# The smooth entropy formalism for von Neumann algebras

## Abstract

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.

- Authors:

- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125 (United States)
- Department of Physics, Graduate School of Science, University of Tokyo, Tokyo, Japan and Institute for Theoretical Physics, Leibniz University Hanover, Hanover (Germany)
- Institute for Theoretical Physics, ETH Zurich, Zurich (Switzerland)

- Publication Date:

- OSTI Identifier:
- 22479617

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 57; Journal Issue: 1; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; AMPLIFICATION; ENTROPY; QUANTUM CRYPTOGRAPHY; QUANTUM INFORMATION; QUANTUM SYSTEMS

### Citation Formats

```
Berta, Mario, E-mail: berta@caltech.edu, Furrer, Fabian, E-mail: furrer@eve.phys.s.u-tokyo.ac.jp, and Scholz, Volkher B., E-mail: scholz@phys.ethz.ch.
```*The smooth entropy formalism for von Neumann algebras*. United States: N. p., 2016.
Web. doi:10.1063/1.4936405.

```
Berta, Mario, E-mail: berta@caltech.edu, Furrer, Fabian, E-mail: furrer@eve.phys.s.u-tokyo.ac.jp, & Scholz, Volkher B., E-mail: scholz@phys.ethz.ch.
```*The smooth entropy formalism for von Neumann algebras*. United States. doi:10.1063/1.4936405.

```
Berta, Mario, E-mail: berta@caltech.edu, Furrer, Fabian, E-mail: furrer@eve.phys.s.u-tokyo.ac.jp, and Scholz, Volkher B., E-mail: scholz@phys.ethz.ch. Fri .
"The smooth entropy formalism for von Neumann algebras". United States. doi:10.1063/1.4936405.
```

```
@article{osti_22479617,
```

title = {The smooth entropy formalism for von Neumann algebras},

author = {Berta, Mario, E-mail: berta@caltech.edu and Furrer, Fabian, E-mail: furrer@eve.phys.s.u-tokyo.ac.jp and Scholz, Volkher B., E-mail: scholz@phys.ethz.ch},

abstractNote = {We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.},

doi = {10.1063/1.4936405},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 1,

volume = 57,

place = {United States},

year = {2016},

month = {1}

}

DOI: 10.1063/1.4936405

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