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Extremal properties of the variance and the quantum Fisher information
Open accessAbstract
We show that the variance is its own concave roof. For rank-2 density matrices and operators with zero diagonal elements in the eigenbasis of the density matrix, we prove analytically that the quantum Fisher information is four times the convex roof of the variance. Strong numerical evidence suggests that this statement is true even for operators with nonzero diagonal elements or density matrices with a rank larger than $2.$ We also find that within the different types of generalized quantum Fisher information considered in Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)], after appropriate normalization, the quantum Fisher information is the largest. Hence, we conjecture that the quantum Fisher information is four times the convex roof of the variance even for the general case.
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“the QFI is the convex roof of the variance [101,102]”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [101]
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- A. S. Holevo, 1982, Probabilistic and Statistical Aspect of Quantum Theory
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Quantum metrology from a quantum information science perspective
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Self-citationQuantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control
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“In this context it is also worth noting that the QFI is the convex roof of the variance [101,102], and hence the relation to the Mandelstam-Tamm bound (12) becomes immediately apparent.”
BackgroundQuantum Limits in Optical Interferometry
Demkowicz-Dobrzański, R., Jarzyna, M., & Kołodyński, J. (2015) · Progress in optics · doi:10.1016/bs.po.2015.02.003 · Cited by 562
“…a very recent and elegant result proving that for unitary parameter encodings, QFI is proportional to the convex roof of the variance of Ĥ (Tóth and Petz, 2013; Yu, 2013): FQ(ρϕ) = 4 min {pi,|ψ(i)〉} ∑ i pi∆ 2H(i), (90) where the minimum is performed over all decompositions of ρ = ∑ i…”
BackgroundQuantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
Deffner, S., & Campbell, S. (2017) · Maryland Shared Open Access Repository (USMAI Consortium) · doi:10.13016/m2heya-i4ys · Cited by 485
Metadata
- Authors
- Tóth, G., & Petz, D.
- Affiliations
- Ikerbasque (ES)
- ORCID
- Tóth, Géza
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2013-03-20
- Volume
- 87
- Issue
- 3
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2037609284
- Cited by
- 157
- References
- 42
- Citations by year
- 2026: 7 · 2025: 11 · 2024: 4 · 2023: 44 · 2022: 9 · 2021: 10 · 2020: 12 · 2019: 10 · 2018: 8 · 2017: 7 · 2016: 10 · 2015: 12 · 2014: 9 · 2013: 4
- FWCI
- 12.90
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Physical Review A (publishedVersion, cc-by)
- arXiv (Cornell University) (publishedVersion)
- Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) (acceptedVersion)
- arXiv (Cornell University)
- Funders
- Fonds National de la Recherche Luxembourg
- Eusko Jaurlaritza (IT4720-10, Project No. IT4720-10)
- Hungarian Scientific Research Fund (K104206, K83858)
- Ministerio de Ciencia e Innovación (FIS2012-36673-C03-03)
- CHIST-ERA
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- quantum Fisher information, variance, extremal properties, quantum metrology, quantum parameter estimation
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