Next Article in Journal
Some Consequences of the Thermodynamic Cost of System Identification
Next Article in Special Issue
Macroscopic Entropy of Non-Equilibrium Systems and Postulates of Extended Thermodynamics: Application to Transport Phenomena and Chemical Reactions in Nanoparticles
Previous Article in Journal
The Interaction Analysis between the Sympathetic and Parasympathetic Systems in CHF by Using Transfer Entropy Method
Previous Article in Special Issue
From Physics to Bioengineering: Microbial Cultivation Process Design and Feeding Rate Control Based on Relative Entropy Using Nuisance Time
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Correction

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

Departement Fysica, Universiteit Antwerpen, Universiteitsplein 1, 2610 Wilrijk Antwerpen, Belgium
Submission received: 6 September 2018 / Accepted: 12 October 2018 / Published: 17 October 2018
(This article belongs to the Special Issue Entropy: From Physics to Information Sciences and Geometry)

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
| | f ρ , K | | = sup A A f ρ , K ( A ) : | | A | | 1 = sup A A ( π ( A ) K Ω ρ , Ω ρ ) : | | A | | 1 = | | | K | 1 / 2 Ω ρ | | 2 | | | K | 1 / 2 | | 2 = | | K | | .
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.
Theorem 3.
The inverse of the map ξ ρ : M B ρ , defined in Theorem 2, is Fréchet-differentiable at ω = ω ρ . The Fréchet derivative is denoted F ρ . It maps K to f ρ , K , where the latter is defined by (10).
Proof. 
Let K = ξ ρ ( ω σ ) . One calculates
| | ω σ ω ρ F ρ K | | = sup A A | ω σ ( A ) ω ρ ( A ) F ρ K ( A ) | : | | A | | 1 = sup A A | ( π ( A ) Ω ρ , [ e K α ( K ) I K ] Ω ρ ) | : | | A | | 1 | | e K α ( K ) I K | | | α ( K ) | + o ( | | K α ( K ) | | ) .
Note that
| α ( K ) | log | | e K | | | | K | |
and
| | K α ( K ) | | 2 | | K | | .
In addition, if | | K | | < 1 then one has
α ρ ( K ) log ( 1 + | | K Ω ρ | | 2 ) | | K Ω ρ | | 2 .
This holds because λ 1 implies exp ( λ ) 1 + λ + λ 2 . One concludes that (11) converges to 0 faster than linearly as | | K | | tends to 0. This proves that F ρ K is the Fréchet derivative of ξ ρ ( ω σ ) ω σ at σ = ρ . ☐

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 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.
Simplify notations by writing ξ 1 and ξ 2 instead of ξ ρ 1 , respectively ξ ρ 2 . Similarly, write Ω 1 and Ω 2 instead of Ω ρ 1 , respectively Ω ρ 2 , and F 1 , F 2 instead of F ρ 1 , respectively F ρ 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 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 X 2 = Y X 1 Y * .
Proof. 
Using the notations of the Appendix of [1], one has
X i = J i ( ρ i 1 / 2 σ ρ i 1 / 2 I ) J i * , i = 1 , 2 .
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
X 2 = Y X 1 Y * with Y = J 2 ( ρ 2 1 / 2 ρ 1 1 / 2 I ) J 1 * .
 ☐
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 . ☐

3. 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.

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.

References

  1. Naudts, J. Quantum Statistical Manifolds. Entropy 2018, 20, 472. [Google Scholar] [CrossRef]
  2. Petz, D.; Sudar, C. Geometries of quantum states. J. Math. Phys. 1996, 37, 2662–2673. [Google Scholar] [CrossRef]
  3. Jenčová, A. Geometry of quantum states: Dual connections and divergence functions. Rep. Math. Phys. 2001, 47, 121–138. [Google Scholar] [CrossRef]
  4. Pistone, G.; Sempi, C. An infinite-dimensional structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 1995, 23, 1543–1561. [Google Scholar] [CrossRef]
  5. Streater, R.F. Quantum Orlicz spaces in information geometry. Open Syst. Inf. Dyn. 2004, 11, 359–375. [Google Scholar] [CrossRef]
  6. Jenčová, A. A construction of a nonparametric quantum information manifold. J. Funct. Anal. 2006, 239, 1–20. [Google Scholar] [CrossRef]

Share and Cite

MDPI and ACS Style

Naudts, J. Correction: Naudts, J. Quantum Statistical Manifolds. Entropy 2018, 20, 472. Entropy 2018, 20, 796. https://0-doi-org.brum.beds.ac.uk/10.3390/e20100796

AMA Style

Naudts J. Correction: Naudts, J. Quantum Statistical Manifolds. Entropy 2018, 20, 472. Entropy. 2018; 20(10):796. https://0-doi-org.brum.beds.ac.uk/10.3390/e20100796

Chicago/Turabian Style

Naudts, Jan. 2018. "Correction: Naudts, J. Quantum Statistical Manifolds. Entropy 2018, 20, 472" Entropy 20, no. 10: 796. https://0-doi-org.brum.beds.ac.uk/10.3390/e20100796

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop