Related papers: Multivariate Trace Inequalities
We consider both known and not previously studied trace functions with applications in quantum physics. By using perspectives we obtain convexity statements for different notions of residual entropy, including the entropy gain of a quantum…
This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen…
In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic…
Uncertainty relations provide constraints on how well the outcomes of incompatible measurements can be predicted, and, as well as being fundamental to our understanding of quantum theory, they have practical applications such as for…
We derive the monotonicity of the quantum relative entropy by an elementary operational argument based on Stein's lemma in quantum hypothesis testing. For the latter we present an elementary and short proof that requires the law of large…
We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2}}^s$, where $\Phi$ and $\Psi$ are positive linear maps. By the same method combined…
We introduce quantum weighted entropy in analogy to an earlier notion of (classical) weighted entropy and derive many of its properties. These include the subadditivity, concavity and strong subadditivity property of quantum weighted…
The Golden-Thompson trace inequality which states that $Tr\, e^{H+K} \leq Tr\, e^H e^K$ has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here…
In this paper, we prove a trace inequality $\text{Tr}[ f(A) A^s B^s ] \leq \text{Tr}[ f(A) (A^{1/2} B A^{1/2} )^s ]$ for any positive and monotone increasing function $f$, $s\in[0,1]$, and positive semi-definite matrices $A$ and $B$. On the…
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions…
Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the…
We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices $A$ and $B$, by giving a lower bound on the quantity $\trace[A^r B^r A^r]^q$ in terms of $\trace[ABA]^{rq}$ for $0\le r\le 1$ and…
We introduce the von Neumann entropy regularization of Unbalanced Non-commutative Optimal Transport, specifically Non-commutative Optimal Transport between semi-definite positive matrices (not necessarily with trace one). We prove the…
We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this…
We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to M. Fannes. In our inequality, equality can be attained for every prescribed value of…
We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations.…
Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace…
In this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities. To a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in…
Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pe{\v} cari\'c and Mi\'ci\'c. Applying…
This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for…