Riemannian metrics on positive definite matrices related to means
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The bound is useful in practical scenarios of noisy quantum metrology, especially in the case of transversal dephasing.
“The bound is useful in practical scenarios of noisy quantum metrology, especially in the case of transversal dephasing [59,60].”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [60]
Transversal dephasing noise is the relevant scenario whereby one can attain precision in estimation of ω0 scaling superclassically with the number of qubits, even if noise applies independently to each qubit; any superclassical advantage is lost for parallel dephasing.
“More precisely, transversal dephasing noise stands as the relevant scenario whereby one can attain a precision in the estimation of the parameter ω 0 that scales superclassically with the number of qubits, even if such noise applies independently to each qubit (while any superclassical advantage is lost in the case of parallel dephasing noise).”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [60]
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“11] that if σ is an operator mean with s := f ′(1) for the corresponding operator monotone function f (see [13]), then”
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“The authors thank the referee for his helpful comments and for calling their attention to the papers [3, 6, 15].”
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“, Ak ∈ Pn was introduced in [26, 9] as the unique minimizer of A ∈ Pn 7→ ∑k j=1 δ (2)(A,Aj).”
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“(2) This is found in [32] but a short proof using the differential formula (see [6]) is given here.”
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BackgroundRiemannian geometry and matrix geometric means
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“This is the so-called exponential metric increasing (EMI ) property in [7, 9].”
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“s=t=0 , of a quasi-entropy [28, 29] SF (D1,D2) defined by a function F on (0,∞) with the relation 1/f(x) = (F (x) + xF (x−1))/(x − 1)2.”
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The Geometric Mean, Matrices, Metrics, and More
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“10) which was called generalized variance in [31].”
BackgroundSelf-citationThe Golden-Thompson trace inequality is complemented
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Self-citationMonotonic properties of the least squares mean
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“The authors thank the referee for his helpful comments and for calling their attention to the papers [3, 6, 15].”
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“It was observed in [38] that the Wigner-Yanase-Dyson skew information I D (p;K) coincides, apart from a constant factor, with a monotone Riemannian metric K fp D (i[D;K]; i[D;K]); where fp is a standard operator monotone function de ned by”
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“When ||| · ||| is the operator norm, such Finsler manifolds have been studied by several authors (see [11] for example).”
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“2) plays the role of a natural Riemannian metric and the corresponding information geometry of the Gaussians was discussed in [32] in detail.”
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“It was recently observed in [10, 22] that the stronger condition (c) is even satis ed for many cases where M ((1));M (2) 2 M0 satisfy (b).”
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“8), fp(0)/fp(x) is increasing in p ∈ (0, 1/2] so that the Wigner-Yanase-Dyson skew information IWYD D (p,K) is increasing in p ∈ (0, 1/2] for fixed D and K (see [5]).”
BackgroundOn infinite divisibility of positive definite functions arising from operator means
Kosaki, H. (2007) · Journal of Functional Analysis · doi:10.1016/j.jfa.2007.09.021 · Cited by 20 · cited 1× · see citing works
“It was recently observed in [10, 22] that the stronger condition (c) is even satis ed for many cases where M ((1));M (2) 2 M0 satisfy (b).”
BackgroundA Geometric Interpretation of Segal's Inequality || e X + Y || ≤ || e X/2 e Y e X/2 ||
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Fujii, J. I., Fujii, M., Miura, T., Takagi, H., & Takahasi, S. (2006) · Journal of Inequalities and Applications · doi:10.1155/jia/2006/75941 · Cited by 5 · see citing works
Flagged first, then by citations
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Generalized Geometric Quantum Speed Limits
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Self-citationConic Geometric Optimization on the Manifold of Positive Definite Matrices
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A new metric on the manifold of kernel matrices with application to matrix geometric means
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Hiai, F. (2010) · Interdisciplinary Information Sciences · doi:10.4036/iis.2010.139 · Cited by 122 · see citing works
“, via the left and right multiplications) were treated in [43, 65] to study certain Riemannian metrics on matrix spaces.”
MethodSelf-citationGeometrical Bounds of the Irreversibility in Markovian Systems
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“Here P(n) forms a differentiable Riemannian manifold [26, 27] with non-positive curvature [28].”
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“This speci c geometry on PD matrices is well studied, see [42, 62, 74, 81].”
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From ƒ-Divergence to Quantum Quasi-Entropies and Their Use
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Self-citationO(n)-invariant Riemannian metrics on SPD matrices
Thanwerdas, Y., & Pennec, X. (2022) · Linear Algebra and its Applications · doi:10.1016/j.laa.2022.12.009 · Cited by 34 · see citing works
“As the goal of this paper is to extend the class of kernel metrics, we selected from [24, 35] the results that we found simple and powerful to be able to generalize them later on.”
