bibkey: alvarezlosadaportesibosyk2023complementarity
authors: Yanet Alvarez; Marcelo Losada; Mariela Portesi; Gustavo M Bosyk
year: 2023
title: "Complementarity between quantum coherence and mixedness: a majorization approach"
doi: 10.1088/1572-9494/acc92f
url: https://ctp.itp.ac.cn/EN/PDF/10.1088/1572-9494/acc92f
claim: "The coherence vector nu(rho) of a d-level state is the majorization supremum of the averages sum_k q_k mu(psi_k) of decreasingly ordered squared-amplitude vectors over pure-state decompositions of rho, and lambda(rho) is its decreasingly ordered spectrum. The paper studies majorization bounds for nu(rho) tensor lambda(rho), finds the optimal qubit bound and numerical qutrit bounds, and conjectures nu_1(rho) lambda_1(rho) >= 1/d for every state."
strata_touched:
  - D5/S3/Quantum/Information/CoherenceVectorSpectrumProductBound
license: citation-only
triage: anchor

Alvarez, Losada, Portesi and Bosyk, Complementarity between quantum coherence and mixedness

Y. Alvarez, M. Losada, M. Portesi and G. M. Bosyk, Communications in Theoretical Physics 75(5), 055102 (2023). No arXiv version was found. The text below is read from the journal PDF; its text layer keeps the prose and garbles the mathematical symbols, so the formulas are transcribed from the rendered pages.

Section 2.2 defines, for a pure state, the coherence vector $\mu (|\psi ⟩⟨\psi |)=(|⟨0|\psi ⟩{|}^2,\dots ,|⟨d-1|\psi ⟩{|}^2)$ (Eq. (9)), the set $\mathcal{D}(\rho )$ of pure-state decompositions of $\rho$ (Eq. (10)), and $${\mathcal{U}}^{\mathrm{psd}}(\rho )=\{\sum _k{q}_k{\mu }^{\downarrow }(|{\psi }_k⟩⟨{\psi }_k|):\rho =\sum _k{q}_k|{\psi }_k⟩⟨{\psi }_k|\}(11)$$ with ${\mu }^{\downarrow }$ the decreasingly ordered vector. Then:

the coherence vector of the state ρ, denoted as ν(ρ), is defined as the supremum (with respect to the majorization relation) of the set

${\mathcal{U}}^{\mathrm{psd}}(\rho )$, that is $\nu (\rho )=\bigvee {\mathcal{U}}^{\mathrm{psd}}(\rho )$ (Eq. (12)). Section 2.3 uses the decreasingly ordered spectrum $\lambda (\rho )$.

The conjecture, at the end of Section 3.4 (“Bounds for qutrit systems: numerical results”):

Finally, we state a conjecture for the greatest component of ν(ρ) ⊗ λ(ρ) that allows us to obtain a nonoptimal but nontrivial bound for the tensor-product majorization relation, which depends only on the dimension of the Hilbert space. Conjecture 5. Let $\rho \in \mathcal{S}({\mathcal{H}}_d)$. Then ${\nu }_1(\rho ){\lambda }_1(\rho )\ge \frac{1}{d}$. (20) […] For d = 2 the conjecture is clearly valid, whereas for d = 3 we have numerical evidence of its validity.

Verified locator

Source: https://doi.org/10.1088/1572-9494/acc92f

• DOI: https://doi.org/10.1088/1572-9494/acc92f (Communications in Theoretical Physics 75(5), 055102; published online 10 May 2023; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://ctp.itp.ac.cn/EN/PDF/10.1088/1572-9494/acc92f. The PDF has 11 pages, 677,541 bytes and md5 7b0ea766599a0a57dedb25ddc2b9b3f6. Locations: Section 2.2, Eqs. (9)–(12) (pp. 3–4); Section 2.3 (p. 4); Section 3.4 and Conjecture 5, Eq. (20) (pp. 6–7).
• arXiv title search for “Complementarity between quantum coherence and mixedness” returned no record.