# Criteria for equality in two entropic inequalities

## Abstract

We obtain a simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel. This shows that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion for the existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy. Bibliography: 20 titles. (paper)

- Authors:

- Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22364928

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Sbornik. Mathematics; Journal Volume: 205; Journal Issue: 7; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; BOSONS; DEPOLARIZATION; ENTROPY; KERNELS; MAPS; MIXED STATE; MIXED STATES; QUANTUM INFORMATION

### Citation Formats

```
Shirokov, M. E., E-mail: msh@mi.ras.ru.
```*Criteria for equality in two entropic inequalities*. United States: N. p., 2014.
Web. doi:10.1070/SM2014V205N07ABEH004409.

```
Shirokov, M. E., E-mail: msh@mi.ras.ru.
```*Criteria for equality in two entropic inequalities*. United States. doi:10.1070/SM2014V205N07ABEH004409.

```
Shirokov, M. E., E-mail: msh@mi.ras.ru. Thu .
"Criteria for equality in two entropic inequalities". United States.
doi:10.1070/SM2014V205N07ABEH004409.
```

```
@article{osti_22364928,
```

title = {Criteria for equality in two entropic inequalities},

author = {Shirokov, M. E., E-mail: msh@mi.ras.ru},

abstractNote = {We obtain a simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel. This shows that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion for the existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy. Bibliography: 20 titles. (paper)},

doi = {10.1070/SM2014V205N07ABEH004409},

journal = {Sbornik. Mathematics},

number = 7,

volume = 205,

place = {United States},

year = {Thu Jul 31 00:00:00 EDT 2014},

month = {Thu Jul 31 00:00:00 EDT 2014}

}