bibkey: haagerupmusatruskai2021extreme
authors: Uffe Haagerup; Magdalena Musat; Mary Beth Ruskai
year: 2021
title: "Extreme Points and Factorizability for New Classes of Unital Quantum Channels"
doi: 10.1007/s00023-021-01071-y
url: https://arxiv.org/abs/2006.03414v4
claim: "The paper introduces a two-parameter family of unital, trace-preserving completely positive maps on 3 by 3 matrices with four Kraus operators. Almost every member is factorizable and extreme among the unital trace-preserving completely positive maps, but extreme neither among the unital nor among the trace-preserving ones. At alpha = beta = 1/sqrt(2) the channel is a convex combination of a unitary conjugation and a Choi-rank-3 map extreme in both larger sets; the paper conjectures the same decomposition whenever |alpha| = |beta| = 1/sqrt(2)."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/QutritChannelUnitaryExtremeDecomposition
license: citation-only
triage: anchor

Haagerup, Musat and Ruskai, Extreme points and factorizability for new classes of unital quantum channels

U. Haagerup, M. Musat and M. B. Ruskai, arXiv:2006.03414v4 [math.OA]; Annales Henri Poincaré 22(10) (2021), 3455–3496. Quotations are from the v4 TeX source, with the source’s macros \1rt2, \kb, \proj, \ovb, \half and \ds expanded.

Theorem thm:UCPT, with its displayed align of the four operators set inline:

Let $\alpha ,\beta \in \mathbf{C}$ with $|\alpha {|}^2+|\beta {|}^2=1$ and let $A_1=\alpha |e_1⟩⟨e_1|+|e_2⟩⟨e_3|$, $A_2=\beta |e_1⟩⟨e_3|+|e_3⟩⟨e_2|$, $A_3=-|e_1⟩⟨e_2|-\overline{\beta }|e_3⟩⟨e_1|$, $A_4=|e_2⟩⟨e_1|+\overline{\alpha }|e_3⟩⟨e_3|$. Then ${\Phi }_{\alpha ,\beta }(\rho )=\frac{1}{2}{\sum }_{k=1}^4{A}_k^{\ast }\rho A_k$ is an extreme point in the set of UCPT maps if $|\alpha {|}^2\ne 0,\frac{1}{2},1$. Moreover, ${\Phi }_{\alpha ,\beta }$ has an exact factorization through $M_3(\mathbf{C})\otimes M_2(\mathbf{C})$ for all $\alpha ,\beta$.

In the subsubsection “Non-extreme cases”, the source mixes the Kraus operators at $\alpha =\beta =\frac{1}{\sqrt{2}}$ by $W=2|1_4⟩⟨1_4|-I_4$. This gives $\Phi =\frac{1}{4}{\stackrel{2\ast \rho X}{X}}_2+\frac{3}{4}\Psi (\rho )$ with ${\tilde{X}}_2$ unitary and $\Psi$ extreme in both the UCP and the CPT maps, of Choi rank 3. It then states:

The argument above easily extends to $\alpha =\pm \overline{\beta }$ with $2X_3$ unitary. It seems reasonable to conjecture that, whenever $|\alpha |=|\beta |=\frac{1}{\sqrt{2}}$, the channel ${\Phi }_{\alpha ,\beta }$ is a convex combination of a unitary conjugation and a map with Choi-rank $3$ which is extreme in both the UCP and CPT maps. However, one would need a different unitary transformation relating $\{A_k\}$ to $\{X_k\}$.

Verified locator

Source: https://arxiv.org/abs/2006.03414v4

• DOI: https://doi.org/10.1007/s00023-021-01071-y (Annales Henri Poincaré 22(10), 3455–3496; published online 12 June 2021; title, authors, journal, volume, issue and pages from Crossref).
• URL: https://arxiv.org/abs/2006.03414v4. The e-print is a single gzipped file HMR2021J.tex (2221 lines, md5 1162bc95019201f2037c415c7b1cf7aa). Locations:
  • macro \tfrac{1}{\sqrt{2}}, l. 56;
  • Subsection sect:UCPT, l. 274; Theorem thm:UCPT, l. 280–290;
  • Subsubsection “Non-extreme cases”, l. 354; the base case at $\alpha =\beta =\frac{1}{\sqrt{2}}$, l. 366–384; the conjecture, l. 386–392.
• The arXiv record lists versions v1 (5 June 2020) to v4 (19 May 2021) (arXiv API metadata).