From Quasi-Entropy to Skew Information

Petz, Dénes; Szabó, Viktor

doi:10.1142/s0129167x09005832

Cited by 12 publications

(8 citation statements)

References 29 publications

(46 reference statements)

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“…The proof of the lemma is elementary. From the lemma and Theorem 6, Theorem 9 follows straightforwardly [40].…”

Section: Skew Informationmentioning

confidence: 93%

From ƒ-Divergence to Quantum Quasi-Entropies and Their Use

Petz

2010

Entropy

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Csiszár's f -divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting, positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy, which is related to some other important concepts as covariance, quadratic costs, Fisher information, Cramér-Rao inequality and uncertainty relation. It is remarkable that in the quantum case theoretically there are several Fisher information and variances. Fisher information are obtained as the Hessian of a quasi-entropy. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting. The von Neumann algebra approach is also discussed for uncertainty relation.

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“…The proof of the lemma is elementary. From the lemma and Theorem 6, Theorem 9 follows straightforwardly [40].…”

Section: Skew Informationmentioning

confidence: 93%

From ƒ-Divergence to Quantum Quasi-Entropies and Their Use

Petz

2010

Entropy

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“…Another proof is in [45] which contains the following theorem. The skew information coming from the minimal Fisher information and it is often denoted as I SLD (D, A).…”

Section: Extended Monotone Metricsmentioning

confidence: 99%

Introduction to Quantum Fisher Information

Petz

Ghinea²

2011

Quantum Probability and Related Topics

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The subject of this paper is a mathematical transition from the Fisher information of classical statistics to the matrix formalism of quantum theory. If the monotonicity is the main requirement, then there are several quantum versions parametrized by a function. In physical applications the minimal is the most popular. There is a one-to-one correspondence between Fisher informations (called also monotone metrics) and abstract covariances. The skew information and the χ 2 -divergence are treated here as particular cases.

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“…Similarly, f (x) = −x t satisfies (SSA) if 1 t 2. In some applications [9] the operator monotone functions…”

Section: Particular Examplesmentioning

confidence: 99%

“…In some applications [9] the operator monotone functions f p (x) = p(1 − p) (x − 1) 2 (x p − 1)(x 1−p − 1) (0 < p < 1) appear.…”

Section: S(α(a)) := −Tr α(A) Log α(A)mentioning

confidence: 99%