BackgroundDynamical role of quantum signatures in quantum thermometry
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“(18) The first half of the following theorem was proved by Hiai and Petz [17], the second is new.”
BackgroundInfluentialSome norm inequalities for matrix means
Bhatia, R., Lim, Y., & Yamazaki, T. (2016) · Linear Algebra and its Applications · doi:10.1016/j.laa.2016.03.018 · Cited by 29 · see citing works
Riemannian metrics on positive definite matrices related to means. II
Hiai, F., & Petz, D. (2011) · Linear Algebra and its Applications · doi:10.1016/j.laa.2011.10.029 · Cited by 29 · see citing works
Self-citationGeometric target detection based on total Bregman divergence
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Self-citationRobust Covariance Estimators Based on Information Divergences and Riemannian Manifold
Hua, X., Cheng, Y., Wang, H., & Qin, Y. (2018) · Entropy · doi:10.3390/e20040219 · Cited by 24 · see citing works
“It is well known that HPD matrices form a differentiable Riemannian manifold [25], that is the most studied example of a manifold with non-positive curvature [26].”
BackgroundThermodynamics and the quantum speed limit in the non-Markovian regime
Das, A., Bera, A., Chakraborty, S., & Chruściński, D. (2021) · Physical Review A · doi:10.1103/physreva.104.042202 · Cited by 20 · see citing works
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“The complex Riemannian interpolation is another ‘‘global’’ interpolation method [26].”
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Hameduddin, I., & Zaki, T. A. (2019) · Journal of Fluid Mechanics · doi:10.1017/jfm.2019.46 · Cited by 16 · see citing works
“In addition, it is not the only one possible for the set of positive-definite tensors, e.g. Hiai & Petz (2009) introduce a family of Riemannian metrics defined by a kernel function and of which the present metric is a member.”
BackgroundPreserver Problems Related to Quasi-Arithmetic Means of Invertible Positive Operators
Gaál, M., & Nagy, G. (2018) · Integral Equations and Operator Theory · doi:10.1007/s00020-018-2426-x · Cited by 14 · see citing works
Ando–Hiai type inequalities for multivariate operator means
Hiai, F., Seo, Y., & Wada, S. (2018) · Linear and Multilinear Algebra · doi:10.1080/03081087.2018.1488938 · Cited by 14 · see citing works
Self-citationA global approach to the refinement of manifold data
Dyn, N., & Sharon, N. (2015) · Mathematics of Computation · doi:10.1090/mcom/3087 · Cited by 13 · see citing works
“Here are several examples: on a sphere (e.g., [14]), on an ellipsoid (e.g., [16]), on the cone of positive definite matrices (e.g., [18]), in the Lie group of orthogonal matrices of the same determinant (e.g., [33, Chapter 3]), in the Heisenberg groups (e.g., [25]).”
Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid
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“For further discussion regarding the metric on the set P(d), see also Hiai and Petz [84] and Hiai and Petz [85].)”
BackgroundOn Interference-Rejection Using Riemannian Geometry for Direction of Arrival Estimation
Bar, A., & Talmon, R. (2023) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2023.3322779 · Cited by 12 · see citing works
“…windows is set to be larger than The incorporation of the Riemannian geometry is realized by viewing each matrix Γ i as a point on the HPD manifold [34] and considering their Riemannian mean, denoted as Γ R and given by In general, there is no closed-form solution to (17) on the HPD manifold for…”
BackgroundStrong monotonicity for various means
Kosaki, H. (2014) · Journal of Functional Analysis · doi:10.1016/j.jfa.2014.07.021 · Cited by 12 · see citing works
On the Ando–Li–Mathias mean and the Karcher mean of positive definite matrices
Fujii, J. I., & Seo, Y. (2014) · Linear and Multilinear Algebra · doi:10.1080/03081087.2014.896359 · Cited by 12 · see citing works
“By Theorem B, Hiai and Petz showed in [28] the following log-majorization for the Karcher mean:”
BackgroundTarget detection in sea clutter via weighted averaging filter on the Riemannian manifold
Hua, X., Cheng, Y., Li, Y., Shi, Y., Wang, H., & Qin, Y. (2017) · Aerospace Science and Technology · doi:10.1016/j.ast.2017.07.042 · Cited by 11 · see citing works
“Here P(n) forms a differentiable Riemannian manifold [12,13] with non-positive curvature [14,15].”
