Erratum: "The quantum ♮α-Rényi divergence of real order" [J. Math. Phys. 63, 072203 (2022)].

Since K(h, )-1 (X) <= (X ) for m <= X <= M and h = M/m by Ref. 1, Lemma 4.3, it follows thatTr[ ( )]=Tr[ ( )1/2 (( -1/2 -1/2) ) ( )1/2]>=K(h2, )-1Tr[ ( )1/2 ( -1/2 -1/2) ( )1/2]=K(h2, )-1Tr[ ( ) ( )],where (X) = ( )-1/2 ( 1/2X 1/2) ( )-1/2, and hence, we have the corrected Theorem 7.4.Furthermore, since Ref. 3, Lemma 8.1 is derived from Ref. 3, Lemma 5.2, it is not valid in general. By the corrected Lemma 5.1, since I t i SP I i sp is operator convex for I i [-1, 0) [1, 2], the condition for inequality (5.2) in Ref. [3], Lemma 5.2 to hold is modified to I i [-1, 0] [1, 2]. (5.3) Therefore, it follows from the corrected Lemma 5.2 that Ref. 3, Theorem 5.3 (data processing inequality for -Rényi divergence) holds for the following restricted conditions. [Extracted from the article]