The DOI matches its Crossref record.
On Equivalence of Infinite Product Measures
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“Accordingly, the generalised angle between arbitrary quantum states is given by the Bures angle [64,65]”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [64]
Citations
Cites
Still being checked.
Cited by
Sorted by citations
The Divergence and Bhattacharyya Distance Measures in Signal Selection
Kailath, T. (1967) · IRE Transactions on Communications Systems · doi:10.1109/tcom.1967.1089532 · Cited by 1,860
The ‘transition probability’ in the state space of a—algebra
Uhlmann, A. (1976) · Reports on Mathematical Physics · doi:10.1016/0034-4877(76)90060-4 · Cited by 1,649
Rényi Divergence and Kullback-Leibler Divergence
van Erven, T., & Harremoës, P. (2014) · IEEE Transactions on Information Theory · doi:10.1109/tit.2014.2320500 · Cited by 1,688
The Selection of Prior Distributions by Formal Rules
Kass, R. E., & Wasserman, L. (1996) · Journal of the American Statistical Association · doi:10.1080/01621459.1996.10477003 · Cited by 1,291
On Choosing and Bounding Probability Metrics
Gibbs, A. L., & Su, F. E. (2002) · International Statistical Review · doi:10.1111/j.1751-5823.2002.tb00178.x · Cited by 1,199
On Divergences and Informations in Statistics and Information Theory
Liese, F., & Vajda, I. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.881731 · Cited by 578
Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control
Deffner, S., & Campbell, S. (2017) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8121/aa86c6 · Cited by 585
Quantum 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
Mathematical Foundations of Infinite-Dimensional Statistical Models
Evarist Gine, M., & Nickl, R. (2015) · Cambridge University Press eBooks · doi:10.1017/cbo9781107337862 · Cited by 401
Bures, D. (1969) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1969-0236719-2 · Cited by 363
Free states of the canonical anticommutation relations
Powers, R. T., & Størmer, E. (1970) · Communications in Mathematical Physics · doi:10.1007/bf01645492 · Cited by 268
Liero, M., Mielke, A., & Savaré, G. (2017) · Inventiones mathematicae · doi:10.1007/s00222-017-0759-8 · Cited by 264
On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates
LeCam, L. M. (1970) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177696960 · Cited by 261
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
Schwab, C., & Gittelson, C. J. (2011) · Acta Numerica · doi:10.1017/s0962492911000055 · Cited by 259
Equivalence and perpendicularity of Gaussian processes
Feldman, J. (1958) · Pacific Journal of Mathematics · doi:10.2140/pjm.1958.8.699 · Cited by 252
Generalized Clausius inequality for nonequilibrium quantum processes
Deffner, S., & Lutz, E. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.170402 · Cited by 245
Informational distance on quantum-state space
Luo, S., & Zhang, Q. (2004) · Physical Review A · doi:10.1103/physreva.69.032106 · Cited by 216
Distributions in Hilbert Space and Canonical Systems of Operators
Segal, I. E. (1958) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1958-0102759-x · Cited by 213
Energy–time uncertainty relation for driven quantum systems
Deffner, S., & Lutz, E. (2013) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8113/46/33/335302 · Cited by 199
The dimensions of individual strings and sequences
Lutz, J. H. (2003) · Information and Computation · doi:10.1016/s0890-5401(03)00187-1 · Cited by 184
