Trace inequality (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Trace inequality" in English language version.

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17doi.org

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14worldcat.org

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9semanticscholar.org

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9harvard.edu

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4arxiv.org

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3nih.gov

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2projecteuclid.org

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1numdam.org

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1archive.org

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archive.org

Marshall, Albert W.; Olkin, Ingram; Arnold, Barry (2011). Inequalities: Theory of Majorization and Its Applications (2nd ed.). New York: Springer. p. 340-341. ISBN 978-0-387-68276-1.

arxiv.org

Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
Hansen, Frank; Pedersen, Gert K. (2003-06-09). "Jensen's Operator Inequality". Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.

doi.org

E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 doi:10.1090/conm/529/10428
Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift (in German). 38 (1). Springer Science and Business Media LLC: 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
Golden, Sidney (1965-02-22). "Lower Bounds for the Helmholtz Function". Physical Review. 137 (4B). American Physical Society (APS): B1127 – B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
Thompson, Colin J. (1965). "Inequality with Applications in Statistical Mechanics". Journal of Mathematical Physics. 6 (11). AIP Publishing: 1812–1813. Bibcode:1965JMP.....6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-x. ISSN 0001-8708.
Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. 16. Canadian Mathematical Society: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X. S2CID 124032721.
Ando, T. (1979). "Concavity of certain maps on positive definite matrices and applications to Hadamard products". Linear Algebra and Its Applications. 26. Elsevier BV: 203–241. doi:10.1016/0024-3795(79)90179-4. ISSN 0024-3795.
Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
Hansen, Frank; Pedersen, Gert K. (2003-06-09). "Jensen's Operator Inequality". Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
Araki, Huzihiro (1990). "On an inequality of Lieb and Thirring". Letters in Mathematical Physics. 19 (2). Springer Science and Business Media LLC: 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). "Perspectives of matrix convex functions". Proceedings of the National Academy of Sciences. 108 (18). Proceedings of the National Academy of Sciences USA: 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.
Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331. S2CID 122252038.
Carlsson, Marcus (2021). "von Neumann's trace inequality for Hilbert-Schmidt operators". Expositiones Mathematicae. 39 (1): 149–157. doi:10.1016/j.exmath.2020.05.001.

harvard.edu

ui.adsabs.harvard.edu

Golden, Sidney (1965-02-22). "Lower Bounds for the Helmholtz Function". Physical Review. 137 (4B). American Physical Society (APS): B1127 – B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
Thompson, Colin J. (1965). "Inequality with Applications in Statistical Mechanics". Journal of Mathematical Physics. 6 (11). AIP Publishing: 1812–1813. Bibcode:1965JMP.....6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
Araki, Huzihiro (1990). "On an inequality of Lieb and Thirring". Letters in Mathematical Physics. 19 (2). Springer Science and Business Media LLC: 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). "Perspectives of matrix convex functions". Proceedings of the National Academy of Sciences. 108 (18). Proceedings of the National Academy of Sciences USA: 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.

nih.gov

ncbi.nlm.nih.gov

Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). "Perspectives of matrix convex functions". Proceedings of the National Academy of Sciences. 108 (18). Proceedings of the National Academy of Sciences USA: 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.

pubmed.ncbi.nlm.nih.gov

Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.

numdam.org

Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-x. ISSN 0001-8708.

projecteuclid.org

Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.

semanticscholar.org

api.semanticscholar.org

Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift (in German). 38 (1). Springer Science and Business Media LLC: 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. 16. Canadian Mathematical Society: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X. S2CID 124032721.
Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
Hansen, Frank; Pedersen, Gert K. (2003-06-09). "Jensen's Operator Inequality". Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
Araki, Huzihiro (1990). "On an inequality of Lieb and Thirring". Letters in Mathematical Physics. 19 (2). Springer Science and Business Media LLC: 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331. S2CID 122252038.

worldcat.org

search.worldcat.org

Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift (in German). 38 (1). Springer Science and Business Media LLC: 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
Golden, Sidney (1965-02-22). "Lower Bounds for the Helmholtz Function". Physical Review. 137 (4B). American Physical Society (APS): B1127 – B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
Thompson, Colin J. (1965). "Inequality with Applications in Statistical Mechanics". Journal of Mathematical Physics. 6 (11). AIP Publishing: 1812–1813. Bibcode:1965JMP.....6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-x. ISSN 0001-8708.
Wigner, Eugene P.; Yanase, Mutsuo M. (1964). "On the Positive Semidefinite Nature of a Certain Matrix Expression". Canadian Journal of Mathematics. 16. Canadian Mathematical Society: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X. S2CID 124032721.
Ando, T. (1979). "Concavity of certain maps on positive definite matrices and applications to Hadamard products". Linear Algebra and Its Applications. 26. Elsevier BV: 203–241. doi:10.1016/0024-3795(79)90179-4. ISSN 0024-3795.
Epstein, H. (1973). "Remarks on two theorems of E. Lieb". Communications in Mathematical Physics. 31 (4). Springer Science and Business Media LLC: 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
Ruskai, Mary Beth (2002). "Inequalities for quantum entropy: A review with conditions for equality". Journal of Mathematical Physics. 43 (9). AIP Publishing: 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP....43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier BV: 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP...60....1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
Lindblad, Göran (1974). "Expectations and entropy inequalities for finite quantum systems". Communications in Mathematical Physics. 39 (2). Springer Science and Business Media LLC: 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
Hansen, Frank; Pedersen, Gert K. (2003-06-09). "Jensen's Operator Inequality". Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
Araki, Huzihiro (1990). "On an inequality of Lieb and Thirring". Letters in Mathematical Physics. 19 (2). Springer Science and Business Media LLC: 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
Effros, E. G. (2009-01-21). "A matrix convexity approach to some celebrated quantum inequalities". Proceedings of the National Academy of Sciences USA. 106 (4). Proceedings of the National Academy of Sciences: 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). "Perspectives of matrix convex functions". Proceedings of the National Academy of Sciences. 108 (18). Proceedings of the National Academy of Sciences USA: 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.