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Article
Log-majorization and matrix norm inequalities with application to quantum information
We are concerned with log-majorization for matrices in connection with the multivariate Golden–Thompson trace inequality and the Karcher mean (i.e., a multivariate extension of the weighted geometric mean). We...
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Article
Quantum Rényi Divergences and the Strong Converse Exponent of State Discrimination in Operator Algebras
The sandwiched Rényi \(\alpha \) α -divergences of two finite-dimensional qu...
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Article
Pusz–Woronowicz Functional Calculus and Extended Operator Convex Perspectives
In this article, we first study, in the framework of operator theory, Pusz and Woronowicz’s functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-varia...
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Article
Nonhomogeneous Karcher equations with vector fields on positive definite matrices
We study a family of Riemannian gradient equations on the Cartan–Hadamard–Riemannian manifold \({\mathbb P}_N\) ...
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Chapter
Maximal f-Divergences
Let M be a von Neumann algebra with its standard form ( M , ℋ , ...
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Chapter
Introduction
The notion of quantum divergences has played a significant role in quantum information, which defines important quantum quantities to discriminate between states of a quantum system. A quantum system is mathem...
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Chapter
Reversibility and Quantum Divergences
Let M and N be von Neumann algebras, whose standard forms are ( M , ℋ ...
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Chapter
Preservation of Maximal f-Divergences
In this chapter we will characterize the preservation of ...
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Chapter
Rényi Divergences and Sandwiched Rényi Divergences
Let M be a general von Neumann algebra given in a standard form ( M , ℋ ...
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Chapter
Measured f-Divergences
Let f be a convex function on (0, ∞), not necessarily operator convex unless we specify that. We use the convention in (2.2). Let M be a general von Neumann algebra. A m...
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Chapter
Reversibility and Measurements
This chapter is concerned with the approximate reversibility (sufficiency) for a sequence of quantum operations α k : M k → M (or quantum channels with input M and outputs M k). Our main problem is to characteriz...
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Chapter
Standard f-Divergences
Let M be a general von Neumann algebra, and M ...
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Article
Operator means deformed by a fixed point method
By means of a fixed point method we discuss the deformation of two-variable and multivariate operator means of positive definite matrices/operators. It is shown that the deformation of any operator mean in the...
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Chapter
Matrix Limit Theorems of Kato Type Related to Positive Linear Maps and Operator Means
We obtain limit theorems for \(\Phi (A^p)^{1/p}\) Φ ( A p ) 1 / p and \((A^p\sigma B)^{1/p}\) ( A p σ B ) 1 / p as \(p\rightarrow \infty \) p → ∞ for positive matrices A, B, where
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Article
Generalized Log-Majorization and Multivariate Trace Inequalities
We show that recent multivariate generalizations of the Araki–Lieb–Thirring inequality and the Golden–Thompson inequality (Sutter et al. in Commun Math Phys, 2016. doi:10.1007/s00220-016-2778-5
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Article
Orbital Free Pressure and Its Legendre Transform
Orbital counterparts of the free pressure and its Legendre transform (or η- entropy) are introduced and studied in comparison with other entropy quantities in free probability theory and in relation to random mul...
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Book
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Chapter
Fundamentals of Operators and Matrices
A linear map** is essentially a matrix if the vector space is finite-dimensional. In this book the vector space is typically a finite-dimensional complex Hilbert space.
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Chapter
Functional Calculus and Derivation
Let \(A\in \mathbb {M}_n({\mathbb C})\) A ∈ ...
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Chapter
Matrix Means and Inequalities
The study of numerical means has been a popular subject for centuries, and the inequalities