Hua, X., Cheng, Y., Wang, H., & Qin, Y. (2018) · Entropy · doi:10.3390/e20040258 · Cited by 11 · see citing works
“It is well known that Hermitian positive-definite (HPD) matrices form a differentiable Riemannian (and also a Finsler) manifold [32,33], that is the most studied example of a manifold with nonpositive curvature [34,35].”
BackgroundNorm inequalities for matrix geometric means of positive definite matrices
Fujii, J. I., Seo, Y., & Yamazaki, T. (2015) · Linear and Multilinear Algebra · doi:10.1080/03081087.2015.1051939 · Cited by 10 · see citing works
“On the other hand, Hiai and Petz [11] showed that∣∣∣∣∣∣∣∣∣GK(ω; Ap1 , . . . , Apn )1/p∣∣∣∣∣∣∣∣∣ ≤ |||♦(ω; A1, . . . , An)||| for all p > 0 and ∣∣∣∣∣∣GK(ω; Ap1 , . . . , Apn )1/p∣∣∣∣∣∣ increases to |||♦(ω; A1, . . . , An)||| as p ↓ 0.”
BackgroundInfluentialRole of flow of information in the speedup of quantum evolution
Wang, J., Wu, Y., & Xie, Z. (2018) · Scientific Reports · doi:10.1038/s41598-018-34890-x · Cited by 10 · see citing works
Path of quasi-means as a geodesic
Fujii, J. I. (2010) · Linear Algebra and its Applications · doi:10.1016/j.laa.2010.09.011 · Cited by 11 · see citing works
Subdivision Schemes for Positive Definite Matrices
Itai, U., & Sharon, N. (2012) · Foundations of Computational Mathematics · doi:10.1007/s10208-012-9131-y · Cited by 9 · see citing works
“It turns out that on the manifold of positive definite matrices, the geometric mean G 1 2 (A,B) is the mid-point of the geodesic joining A and B , defined by the Riemannian metric d(A,B) = ‖log(A− (1)2 BA− (1)2 )‖ (for more details see [18]).”
BackgroundAn order inequality characterizing invariant barycenters on symmetric cones
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From quasi-entropyPreprint
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Self-citationGeometry-Aware Discriminative Dictionary Learning for PolSAR Image Classification
Zhang, Y., Lai, X., Xie, Y., Qu, Y., & Li, C. (2021) · Remote Sensing · doi:10.3390/rs13061218 · Cited by 8 · see citing works
Convexity of quasi-entropy type functions: Lieb’s and Ando’s convexity theorems revisited
Hiai, F., & Petz, D. (2013) · Cited by 8 · see citing works
Self-citationMultivariate Trace Inequalities
Sutter, D. (2018) · SpringerBriefs in mathematical physics · doi:10.1007/978-3-319-78732-9_4 · Cited by 8 · see citing works
Yan, Y., Hu, C., Jiang, Q., Yang, H., Li, W., & Wang, R. (2025) · IEEE Transactions on Aerospace and Electronic Systems · doi:10.1109/taes.2025.3581895 · Cited by 7 · see citing works
“For two points P , Q ∈ T Y M , the Rao–Fisher metric is defined as [37] g Then, the squared length of a small displacement d Y on the SPD manifold is The geodesic is the curve connecting two points on the manifold that has the minimum length.”
Convex multivariate operator means
Hansen, F. (2018) · Linear Algebra and its Applications · doi:10.1016/j.laa.2018.11.032 · Cited by 7 · see citing works
“A number of authors studied the properties of the Karcher mean, see [11, 12, 14, 15, 3, 4, 10] and the references therein.”
BackgroundThe geometry of mixed-Euclidean metrics on symmetric positive definite matrices
Thanwerdas, Y., & Pennec, X. (2022) · Differential Geometry and its Applications · doi:10.1016/j.difgeo.2022.101867 · Cited by 7 · see citing works
“Secondly, we recall the main facts about classes of kernel and mean kernel metrics introduced in [17, 18].”