The Burbea-Rao and Bhattacharyya Centroids
Nielsen, F., & Boltz, S. (2011) · IEEE Transactions on Information Theory · doi:10.1109/tit.2011.2159046 · Cited by 162
On Choosing and Bounding Probability Metrics
Gibbs, A. L., & Su, F. E. (2002) · International Statistical Review · doi:10.2307/1403865 · Cited by 136
Harmonic analysis on the infinite symmetric group
Kerov, S., Olshanski, G., & Vershik, A. (2004) · Inventiones mathematicae · doi:10.1007/s00222-004-0381-4 · Cited by 131
Probabilistic well-posedness for the cubic wave equation
Burq, N., & Tzvetkov, N. (2013) · Journal of the European Mathematical Society · doi:10.4171/jems/426 · Cited by 129
Iterated function systems, Ruelle operators, and invariant projective measures
Dutkay, D. E., & Jørgensen, P. E. T. (2006) · Mathematics of Computation · doi:10.1090/s0025-5718-06-01861-8 · Cited by 126
Infinite Random Matrices and Ergodic Measures
Borodin, A., & Olshanski, G. (2001) · Communications in Mathematical Physics · doi:10.1007/s002200100529 · Cited by 114
Effective Strong Dimension in Algorithmic Information and Computational Complexity
Athreya, K. B., Hitchcock, J. M., Lutz, J. H., & Mayordomo, E. (2007) · SIAM Journal on Computing · doi:10.1137/s0097539703446912 · Cited by 113
ABSOLUTE CONTINUITY AND SINGULARITY OF LOCALLY ABSOLUTELY CONTINUOUS PROBABILITY DISTRIBUTIONS. I
Kabanov, J. M., Lipcer, R. Š., & Širjaev, A. N. (1979) · Mathematics of the USSR-Sbornik · doi:10.1070/sm1979v035n05abeh001615 · Cited by 101
Mathematical metaphysics of randomness
Muchnik, A. A., Semenov, A. L., & Uspensky, V. (1998) · Theoretical Computer Science · doi:10.1016/s0304-3975(98)00069-3 · Cited by 95
Geometric phases and related structures
Uhlmann, A. (1995) · Reports on Mathematical Physics · doi:10.1016/0034-4877(96)83640-8 · Cited by 94
Analytic properties of infinite-dimensional distributions
Bogachev, V. I., & Smolyanov, O. G. (1990) · Russian Mathematical Surveys · doi:10.1070/rm1990v045n03abeh002364 · Cited by 90
Density operators as an arena for differential geometry
Uhlmann, A. (1993) · Reports on Mathematical Physics · doi:10.1016/0034-4877(93)90060-r · Cited by 88
Liero, M., Mielke, A., & Savaré, G. (2015) · doi:10.20347/wias.preprint.2160 · Cited by 86
On non-singular transformations of a measure space. I
Krieger, W. (1969) · Probability Theory and Related Fields · doi:10.1007/bf00531811 · Cited by 86
Information geometry in vapour–liquid equilibrium
Brody, D. C., & Hook, D. W. (2008) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/42/2/023001 · Cited by 86
On thef-divergence and singularity of probability measures
Vajda, I. (1972) · Periodica Mathematica Hungarica · doi:10.1007/bf02018663 · Cited by 82
Distinguishing a Sequence of Random Variables from a Translate of Itself
Shepp, L. A. (1965) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177699985 · Cited by 80
Additive properties of random sequences of positive integers
Erdös, P., & Rényi, A. (1960) · Acta Arithmetica · doi:10.4064/aa-6-1-83-110 · Cited by 79
Von Mises' definition of random sequences reconsidered
van Lambalgen, M. (1987) · Journal of Symbolic Logic · doi:10.1017/s0022481200029728 · Cited by 79
Liero, M., Mielke, A., & Savaré, G. (2016) · SIAM Journal on Mathematical Analysis · doi:10.1137/15m1041420 · Cited by 78
On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space
Krieger, W. (1970) · Lecture notes in mathematics · doi:10.1007/bfb0060653 · Cited by 70
On quasiequivalence of locally normal states
