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Inequalities on partial traces of positive semidefinite block matrices

Part of: Basic linear algebra Special classes of linear operators

Published online by Cambridge University Press:  18 December 2020

Xiaohui Fu ,
Pan-Shun Lau  and
Tin-Yau Tam
Xiaohui Fu
Affiliation:
Department of Mathematics and Statistics, Hainan Normal University, Haikou, ChinaKey Laboratory of Data Science and Intelligence Education (Hainan Normal University), Ministry of Education, Haikou, China and Key Laboratory of Computational Science and Application of Hainan Province, Haikou, Chinae-mail:[email protected]
Pan-Shun Lau*
Affiliation:
Department of Mathematics & Statistics, University of Nevada, Reno, NV89557-0084, USAe-mail:[email protected]
Tin-Yau Tam
Affiliation:
Department of Mathematics & Statistics, University of Nevada, Reno, NV89557-0084, USAe-mail:[email protected]
*
e-mail:[email protected]
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Abstract

Inequalities on partial traces of positive semidefinite matrices are studied. Extensions of several existing inequalities on the determinant of partial traces are then obtained. Particularly, we improve a determinantal inequality given by Lin [Canad. Math. Bull. 59(2016)].

Keywords

Partial tracespositive semidefinitedeterminant

MSC classification

Primary: 15A42: Inequalities involving eigenvalues and eigenvectors 15A45: Miscellaneous inequalities involving matrices 47B65: Positive operators and order-bounded operators
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 964 - 969
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Copyright
© Canadian Mathematical Society 2020

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References

Ando, T. and Choi, M. D., Non-linear completely positive maps . In: Aspects of positivity in functional analysis, Elsevier Science Publishers B.V. (North Holland), Amsterdam, 1986.Google Scholar
Audenaert, K. M. R., Subadditivity of $q$ -entropies for $q>1$ . J. Math. Phys. 48(2007), no. 8, 083507.CrossRefGoogle Scholar
Bhatia, R., Partial traces and entropy inequalities . Linear Algebra Appl. 370(2003), 125–132.CrossRefGoogle Scholar
Bhatia, R., Positive definite matrices. Princeton University Press, Princeton, 2007.Google Scholar
Choi, D., Inequalities related to partial transpose and partial trace . Linear Algebra Appl. 516(2017), 1–7.CrossRefGoogle Scholar
Choi, D., Inequalities related to trace and determinant of positive semidefinite block matrices . Linear Algebra Appl. 523(2017), 1–7.Google Scholar
Choi, D., Inequalities about partial transpose and partial traces . Linear Multilinear Algebra 66(2018), 1619–1625.CrossRefGoogle Scholar
de Pillis, J., Transformations on partitioned matrices . Duke Math. J. 36(1969), 511–515.CrossRefGoogle Scholar
Fiedler, M. and Markham, T. L., On a theorem of Everitt, Thompson, and de Pillis . Math. Slovaca 44(1994), 441–444.Google Scholar
Lau, P. S. and Tam, T. Y., Weak log-majorization of unital trace-preserving completely positive maps . Electron. J. Linear Algebra 35(2019), 524–532.CrossRefGoogle Scholar
Lin, M., A determinantal inequality involving partial traces . Canad. Math. Bull. 59(2016), 585–591.CrossRefGoogle Scholar
Lin, M., A treatment of a determinant inequality of Fiedler and Markham . Czech. Math. J. 66(2016), 737–742.CrossRefGoogle Scholar
Nielsen, M. A. and Chuang, I. L., Quantum computation and quantum information. Cambridge University Press, Cambridge, 2000.Google Scholar
Petz, D., Quantum information theory and quantum statistics, Springer-Verlag, Berlin, 2008.Google Scholar
Thompson, R., A determinantal inequality for positive definite matrices . Canad. Math. Bull. 4(1961), 57–62.CrossRefGoogle Scholar
Umegaki, H., Conditional expectation in an operator algebra . Tohoku Math. J. 6(1954), 177–181.CrossRefGoogle Scholar
Zhang, F., Matrix Theory: Basic Results and Techniques. 2nd ed., Springer, New York, 2011.CrossRefGoogle Scholar