Continuity of characteristics of composite quantum systems: a review

Abstract

General methods of quantitative and qualitative continuity analysis of characteristics of composite quantum systems are described. Several modifications of the Alicki-Fannes-Winter method are considered, which make it applicable to a wide class of characteristics in both finite-dimensional and infinite-dimensional cases. A new approximation method for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This method allows us to prove several general results (Simon-type dominated convergence theorem, the theorem about preserving continuity under convex mixtures, etc.).Uniform continuity bounds and local continuity conditions for basic characteristics of composite quantum systems are presented. Along with the results obtained earlier by different authors, a number of new results proved by the proposed methods are described.

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

“…• the constrained Holevo capacity and the mutual information of a quantum channel (as a function of input state) [2,6], [1,Section 5.5].…”

Quantum relative entropy: general convergence criterion and preservation of convergence under completely positive linear maps

“…Since D(ρ n σ n ) < +∞, we have suppρ n ⊆ suppσ n . So, the second and third conditions in (22) imply, by Lemma 4 in [7]…”

“…Since P n m ρ n P n m tends to P 0 m ρ 0 P 0 m as n → +∞ and sup n≥0 rankP n m ρ n P n m < +∞ by the conditions in (22)…”

“…Let m 0 be the multiplicity of λ σ 0 1 . For each m ≥ m 0 let P n m be the spectral projector of σ n corresponding to its maximal m eigenvalues 8 , where m is maximal number in M * σ 0 not exceeding m. By using the arguments presented at the end of Section 4.2.1 in [22] (based on Theorem VIII.23 in [13] and the Mirsky inequality) it is easy to show that…”

“…Then P r 0 is the projector onto the support of σ 0 . By using the arguments presented at the end of Section 4.2.1 in [22] (based on Theorem VIII.23 in [13] and the Mirsky inequality) it is easy to show that…”