Холево, А. С. (1972) · Theoretical and Mathematical Physics · doi:10.1007/bf01035528 · Cited by 62
Non singular transformations and spectral analysis of measures
Host, B., Méla, J.-F., & Parreau, F. (1991) · Bulletin de la Société mathématique de France · doi:10.24033/bsmf.2157 · Cited by 62
Transient and recurrent spectrum
Avron, J. E., & Barry Simon, A. (1981) · Journal of Functional Analysis · doi:10.1016/0022-1236(81)90034-3 · Cited by 62
Endomorphisms of B(H). II. Finitely Correlated States on On
Bratteli, O., & Jørgensen, P. E. T. (1997) · Journal of Functional Analysis · doi:10.1006/jfan.1996.3033 · Cited by 61
Thermodynamic length for far-from-equilibrium quantum systems
Deffner, S., & Lutz, E. (2013) · Physical Review E · doi:10.1103/physreve.87.022143 · Cited by 56
Injections de Sobolev probabilistes et applications
Burq, N., & Lebeau, G. (2013) · Annales Scientifiques de l École Normale Supérieure · doi:10.24033/asens.2206 · Cited by 55
Conditions for absolute continuity between a certain pair of probability measures
Kadota, T., & Shepp, L. A. (1970) · Probability Theory and Related Fields · doi:10.1007/bf00534599 · Cited by 54
Approximation of Improper Prior Measures by Prior Probability Measures
Stein, C. (1965) · Springer eBooks · doi:10.1007/978-3-642-99884-3_13 · Cited by 53
Gaussian Measures on Separable Hilbert Spaces and Applications
Maniglia, S., & Rhandi, A. (2004) · Università del Salento · Cited by 52
Dimension-free path-integral molecular dynamics without preconditioning
Korol, R., Rosa-Raíces, J. L., Bou-Rabee, N., & Miller, T. F. (2020) · The Journal of Chemical Physics · doi:10.1063/1.5134810 · Cited by 52
On the theory of statistical decision functions
Matusita, K. (1951) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf02949773 · Cited by 52
On the Valuation of Long-Dated Assets
MARTIN, I. W. R. (2012) · Journal of Political Economy · doi:10.1086/666527 · Cited by 45
Representation Formulae for the Fractional Brownian Motion
Picard, J. (2010) · Lecture notes in mathematics · doi:10.1007/978-3-642-15217-7_1 · Cited by 50
Non-Gaussian statistical inverse problems. Part I: Posterior distributions
Lasanen, S. (2012) · Inverse Problems and Imaging · doi:10.3934/ipi.2012.6.215 · Cited by 46
ON THE QUESTION OF ABSOLUTE CONTINUITY AND SINGULARITY OF PROBABILITY MEASURES
Kabanov, J. M., Lipcer, R. Š., & Širjaev, A. N. (1977) · Mathematics of the USSR-Sbornik · doi:10.1070/sm1977v033n02abeh002421 · Cited by 46
Equivalence and Singularity of Gaussian Measures and Applications
Chatterji, S. D., & Mandrekar, V. S. (1978) · Elsevier eBooks · doi:10.1016/b978-0-12-095601-2.50009-2 · Cited by 45
Potential theory on the infinite dimensional torus
Berg, C. (1976) · Inventiones mathematicae · doi:10.1007/bf01389771 · Cited by 45
Perturbation theory of dense point spectra
Howland, J. S. (1987) · Journal of Functional Analysis · doi:10.1016/0022-1236(87)90038-3 · Cited by 45
Stochastic Processes Induced by Singular Operators
Alpay, D., & Jørgensen, P. E. T. (2012) · Numerical Functional Analysis and Optimization · doi:10.1080/01630563.2012.682132 · Cited by 44
The Transition Probability for States of *‐Algebras
Uhlmann, A. (1985) · Annalen der Physik · doi:10.1002/andp.19854970419 · Cited by 43
THEORIES AND METHODS IN CLASSIFICATION: A REVIEW
Gupta, S. D. (1973) · Elsevier eBooks · doi:10.1016/b978-0-12-154050-0.50012-5 · Cited by 42
Self-consistent theory for random alloys with short-range order
