# α-z-Rényi relative entropies

## Abstract

We consider a two-parameter family of Rényi relative entropies D{sub α,z}(ρ ∥ σ) that are quantum generalisations of the classical Rényi divergence D{sub α}(p ∥ q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi’s axioms for a divergence. We consider the range of the parameters α, z for which the data-processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them.

- Authors:

- Department of Mathematics, Royal Holloway University of London, Egham TW20 0EX (United Kingdom)
- (Belgium)
- Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22403101

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DATA PROCESSING; ENERGY QUALITY; ENTROPY; QUANTUM MECHANICS

### Citation Formats

```
Audenaert, Koenraad M. R., Department of Physics and Astronomy, Ghent University, S9, Krijgslaan 281, B-9000 Ghent, and Datta, Nilanjana.
```*α-z-Rényi relative entropies*. United States: N. p., 2015.
Web. doi:10.1063/1.4906367.

```
Audenaert, Koenraad M. R., Department of Physics and Astronomy, Ghent University, S9, Krijgslaan 281, B-9000 Ghent, & Datta, Nilanjana.
```*α-z-Rényi relative entropies*. United States. doi:10.1063/1.4906367.

```
Audenaert, Koenraad M. R., Department of Physics and Astronomy, Ghent University, S9, Krijgslaan 281, B-9000 Ghent, and Datta, Nilanjana. Sun .
"α-z-Rényi relative entropies". United States.
doi:10.1063/1.4906367.
```

```
@article{osti_22403101,
```

title = {α-z-Rényi relative entropies},

author = {Audenaert, Koenraad M. R. and Department of Physics and Astronomy, Ghent University, S9, Krijgslaan 281, B-9000 Ghent and Datta, Nilanjana},

abstractNote = {We consider a two-parameter family of Rényi relative entropies D{sub α,z}(ρ ∥ σ) that are quantum generalisations of the classical Rényi divergence D{sub α}(p ∥ q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi’s axioms for a divergence. We consider the range of the parameters α, z for which the data-processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them.},

doi = {10.1063/1.4906367},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 56,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}

DOI: 10.1063/1.4906367

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