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On Equivalence of Infinite Product Measures
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“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]
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Bruin, H., & Hawkins, J. (1998) · Israel Journal of Mathematics · doi:10.1007/bf02783043 · Cited by 7
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Koopmans, L. H. (1969) · American Mathematical Monthly · doi:10.1080/00029890.1969.12000204 · Cited by 7
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Breutzmann, J. M., & Lutz, J. H. (1997) · Lecture notes in computer science · doi:10.1007/bfb0023487 · Cited by 7
Parreau, F. (1989) · Inventiones mathematicae · doi:10.1007/bf01388856 · Cited by 6
Merging of opinions in game-theoretic probability
Vovk, V. (2008) · Annals of the Institute of Statistical Mathematics · doi:10.1007/s10463-007-0165-x · Cited by 6
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Kosloff, Z. (2009) · Explore Bristol Research · doi:10.48550/arxiv.0912.1094 · Cited by 6
Hirai, T., & Shimomura, H. (1997) · Kyoto journal of mathematics · doi:10.1215/kjm/1250518335 · Cited by 6
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Kautz, S. M. (1997) · Lecture notes in computer science · doi:10.1007/3-540-63248-4_17 · Cited by 6
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Džamonja, M., & Plebanek, G. (2004) · Topology and its Applications · doi:10.1016/j.topol.2004.04.002 · Cited by 6
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Li, G., Oh, T., & Zheng, G. (2025) · Transactions of the London Mathematical Society · doi:10.1112/tlm3.70005 · Cited by 5
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Kosloff, Z., & Soo, T. (2021) · Studia Mathematica · doi:10.4064/sm201013-7-4 · Cited by 5
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Shale, D. (1973) · Journal of Functional Analysis · doi:10.1016/0022-1236(73)90083-9 · Cited by 5
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Koshmanenko, V. D. (2007) · Ukrainian Mathematical Journal · doi:10.1007/s11253-007-0054-8 · Cited by 5
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Singular measures and the key of G
Buckley, S. M., & MacManus, P. (2000) · Publicacions Matemàtiques · doi:10.5565/publmat_44200_07 · Cited by 5
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Ritter, G. (1979) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100055523 · Cited by 5
Some remarks on integral operators and equimeasurable sets
Walter Schachermayer, B. (1986) · Lecture notes in mathematics · doi:10.1007/bfb0076303 · Cited by 5
Representation formulae for the fractional Brownian motionPreprint
Picard, J. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0912.3168 · Cited by 5
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Alberti, P. M., & Peltri, G. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0008164 · Cited by 5
Sample size, parameter rates and contiguity the I.I.D. case
Akritas, M. G., PURI, M. L., & Roussas, G. G. (1979) · Communication in Statistics- Theory and Methods · doi:10.1080/03610927808827738 · Cited by 5
Unendliche Produkte unkorrelierter Funktionen auf kompakten, Abelschen Gruppen.
Ritter, G. (1978) · MATHEMATICA SCANDINAVICA · doi:10.7146/math.scand.a-11752 · Cited by 5
Approximation of Improper Prior Measures by Prior Probability Measures
Stein, C. (1965) · Springer eBooks · doi:10.1007/978-3-642-49750-6_13 · Cited by 5
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Danilenko, A. I., & Silva, C. E. (2023) · Encyclopedia of complexity and systems science series · doi:10.1007/978-1-0716-2388-6_183 · Cited by 5
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Oh, T., & Quastel, J. (2015) · Proceedings of the Edinburgh Mathematical Society · doi:10.1017/s0013091515000218 · Cited by 4
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Lev, N., Peled, R., & Peres, Y. (2015) · Proceedings of the London Mathematical Society · doi:10.1112/plms/pdu057 · Cited by 4
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uniform integrability of an additive martingale and its exponential
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Hu, T. (2012) · Cited by 4
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REMARKS ON A RADON-NIKODYM THEOREM FOR VECTOR MEASURES**Supported under the NSF Grant GP-30672.
Rao, M. M. (1973) · Elsevier eBooks · doi:10.1016/b978-0-12-702450-9.50033-6 · Cited by 4
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Brunk, H. D. (1955) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1955-0067398-5 · Cited by 3
On the cramer-frechet-rao inequality for translation parameter in the case of finite support
PURI, M. L., & Vlncze, I. (1985) · Statistics · doi:10.1080/02331888508801882 · Cited by 3
Linear Transformation of Quasi-Invariant Measures
Shimomura, H. (1976) · Publications of the Research Institute for Mathematical Sciences · doi:10.2977/prims/1195190378 · Cited by 3
4 DENSE DEFINITENESS AND BOUNDEDNESS OF COMPOSITION OPERATORS IN L2-SPACES VIA INDUCTIVE LIMITSPreprint
Artur, & Płaneta, A. (2016) · Cited by 3
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Devroye, L., & Gravel, C. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1502.02539 · Cited by 3
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Freedman, D. A., & Pitman, J. (1990) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1990-0990427-5 · Cited by 2
Prats’ovyta, I. M. (2009) · Ukrainian Mathematical Journal · doi:10.1007/s11253-009-0266-1 · Cited by 2
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
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