Gray, L. J., & Kaplan, T. (1981) · Physical review. B, Condensed matter · doi:10.1103/physrevb.24.1872 · Cited by 42
The quasi-invariance property for the Gamma kernel determinantal measure
Olshanski, G. (2010) · Advances in Mathematics · doi:10.1016/j.aim.2010.09.015 · Cited by 41
Basic Results on Lévy Processes
Sato, K. (2001) · Birkhäuser Boston eBooks · doi:10.1007/978-1-4612-0197-7_1 · Cited by 41
Asymptotic behaviour of stochastic flows of diffeomorphisms
Baxendale, P. H. (1986) · Lecture notes in mathematics · doi:10.1007/bfb0076869 · Cited by 40
Alpay, D., & Jørgensen, P. E. T. (2015) · Numerical Functional Analysis and Optimization · doi:10.1080/01630563.2015.1062777 · Cited by 39
On orthogonality of Riesz products
Brown, G., & Moran, W. (1974) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100048830 · Cited by 39
Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns
Lasanen, S. (2012) · Inverse Problems and Imaging · doi:10.3934/ipi.2012.6.267 · Cited by 38
The two-sided infinite extension of the Mallows model for random permutations
GNEDIN, A. V., & Olshanski, G. (2012) · Advances in Applied Mathematics · doi:10.1016/j.aam.2012.01.001 · Cited by 38
On a Bernoulli shift with non-identical factor measures
Hamachi, T. (1981) · Ergodic Theory and Dynamical Systems · doi:10.1017/s0143385700001255 · Cited by 38
A JKO Splitting Scheme for Kantorovich--Fisher--Rao Gradient Flows
Gallouët, T., & Monsaingeon, L. (2017) · SIAM Journal on Mathematical Analysis · doi:10.1137/16m106666x · Cited by 38
Ergotropy from quantum and classical correlations
Touil, A., Çakmak, B., & Deffner, S. (2021) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ac3eba · Cited by 37
Decreasing sequences of $\sigma$-fields and a measure change for Brownian motion
Dubins, L. E., Feldman, J., Smorodinsky, M., & Tsirelson, B. S. (1996) · The Annals of Probability · doi:10.1214/aop/1039639367 · Cited by 37
Les Martingales et leurs Applications Analytiques
Chatterji, S. D. (1973) · Lecture notes in mathematics · doi:10.1007/bfb0059708 · Cited by 36
Absolute continuity and singularity of Palm measures of the Ginibre point process
Osada, H., & Shirai, T. (2015) · Probability Theory and Related Fields · doi:10.1007/s00440-015-0644-6 · Cited by 35
Measures of similarity between distributions
Vegelius, J., Janson, S., & Johansson, F. (1986) · Quality & Quantity · doi:10.1007/bf00123091 · Cited by 35
INVARIANT MEASURES ON PRODUCT SPACES
Moore, C. C. (1967) · University of California Press eBooks · doi:10.1525/9780520325340-031 · Cited by 35
Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions
Gunaratnam, T. S., Oh, T., Tzvetkov, N., & Weber, H. (2022) · Probability and Mathematical Physics · doi:10.2140/pmp.2022.3.343 · Cited by 34
Equivalent-Model Augmentation for Variable-Structure Multiple-Model Estimation
Lan, J., & Li, X. R. (2013) · IEEE Transactions on Aerospace and Electronic Systems · doi:10.1109/taes.2013.6621840 · Cited by 33
On a class of KMS states for the unitary groupU(∞)
StrĂtilĂ, Ş., & Voiculescu, D. (1978) · Mathematische Annalen · doi:10.1007/bf01421594 · Cited by 33
Measures of non-compactness of operators in Banach lattices
de Pagter, B., & Schep, A. R. (1988) · Journal of Functional Analysis · doi:10.1016/0022-1236(88)90131-0 · Cited by 33
Transformations without finite invariant measure have finite strong generators
Krengel, U. (1970) · Lecture notes in mathematics · doi:10.1007/bfb0060652 · Cited by 32
Unitary equivalence of Fock representations on the Weyl algebra
