Weighted entropy

Harold Falk

Yu. M. Suhov, Stefan Zohren

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 entropy, as well as an analog of the Araki-Lieb inequality. Interesting byproducts of the proofs are a weighted analog of Klein's inequality and non-negativity of quantum weighted relative entropy. A main difficulty is the fact that the weights in general do not commute with the density matrices.

Vladimir I. Man’ko, Zhanat Batyrbekuly Seilov

The notion of weighted quantum entropy is reviewed and considered for bipartite and noncomposite quantum systems. The known for the weighted entropy information inequality (subadditivity condition) is extended to the case of indivisible qudit system on example of qutrit. This new inequality for qutrit density matrix is discussed for different cases of weights and states. The role of weighted entropy is discussed in connection with studies of nonlinear quantum channels.

Huzihiro Araki, Élliott H. Lieb

Rodney W. Benoist, Jean-Paul Marchand, Walter Wyss

H. P. Heinig, M. Smith

In this paper a number of generalizations of the classical Heisenberg‐Weyl uncertainty inequality are given. We prove the n‐dimensional Hirschman entropy inequality (Theorem 2.1) from the optimal form of the Hausdorff‐Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted form of the Hausdorff‐Young theorem, a one‐dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg‐Weyl inequalities are given.