BackgroundMethodGeneralized Pólya–Szegö type inequalities for some non-commutative geometric means
Seo, Y. (2011) · Linear Algebra and its Applications · doi:10.1016/j.laa.2011.10.036 · Cited by 7 · see citing works
A Characterization of Unitary-Antiunitary Similarity Transformations via Kubo–Ando Means
Gaál, M., & Nagy, G. (2018) · Analysis Mathematica · doi:10.1007/s10476-018-0401-z · Cited by 7 · see citing works
Riemannian frameworks for the harmonization of resting-state functional MRI scans
Honnorat, N., Seshadri, S. S., Killiany, R., Blangero, J., Glahn, D. C., Fox, P. T., & Habes, M. (2023) · Medical Image Analysis · doi:10.1016/j.media.2023.103043 · Cited by 6 · see citing works
Weighted multivariable operator means of positive definite operators
Pálfia, M., & Petz, D. (2014) · Linear Algebra and its Applications · doi:10.1016/j.laa.2014.08.025 · Cited by 6 · see citing works
Self-citationTHE GEOMETRIC MEAN AND THE GEODESIC FITTING PROBLEM ON THE GRASSMANN MANIFOLD
Batzies, E., Machado, L., & Leite, F. S. (2013) · Cited by 6 · see citing works
Preserving problems of geodesic-affine maps and related topics on positive definite matrices
Szokol, P., Tsai, M.-C., & Zhang, J. (2015) · Linear Algebra and its Applications · doi:10.1016/j.laa.2015.06.009 · Cited by 6 · see citing works
“Among others in [5, 6, 16, 18], the authors determined the geodesic curves and geodesic distances between any pairs of positive definite matrices in Riemannian manifolds.”
BackgroundConvexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited
Hiai, F., & Petz, D. (2013) · Journal of Mathematical Physics · doi:10.1063/1.4810781 · Cited by 6 · see citing works
Self-citationThe Ando-Hiai inequalities for the solution of the generalized Karcher equation and related results
Yamazaki, T. (2019) · Journal of Mathematical Analysis and Applications · doi:10.1016/j.jmaa.2019.06.036 · Cited by 6 · see citing works
“It was obtained in [15].”
Isometries of the geodesic distances for the space of invertible positive operators and matrices
Honma, S., & Nogawa, T. (2013) · Linear Algebra and its Applications · doi:10.1016/j.laa.2013.11.037 · Cited by 5 · see citing works
Hiai, F., & Lim, Y. (2017) · Linear Algebra and its Applications · doi:10.1016/j.laa.2017.07.017 · Cited by 5 · see citing works
Self-citationA Universal Framework for Learning the Elliptical Mixture Model
Li, S., Yu, Z., & Mandic, D. P. (2020) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2020.3010198 · Cited by 5 · see citing works
“The entropy related metric was later used by Hiai and Petz to define the Riemannian metric for positive definite matrices [53].”
MethodInfluentialBiran, A., & Jeremic, A. M. (2022) · Biomedical Signal Processing and Control · doi:10.1016/j.bspc.2023.104575 · Cited by 5 · see citing works
Structure of Hiai–Petz parametrized geometry for positive definite matrices
Fujii, J. I. (2009) · Linear Algebra and its Applications · doi:10.1016/j.laa.2009.08.009 · Cited by 5 · see citing works
Matrix Analysis with some Applications
Petz, D. (2011) · Cited by 5 · see citing works
Self-citationRiemannian competitive learning for symmetric positive definite matrices clustering
Zheng, L., Qiu, G., & Huang, J. (2018) · Neurocomputing · doi:10.1016/j.neucom.2018.03.015 · Cited by 4 · see citing works
Nagy, G., & Szokol, P. (2019) · Electronic Journal of Linear Algebra · doi:10.13001/1081-3810.3897 · Cited by 4 · see citing works
“(10) until the end of the proof of Lemma 3 (for g ◦ (−idR) in the case where g is decreasing), we conclude that T2 is unitary or antiunitary and Y1 + Y2 = 0.”
BackgroundPermutation-Invariant Log-Euclidean Geometries on Full-Rank Correlation Matrices
Thanwerdas, Y. (2024) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/22m1538144 · Cited by 3 · see citing works
Information geometry of operator scaling
Matsuda, T., & Soma, T. (2022) · Linear Algebra and its Applications · doi:10.1016/j.laa.2022.04.022 · Cited by 3 · see citing works
Generalized Karcher equation, relative operator entropy and the Ando-Hiai inequality
Yamazaki, T. (2018) · Cited by 3 · see citing works
Non-monotone metric on the quantum parametric model
Suzuki, J. (2021) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-021-01101-y · Cited by 3 · see citing works
Speed of qubit states during thermalisationPreprint
Feyles, M. M., Mancino, L., Sbroscia, M., Gianani, I., & Barbieri, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1901.05015 · Cited by 6 · see citing works
“[30] F.”
BackgroundDomain Adaptation for DoA Estimation in Multipath Channels With Interferences
Bar, A., Picard, J. S., Cohen, I., & Talmon, R. (2025) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2025.3577257 · Cited by 2 · see citing works
“We show that the map we built could be described analogously as parallel transport in the manifold of HPD matrices equipped with the Affine Invariant metric [31], [32], [33], [34], [35].”
