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 quantum Fisher information is the convex roof of the variance.
“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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“More precisely, the SLDF is the variance convex roof, F (∑i pi|ψi〉,H) = inf {pi,|ψi〉}∑i piV(|ψi〉,H) [18,19].”
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“Thus, we can show that the scaling of the associated QFI for these states is the same as the one for separable states, saturating the bounds found in [56, 57].”
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“The variance is the concave roof of itself [31]: (∆A)(2)ρ = max {pk ,|Ψk〉} ∑”
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Robustness of multiparticle entanglement: specific entanglement classes and random statesPreprint
Ali, M. (2016) · Cited by 17 · see citing works
Gibilisco, P., Girolami, D., & Hansen, F. (2021) · Entropy · doi:10.3390/e23030263 · Cited by 16 · see citing works
Lambert, J., & Sørensen, E. S. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.045117 · Cited by 16 · see citing works
Bertlmann, R. A., & Friis, N. (2023) · doi:10.1093/oso/9780199683338.001.0001 · Cited by 15 · see citing works
Wigner-Araki-Yanase Theorem for Continuous and Unbounded Conserved Observables
Kuramochi, Y., & Tajima, H. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.210201 · Cited by 15 · see citing works
Experimental investigation of geometric quantum speed limits in an open quantum system
Pires, D. P., DeAzevedo, E. R., Soares-Pinto, D. O., Brito, F., & Filgueiras, J. G. (2024) · Communications Physics · doi:10.1038/s42005-024-01634-5 · Cited by 17 · see citing works
“However, for nonunitary dynamics and mixed states, bounds on the variance are found that provide, at most, estimates of these quantities [74, 75].”
BackgroundAnti- PT -symmetric qubit: Decoherence and entanglement entropy
Cen, J., & Saxena, A. (2022) · Physical Review A · doi:10.1103/physreva.105.022404 · Cited by 14 · see citing works
“an or PT-symmetric qubit. We note that this calculation complements the analysis of decoherence function. 5 Fisher Information Quantum Fisher information is an important quantity in quantum metrology [60]. It quanties the precision that can be achieved in estimating a parameter for a given quantum state. Thus, it can…”
BackgroundOn a Conjecture regarding Fisher Information
Plastino, A., Bellomo, G., & Plastino, A. R. (2015) · Advances in Mathematical Physics · doi:10.1155/2015/120698 · Cited by 14 · see citing works
“In particular, it allows for the formulation of new quantum uncertainty principles [20–24].”
BackgroundPrecision bounds for noisy nonlinear quantum metrology
Zwierz, M., & Wiseman, H. M. (2014) · Physical Review A · doi:10.1103/physreva.89.022107 · Cited by 13 · see citing works
Falaye, B. J., Oyewumi, K. J., Sadikoglu, F. M., Hamzavi, M., & Ikhdair, S. M. (2015) · Journal of Theoretical and Computational Chemistry · doi:10.1142/s0219633615500364 · Cited by 14 · see citing works
Hall, M. J. W. (2023) · Physical Review A · doi:10.1103/physreva.107.062215 · Cited by 12 · see citing works
Unified entropies and quantum speed limits for nonunitary dynamics
Pires, D. P. (2022) · Physical Review A · doi:10.1103/physreva.106.012403 · Cited by 12 · see citing works
Quantum Wasserstein distance based on an optimization over separable states
Tóth, G., & Pitrik, J. (2023) · Quantum · doi:10.22331/q-2023-10-16-1143 · Cited by 12 · see citing works
Self-citationGeneralized-mean Cramér-Rao bounds for multiparameter quantum metrology
Lu, X.-M., Ma, Z., & Zhang, C. (2020) · Physical Review A · doi:10.1103/physreva.101.022303 · Cited by 12 · see citing works
“where the last equality is due to the equivalence between the QFI and the convex roof of variance [50, 51].”
BackgroundRevealing divergent length scales using quantum Fisher information in the Kitaev honeycomb model
Lambert, J., & Sørensen, E. S. (2020) · Physical review. B./Physical review. B · doi:10.1103/physrevb.102.224401 · Cited by 11 · see citing works
Kudo, D., & Tajima, H. (2023) · Physical Review A · doi:10.1103/physreva.107.062418 · Cited by 10 · see citing works
Geometry of quantum state space and quantum correlations
Deb, P. (2016) · Quantum Information Processing · doi:10.1007/s11128-015-1227-2 · Cited by 10 · see citing works
Measurement disturbance and conservation laws in quantum mechanics
Mohammady, M., Miyadera, T., & Loveridge, L. (2023) · Quantum · doi:10.22331/q-2023-06-05-1033 · Cited by 9 · see citing works
Photonic quantum metrology with variational quantum optical nonlinearities
Heras, A. M. D. L., Tabares, C., Schneider, J. T., Tagliacozzo, L., Porras, D., & González-Tudela, A. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013299 · Cited by 9 · see citing works
Limited-control metrology approaching the Heisenberg limit without entanglement preparation
Tratzmiller, B., Chen, Q., Schwartz, I., Huelga, S. F., & Plenio, M. B. (2020) · Physical Review A · doi:10.1103/physreva.101.032347 · Cited by 9 · see citing works
“[7] G.”
MethodAlternatives of entanglement depth and metrological entanglement criteria
Szalay, S., & Tóth, G. (2025) · Quantum · doi:10.22331/q-2025-04-18-1718 · Cited by 9 · see citing works
“For pure states ρ = | ϕ ⟩⟨ ϕ | ∈ P , this reduces to (four times) the variance [22], it is convex in the quantum state [22, 74], moreover, it is the convex roof extension of (four times) the variance [25, 26], These properties have also led to new types of uncertainty relations [75].”
