Correction: Naudts, J. Quantum Statistical Manifolds. Entropy 2018, 20, 472

Section 4 of “Naudts J. Quantum Statistical Manifolds. Entropy 2018, 20, 472” contains errors. They have limited consequences for the remainder of the paper. A new version of this Section is found here. Some smaller shortcomings of the paper are taken care of as well. In particular, the proof of Theorem 3 was not complete, and is therefore amended. Also, a few missing references are added.

The operator K is replaced by |K| because K need not be positive. The sentence "This is a prerequisite for proving in the next Theorem that this map is the Fréchet derivative of the chart ξ ρ ." should read "This is a prerequisite for proving in the next Theorem that this map is the Fréchet derivative of the inverse of the chart ξ ρ ." The proof of the following Theorem is amended.
The Fréchet derivative is denoted F ρ . It maps K to f ρ,K , where the latter is defined by (10).

New Version of Section 4
Propositions 1 and 2 of [1] are not correct. This only has consequences for one sentence in the Introduction of [1] and for the results reported in Section 4 of [1]. The text in the Introduction "Next, an atlas is introduced which contains a multitude of charts, one for each element of the manifold. Theorem 4 proves that the manifold is a Banach manifold and that the cross-over maps are linear operators." should be changed to "Next, an atlas is introduced which contains a multitude of charts, one for each element of the manifold. Theorem 4 proves that the manifold is a Banach manifold and that the cross-over maps are continuous." A new version of Section 4 follows below:

The Atlas
Following the approach of Pistone and collaborators [1,3,4,24], we build an atlas of charts ξ ρ , one for each strictly positive density matrix ρ. The compatibility of the different charts requires the study of the cross-over map ξ ρ 1 (σ) → ξ ρ 2 (σ), where ρ 1 , ρ 2 , σ are arbitrary strictly positive density matrices.

Proposition 1. RETRACTED
Continuity of the cross-over map follows from the continuity of the exponential and logarithmic functions and from the following result. Proposition 2. Fix strictly positive density matrices ρ 1 and ρ 2 . There exists a linear operator Y such that for any strictly positive density matrix σ and corresponding positive operators X 1 , X 2 in the commutant A one has Proof. Using the notations of the Appendix of [1], one has Note that the isometry J depends on the reference state with density matrix ρ. Therefore, it carries an index i. The above expression for X i implies that Theorem 4. The set M of faithful states on the algebra A of square matrices, together with the atlas of charts ξ ρ , where ξ ρ is defined by Theorem 1, is a Banach manifold. For any pair of strictly positive density matrices ρ 1 and ρ 2 , the cross-over map ξ 2 • ξ −1 1 is continuous.
Proof. The continuity of the map X 1 → X 2 follows from the previous Proposition. The continuity of the maps K 1 → X 1 and X 2 → K 2 follows from the continuity of the exponential and logarithmic functions and the continuity of the function α.

Corrections in Section 9
In the proof of Proposition 4, the symbol Ω ρ is missing five times in obvious places. It has been added.

Added References
In the overview of papers devoted to the study of the quantum statistical manifold in the finite-dimensional case, the references [2,3] should be added. A quantum version of the work of Pistone and Sempi [4], alternative to [5], is found in [6]. Reference [7] to the work of Ciaglia et al. has been updated.