Approximating Sets of Symmetric and Positive-Definite Matrices by Geodesics
Machado, L., & Leite, F. S. (2013) · Conference Papers in Mathematics · doi:10.1155/2013/425608 · Cited by 2 · see citing works
Clustering Symmetric Positive Definite Matrices on the Riemannian Manifolds
Zheng, L., Qiu, G., & Huang, J. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-54181-5_26 · Cited by 2 · see citing works
Semigroups of operator means and generalized Karcher equationsPreprint
Pálfia, M. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1208.5603 · Cited by 2 · see citing works
RTF Estimation Using Riemannian Geometry for Speech Enhancement in the Presence of Interferences
Ronai, O., Sitton, Y., Bar, A., & Talmon, R. (2025) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp49660.2025.10888479 · Cited by 1 · see citing works
“Specifically, we propose to use the Riemannian mean of the spatial correlation matrices, considering the Affine-Invariant (AI) metric [24], [25].”
Application of joint domain localised matrix CFAR detector for HFSWR
Ye, L., Yang, Q., Chen, Q., & Deng, W. (2019) · The Journal of Engineering · doi:10.1049/joe.2019.0744 · Cited by 1 · see citing works
Predicting precision matrices for color matching problem
Nakamoto, T., Nishii, R., & Eguchi, S. (2019) · International Journal of Mathematics for Industry · doi:10.1142/s2661335219500023 · Cited by 1 · see citing works
The use of systems of stochastic PDEs as priors for multivariate models with discrete structuresPreprint
Aune, E., & Simpson, D. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1208.1717 · Cited by 1 · see citing works
“This exposition is based on Hiai and Petz (2009); Bhatia (2007).”
BackgroundOn some isometries and other preservers
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Matrix means and geometric constructions
Pálfia, M. (2011)
Nonparametric Bayes for Big Data
Yang, Y. (2014) · DukeSpace (Duke University)
On the construction of local interaction rules that perform global linear computation
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“The geodesic from I to PT in the space of positive definite matrices (in Frobenius norm) [7] is”
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Aune, E., & Simpson, D. (2012)
Geometria da informação quântica: uma abordagem geral acerca do tempo de evolução
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Groups of Nonlinear Congruences and Geodesic Lines on the Ordered Set of Positive Definite Matrices
Bessmertnyi (2019) · Intellectual Archive · doi:10.32370/ia_2019_01_19
Chansangiam, P., & Sabău, S. V. (2019) · Mathematics · doi:10.3390/math7121162
“It turns out that a differentiable monotone metric on Dn determines a Riemannian metric; see more information in [10].”
BackgroundBasic Entropies for Positive-Definite Matrices
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On a structure of the Hiai-Petz geometry (Prospects of non-commutative analysis in operator theory)
Fujii, J. I. (2010) · Kyoto University Research Information Repository (Kyoto University)
Segman, Y.-I., Picard, J. S., & Talmon, R. (2026) · IEEE Open Journal of Signal Processing · doi:10.1109/ojsp.2026.3733589
“…in the case of two matrices, there is a closed-form expression for (8), which is the midpoint of the unique geodesic curve between R 1 and R 2 [35]: Several works have shown that covariance matrices are more naturally handled through their intrinsic Riemannian geometry than through Euclidean…”
- Authors
- Hiai, F., & Petz, D.
- Affiliations
- Tohoku University (JP)
- Corresponding author
- Hiai, Fumio
- ORCID
- Hiai, Fumio
- Venue
- Linear Algebra and its Applications
- Publisher
- Elsevier BV
- Published
- 2009-06
- Volume
- 430
- Issue
- 11-12
- Pages
- 3105-3130
- ISSN
- 0024-3795 (ISSN-L)
- OpenAlex
- W2029258199
- Cited by
- 92
- References
- 49
- Citations by year
- 2026: 1 · 2025: 3 · 2024: 1 · 2023: 4 · 2022: 4 · 2021: 5 · 2020: 2 · 2019: 12 · 2018: 14 · 2017: 6 · 2016: 3 · 2015: 9 · 2014: 6 · 2013: 5 · 2012: 6
- FWCI
- 8.72
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Linear Algebra and its Applications (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Hungarian Scientific Research Fund
- Japan Society for the Promotion of Science (21540208, 17340043)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Inequalities and Applications
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- Riemannian metrics, positive definite matrices, mean, geometric mean, logarithmic mean, geodesics, geodesic distance, statistical indicators, quantum Fisher information
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