BackgroundSelf-citationUniversal limitation of quantum information recovery: symmetry versus coherencePreprint
Tajima, H., & Saito, K. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.01876 · Cited by 8 · see citing works
Yamaguchi, K., & Tajima, H. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.200203 · Cited by 9 · see citing works
Standard symmetrized variance with applications to coherence, uncertainty, and entanglement
Zhao, M., Zhang, L., & Fei, S.-M. (2022) · Physical Review A · doi:10.1103/physreva.106.012417 · Cited by 7 · see citing works
Ge, W., Jacobs, K. A., & Zubairy, M. S. (2023) · npj Quantum Information · doi:10.1038/s41534-022-00670-9 · Cited by 7 · see citing works
“Performing the analysis in Eq.(35), but splitting the QFI into its quantum and classical parts, we have the equivalent result for the metrological power for a mixed state”
MethodA sufficient entanglement criterion based on quantum fisher information and variance
Zhang, Q., & Fei, S.-M. (2020) · Laser Physics Letters · doi:10.1088/1612-202x/ab8793 · Cited by 7 · see citing works
“[18] G.”
BackgroundUnified framework for calculating convex roof resource measures
Zhu, X., Zhang, C., An, Z., & Zeng, B. (2025) · npj Quantum Information · doi:10.1038/s41534-025-01012-1 · Cited by 7 · see citing works
“It was firstly conjectured that QFI can also be expressed by convex roof extension [60] and then proved in [61], i.e., where (∆ H ) 2 ψ = ⟨ ψ | H 2 | ψ ⟩ − ⟨ ψ | H | ψ ⟩ 2 is the variance of H by measuring a pure state | ψ ⟩ .”
Marić, V., Ferro, F., & Fagotti, M. (2025) · Physical review. B./Physical review. B · doi:10.1103/physrevb.111.205118 · Cited by 10 · see citing works
Quantum information inequalities via tracial positive linear maps
Dadkhah, A. R., & Moslehian, M. S. (2016) · Journal of Mathematical Analysis and Applications · doi:10.1016/j.jmaa.2016.10.027 · Cited by 6 · see citing works
Smooth Metric Adjusted Skew Information Rates
Yamaguchi, K., & Tajima, H. (2023) · Quantum · doi:10.22331/q-2023-05-22-1012 · Cited by 6 · see citing works
Some inequalities for central moments of matrices
Léka, Z. (2016) · Linear Algebra and its Applications · doi:10.1016/j.laa.2016.01.044 · Cited by 6 · see citing works
“Recently, extremal properties of the quantum variance were studied in [20].”
BackgroundOn a Conjecture regarding Fisher Information
Plastino, A. R., Bellomo, G., & Plastino, A. R. (2015) · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · Cited by 5 · see citing works
Average number is an insufficient metric for interferometry
Branford, D., & Rubio, J. (2021) · New Journal of Physics · doi:10.1088/1367-2630/ac3571 · Cited by 5 · see citing works
Can multipartite entanglement be characterized by two-point connected correlation functions?
Lepori, L., Trombettoni, A., Giuliano, D., Kombe, J., Malo, J. Y., Daley, A. J., Smerzi, A., & Chiofalo, M. L. (2023) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/acdd36 · Cited by 5 · see citing works
“It turns out that {V̄ [ρ, Ô]N , V [ρ, Ô]N} ≥ F [ρ, Ô]N , as formalized in the so-called ”convex roof theorem” [65, 66] : for chosen ρ and Ô, the resulting QFI is the minimum average variance between all the possible decompositions {pλ, |λ〉} of ρ.”
Detecting macroscopic indefiniteness of cat states in bosonic interferometers
Kelly, S. P., Timmermans, E., & Tsai, S.-W. (2019) · Physical Review A · doi:10.1103/physreva.100.032117 · Cited by 5 · see citing works
Anti-PT-symmetric qubit: Decoherence and Entanglement Entropy
Los Alamos National Lab. (LANL), L. A., Saxena, A., & Cen, J. (2020) · doi:10.2172/1647202 · Cited by 5 · see citing works
Generalized entropic quantum speed limits
de Sousa, J. F., & Pires, D. P. (2025) · Physical Review A · doi:10.1103/ghr4-d2vb · Cited by 4 · see citing works
Symmetry hinders quantum information recoveryPreprint
Tajima, H., & Saito, K. (2021) · arXiv (Cornell University) · Cited by 4 · see citing works
Uncertainty Relations between Quantum Fisher Information and Entanglement Monotones
Du, S., Liu, S., Fadel, M., Vitagliano, G., & He, Q. (2026) · Physical Review Letters · doi:10.1103/54mc-2yl3 · Cited by 3 · see citing works
“Moreover, the QFI, which can be seen as a generalized and convex variance [10, 11], is connected to statistical and information-theoretic concepts of quantum theory, such as parameter estimation [12–16], uncertainty relations [17, 18], and quantum resources [19–22], especially entanglement [23–31].”
Measurement-based quantum machine learning
Calderón, L. M., Raussendorf, R., Feldmann, P., & Bondarenko, D. (2026) · Physical Review A · doi:10.1103/2snk-m8c6 · Cited by 5 · see citing works
Some Decompositions of Matrix Variances
Léka, Z., & Petz, D. (2013) · Probability and Mathematical Statistics · Cited by 3 · see citing works
Self-citation
- 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 › Computer Science › Artificial Intelligence › 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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