ﻻ يوجد ملخص باللغة العربية
The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(rho||sigma) := {rm Tr}[rho(log rho - log sigma)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${mathcal M}$ be a finite dimensional von Neumann algebra and ${mathcal N}$ a von Neumann subalgebra if it. Let ${mathcal E}_tau$ be the tracial conditional expectation from ${mathcal M}$ onto ${mathcal N}$. For density matrices $rho$ and $sigma$ in ${mathcal N}$, let $rho_{mathcal N} := {mathcal E}_tau rho$ and $sigma_{mathcal N} := {mathcal E}_tau sigma$. Since ${mathcal E}_tau$ is CPTP, the DPI says that $S(rho||sigma) geq S(rho_{mathcal N}||sigma_{mathcal N})$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if $sigma = {mathcal R}_rho(sigma_{mathcal N} )$, where ${mathcal R}_rho$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator ${mathcal A}_rho$ from ${mathcal M} $ to ${mathcal N} $. In it simplest form, our bound is $$S(rho||sigma) - S(rho_{mathcal N} ||sigma_{mathcal N} ) geq left(frac{1}{8pi}right)^{4} |Delta_{sigma,rho}|^{-2} | {mathcal R}_{rho_{mathcal N}} -sigma|_1^4 $$ where $Delta_{sigma,rho}$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation $sigma = {mathcal R}_rho(sigma_{mathcal N} )$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.
In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard
Several works have shown that perturbation stable instances of the MAP inference problem in Potts models can be solved exactly using a natural linear programming (LP) relaxation. However, most of these works give few (or no) guarantees for the LP sol
We study the symmetrized noncommutative arithmetic geometric mean inequality introduced(AGM) by Recht and R{e} $$ |frac{(n-d)!}{n!}sumlimits_{{ j_1,...,j_d mbox{ different}} }A_{j_{1}}^*A_{j_{2}}^*...A_{j_{d}}^*A_{j_{d}}...A_{j_{2}}A_{j_{1}} | leq
In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $phi:mathcal A rightarrow B(H)$, where $mathcal A$ is a unital C*-algebra and H is a Hilbert space, if $n ge 3$. They also demonstrate
Two new proofs of the Fisher information inequality (FII) using data processing inequalities for mutual information and conditional variance are presented.