Daele, A. V., & Verbeure, A. F. (1971) · Communications in Mathematical Physics · doi:10.1007/bf01646623 · Cited by 32
Random Coverings and Multiplicative Processes
Kahane, J. (2000) · Birkhäuser Basel eBooks · doi:10.1007/978-3-0348-8380-1_6 · Cited by 29
On non-singular transformations of a measure space. II
Krieger, W. (1969) · Probability Theory and Related Fields · doi:10.1007/bf00531812 · Cited by 29
Ergodic Theory : Non-singular Transformations
(2011) · SpringerReference · doi:10.1007/springerreference_60355 · Cited by 28
A Weighted Similarity Measure Between Z-Numbers and Bow-Tie Quantification
Das, S., Garg, A., Pal, S. K., & Maiti, J. (2019) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2019.2930935 · Cited by 27
Connections and metrics respecting purification of quantum states
Dittmann, J., & Uhlmann, A. (1999) · Journal of Mathematical Physics · doi:10.1063/1.532884 · Cited by 27
The Kakutani theorem for tensor products ofW∗-algebras
Promislow, D. (1971) · Pacific Journal of Mathematics · doi:10.2140/pjm.1971.36.507 · Cited by 27
Multivariate mixtures of Polya trees for modeling ROC data
Hanson, T., Branscum, A. J., & Gardner, I. A. (2008) · Statistical Modelling · doi:10.1177/1471082x0700800106 · Cited by 26
Uncertainty in prediction and in inference
Hilgevoord, J., & Uffink, J. (1991) · Foundations of Physics · doi:10.1007/bf01883638 · Cited by 26
Gaussian measures in function space
Shepp, L. A. (1966) · Pacific Journal of Mathematics · doi:10.2140/pjm.1966.17.167 · Cited by 26
Gaussian Distributions on Locally Compact Abelian Groups
Feldman, G. M. (1979) · Theory of Probability and Its Applications · doi:10.1137/1123062 · Cited by 25
ABSOLUTE CONTINUITY AND SINGULARITY OF LOCALLY ABSOLUTELY CONTINUOUS PROBABILITY DISTRIBUTIONS. II
Kabanov, J. M., Lipcer, R. Š., & Širjaev, A. N. (1980) · Mathematics of the USSR-Sbornik · doi:10.1070/sm1980v036n01abeh001760 · Cited by 25
Certain induced measures and the fractional dimensions of their ?supports?
Chatterji, S. D. (1964) · Probability Theory and Related Fields · doi:10.1007/bf00534907 · Cited by 25
On the infinite product construction of non-singular transformations of a measure space
Krieger, W. (1971) · Inventiones mathematicae · doi:10.1007/bf01418932 · Cited by 25
Monic representations of the Cuntz algebra and Markov measures
Dutkay, D. E., & Jørgensen, P. E. T. (2014) · Journal of Functional Analysis · doi:10.1016/j.jfa.2014.05.016 · Cited by 6
Iterated function systems, representations, and Hilbert spacePreprint
Jørgensen, P. E. T. (2012) · Cited by 24
Schiavo, L. D. (2022) · The Annals of Probability · doi:10.1214/21-aop1541 · Cited by 23
Metadata
- Authors
- Kakutani, S.
- Corresponding author
- Kakutani, Shizuo
- Venue
- The Annals of Mathematics
- Publisher
- JSTOR
- Published
- 1948-01
- Volume
- 49
- Issue
- 1
- Pages
- 214
- ISSN
- 0003-486X (ISSN-L)
- OpenAlex
- W2320259161
- Cited by
- 424
- References
- 0
- Citations by year
- 2026: 5 · 2025: 9 · 2024: 10 · 2023: 8 · 2022: 10 · 2021: 9 · 2020: 12 · 2019: 11 · 2018: 16 · 2017: 7 · 2016: 8 · 2015: 15 · 2014: 11 · 2013: 12 · 2012: 13
- FWCI
- 3.18
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Annals of Mathematics (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Physical Sciences › Mathematics › Mathematical Physics › p-Adic Analysis and Mathematical Physics
- Keywords
- measurement invariance, product probability spaces
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
