bibkey: zhaopereiramafei2021coherenceassistance
authors: Ming-Jing Zhao; Rajesh Pereira; Teng Ma; Shao-Ming Fei
year: 2021
title: "Coherence of assistance and assisted maximally coherent states"
doi: 10.1038/s41598-021-85273-8
url: https://arxiv.org/abs/2103.08818v1
claim: "For a real symmetric concave function f on the probability simplex vanishing at the vertices, the pure-state coherence C_f is f of the coherence vector. Its convex roof C_c and its coherence of assistance C_a take the minimum and maximum of the decomposition averages over pure-state decompositions of a mixed state. The paper proves that equal roofs force the pure-state function to be a quadratic form on the range of the state, deduces C_c < C_a for full-rank states, and asks whether C_c < C_a holds for all mixed states."
strata_touched:
  - D5/S3/Quantum/Information/CoherenceAssistanceRoofEquality
license: citation-only
triage: anchor

Zhao, Pereira, Ma and Fei, Coherence of assistance and assisted maximally coherent states

M.-J. Zhao, R. Pereira, T. Ma and S.-M. Fei, arXiv:2103.08818v1 [quant-ph]; Scientific Reports 11, 5935 (2021). Quotations are from the v1 TeX source.

The class of functions:

Let $\mathfrak{F}=\{f\}$ be the set of functions on the probability simplex $\Omega =\{\vec{x}=(x_0,x_1,\cdots ,x_{n-1}{)}^T|{\sum }_{i=0}^{n-1}{x}_i=1\text{and}{x}_i\ge 0\}$ such that (i) $f$ is a real symmetric concave function; (ii) $f((1,0,\cdots ,0{)}^T)=0$.

For $f\in \mathfrak{F}\setminus \{0\}$ the pure-state measure is $C_f(|\psi ⟩)=f(\mu (|\psi ⟩))$, with the coherence vector $\mu (|\psi ⟩)=(|{\psi }_0{|}^2,\dots ,|{\psi }_{n-1}{|}^2{)}^T$. The coherence of convex roof $C_c(\rho )$ and the coherence of assistance $C_a(\rho )$ are the minimum and the maximum of ${\sum }_k{p}_k{C}_f(|{\psi }_k⟩)$ over all pure-state decompositions $\rho ={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$.

Theorem th ca=cc:

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $F$ a nonnegative function defined on the pure states in $\mathcal{H}$. Let ${\rho }_o$ be a density matrix on $\mathcal{H}$ and $R({\rho }_o)$ the range of ${\rho }_o$. If $F_a({\rho }_o)=F_c({\rho }_o)$, then there exists a positive semidefinite operator $Q$ on $R({\rho }_o)$ such that $F(|\psi ⟩)=⟨\psi |Q|\psi ⟩$ for all pure states $|\psi ⟩$ in $R({\rho }_o)$.

Corollary cor coh:

For full rank quantum states $\rho$, the coherence of assistance is strictly larger than the coherence of convex roof, $C_c(\rho )<C_a(\rho )$.

The question, in the paragraph “Relation between the convex roof extension and the least concave majorant extension”:

The strict relation between the coherence of convex roof and the coherence of assistance, that is, whether $C_c(\rho )<C_a(\rho )$ holds for all mixed quantum states is an interesting topic.

Verified locator

Source: https://arxiv.org/abs/2103.08818v1

• DOI: https://doi.org/10.1038/s41598-021-85273-8 (Scientific Reports 11, 5935; published online 15 March 2021; title, authors, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/2103.08818v1. The e-print is a single gzipped file ZhaoMJ-AMC4.tex (502 lines, md5 91c15498f4ee964803bd2694ee500287). Locations:
  • the class $\mathfrak{F}$, l. 189;
  • $C_f$, the coherence vector and the two roofs, l. 207–236;
  • the question, l. 303;
  • Theorem th ca=cc, l. 314–317;
  • Corollary cor coh, l. 324–326.
• The arXiv record lists one version (v1, 16 March 2021) and the journal reference “Scientific Reports (2021) 11:5935” (arXiv API metadata).