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

Abstract

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.

1. Corrections in Section 3

The display on top of page 5 should read

$$\begin{array}{ccc}\hfill ||f_{\rho ,K}||& =& \underset{A\in \mathcal{A}}{sup}\left\{f_{\rho ,K}(A):||A||\le 1\right\}\hfill \\ & =& \underset{A\in \mathcal{A}}{sup}\left\{(\pi (A)K{\Omega }_{\rho },{\Omega }_{\rho }):||A||\le 1\right\}\hfill \\ & =& {|||K|}^{1/2}{\Omega }_{\rho }{||}^2\hfill \\ & \le & {|||K|}^{1/2}{||}^2=||K||.\hfill \end{array}$$

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 ${\xi }_{\rho }$.” 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 ${\xi }_{\rho }$.”

The proof of the following Theorem is amended.

Theorem 3.

The inverse of the map ${\xi }_{\rho }:\mathbb{M}\mapsto {\mathcal{B}}_{\rho }$, defined in Theorem 2, is Fréchet-differentiable at $\omega ={\omega }_{\rho }$. The Fréchet derivative is denoted $F_{\rho }$. It maps K to $f_{\rho ,K}$, where the latter is defined by (10).

Proof. 

Let $K={\xi }_{\rho }({\omega }_{\sigma })$. One calculates

$$\begin{array}{ccc}\hfill ||{\omega }_{\sigma }-{\omega }_{\rho }-F_{\rho }K||& =& \underset{A\in \mathcal{A}}{sup}\left\{|{\omega }_{\sigma }(A)-{\omega }_{\rho }(A)-F_{\rho }K(A)|:||A||\le 1\right\}\hfill \\ & =& \underset{A\in \mathcal{A}}{sup}\left\{|(\pi (A){\Omega }_{\rho },[e^{K-\alpha (K)}-\mathbb{I}-K]{\Omega }_{\rho })|:||A||\le 1\right\}\hfill \\ & \le & ||e^{K-\alpha (K)}-\mathbb{I}-K||\hfill \\ & \le & |\alpha (K)|+\mathrm{o}(||K-\alpha (K)||).\hfill \end{array}$$

(11)

Note that

$$\begin{array}{c}\hfill |\alpha (K)|\le log||e^K||\le ||K||\end{array}$$

and

$$\begin{array}{c}\hfill ||K-\alpha (K)||\le 2||K||.\end{array}$$

In addition, if $||K||<1$ then one has

$$\begin{array}{c}\hfill {\alpha }_{\rho }(K)\le log(1+||K{\Omega }_{\rho }{||}^2)\le ||K{\Omega }_{\rho }{||}^2.\end{array}$$

This holds because $\lambda \le 1$ implies $exp(\lambda )\le 1+\lambda +{\lambda }^2$. One concludes that (11) converges to 0 faster than linearly as $||K||$ tends to 0. This proves that $F_{\rho }K$ is the Fréchet derivative of ${\xi }_{\rho }({\omega }_{\sigma })\mapsto {\omega }_{\sigma }$ at $\sigma =\rho$. ☐

2. 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:

3. Corrections in Section 9

In the proof of Proposition 4, the symbol ${\Omega }_{\rho }$ is missing five times in obvious places. It has been added.

4. 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.