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.

Theorem 1.

 

Theorem 2.

 

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:

4. The Atlas

Following the approach of Pistone and collaborators [1,3,4,24], we build an atlas of charts ${\xi }_{\rho }$, one for each strictly positive density matrix $\rho$. The compatibility of the different charts requires the study of the cross-over map ${\xi }_{{\rho }_1}(\sigma )\mapsto {\xi }_{{\rho }_2}(\sigma )$, where ${\rho }_1,{\rho }_2,\sigma$ are arbitrary strictly positive density matrices.

Simplify notations by writing ${\xi }_1$ and ${\xi }_2$ instead of ${\xi }_{{\rho }_1}$, respectively ${\xi }_{{\rho }_2}$. Similarly, write ${\Omega }_1$ and ${\Omega }_2$ instead of ${\Omega }_{{\rho }_1}$, respectively ${\Omega }_{{\rho }_2}$, and $F_1,F_2$ instead of $F_{{\rho }_1}$, respectively $F_{{\rho }_2}$.

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 ${\rho }_1$ and ${\rho }_2$. 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 ${\mathcal{A}}^{\prime }$ one has $X_2=Y{X}_1{Y}^{*}$.

Proof. 

Using the notations of the Appendix of [1], one has

$$\begin{array}{ccc}\hfill X_i& =& J_i({\rho }_i^{-1/2}\sigma {\rho }_i^{-1/2}\otimes \mathbb{I})J_i^{*},i=1,2.\hfill \end{array}$$

Note that the isometry J depends on the reference state with density matrix $\rho$. Therefore, it carries an index i. The above expression for $X_i$ implies that

$$\begin{array}{c}\hfill X_2=Y{X}_1{Y}^{*}\mathrm{with}Y=J_2({\rho }_2^{-1/2}{\rho }_1^{1/2}\otimes \mathbb{I})J_1^{*}.\end{array}$$

 ☐

Theorem 4.

The set $\mathbb{M}$ of faithful states on the algebra $\mathcal{A}$ of square matrices, together with the atlas of charts ${\xi }_{\rho }$, where ${\xi }_{\rho }$ is defined by Theorem 1, is a Banach manifold. For any pair of strictly positive density matrices ${\rho }_1$ and ${\rho }_2$, the cross-over map ${\xi }_2\circ {\xi }_1^{-1}$ is continuous.

Proof. 

The continuity of the map $X_1\mapsto X_2$ follows from the previous Proposition. The continuity of the maps $K_1\mapsto X_1$ and $X_2\mapsto K_2$ follows from the continuity of the exponential and logarithmic functions and the continuity of the function $\alpha$. ☐

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.