. eq:def_MD, l. 926–931;
  • Section Conclusion and Outlook, l. 1055; the question, l. 1058–1064.
• The arXiv record lists four versions (v1, 12 April 2020; v4, 17 June 2020) and the journal reference “Lin. Multilin. Alg., 70:19 (2022) 4023-4048” (arXiv API metadata).