Convergence conditions for the quantum relative entropy and other applications of the deneralized quantum Dini lemma

Abstract

We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows us to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual information of a quantum channel considered as a function of a pair (channel, input state). A simple convergence criterion for the von Neumann entropy is also obtained.

32 pages, any comments are welcome. arXiv admin note: text overlap with arXiv:2201.11477

References in corpus (1)

• On the construction of a quantum channel corresponding to non-commutative graph for a qubit interacting with quantum oscillator

Cited by in corpus (4)

• Continuity of the relative entropy of resource
• Quantum relative entropy: general convergence criterion and preservation of convergence under completely positive linear maps
• Lower semicontinuity of the relative entropy disturbance and its corollaries
• New monotonicity property of the quantum relative entropy