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 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).

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

“…= D(̺ σ n ), n ≥ 0, on S(H). The arguments from the proof of Proposition 2 in [9] show the validity of all the conditions of Corollary 1 in [9] for the sequence {f n } with S 0 = S(H). By using (7) we see that the homogeneous extension of the function f n to the cone T + (H) has the form…”

“…By the first condition in (23) we have σ n = P n m σ n + P n m σ n for all n ≥ 0 and m ≥ m 0 . So, by using identities ( 7), ( 8) and ( 13) we obtain Hence, by Corollary 1 in [9] (its applicability to the sequence {f n } is mentioned before) to prove (25) it suffices to show that…”

“…By the remark at the begin of the proof S(ρ n ) is finite for all n ≥ 0. So, Corollary 5B in [9] shows that lim m→+∞ sup n≥0 S( Pm ρ n Pm ) = 0.…”

“…Theorem 1 is used in Section 4 below to generalize the implication (39)⇒(38) to the case of countable sums of operators (under a certain condition). By applying Theorem 1 one can also obtain simple proofs of Propositions 2 and 3 in [9]. Example 3.…”

“…holds instead of (1). This question, which is important for applications, has been studied in the literature [3,4,7,8,9]. It seems that the first result in this direction is Lemma 4 in [7] that states, in particular, that the limit relation (2) holds for the sequences of states ρ n = P n ρ 0 P n , σ n = P n σ 0 P n ,…”