# Subnormalized states and trace-nonincreasing maps

## Abstract

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M{sub N} of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (HS) (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M{sub N} induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in M{sub N{sup 2}}.

- Authors:

- Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnikow 32/44, 02-668 Warsaw (Poland) and 'Mark Kac' Complex Systems Research Centre, Uniwersytet Jagiello ski, ul. Reymonta 4, 30-059 Cracow (Poland)
- (Germany)
- (Poland) and Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiellonski, ul. Reymonta 4, 30-059 Cracow (Poland)

- Publication Date:

- OSTI Identifier:
- 20929703

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2738359; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; EUCLIDEAN SPACE; HILBERT SPACE; MAPS; MATRICES; MEASURE THEORY; QUANTUM MECHANICS

### Citation Formats

```
Cappellini, Valerio, Sommers, Hans-Juergen, Zyczkowski, Karol, Fachbereich Physik, Universitaet Duisburg-Essen, Campus Duisburg, 47048 Duisburg, and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnikow 32/44, 02-668 Warsaw.
```*Subnormalized states and trace-nonincreasing maps*. United States: N. p., 2007.
Web. doi:10.1063/1.2738359.

```
Cappellini, Valerio, Sommers, Hans-Juergen, Zyczkowski, Karol, Fachbereich Physik, Universitaet Duisburg-Essen, Campus Duisburg, 47048 Duisburg, & Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnikow 32/44, 02-668 Warsaw.
```*Subnormalized states and trace-nonincreasing maps*. United States. doi:10.1063/1.2738359.

```
Cappellini, Valerio, Sommers, Hans-Juergen, Zyczkowski, Karol, Fachbereich Physik, Universitaet Duisburg-Essen, Campus Duisburg, 47048 Duisburg, and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnikow 32/44, 02-668 Warsaw. Tue .
"Subnormalized states and trace-nonincreasing maps". United States.
doi:10.1063/1.2738359.
```

```
@article{osti_20929703,
```

title = {Subnormalized states and trace-nonincreasing maps},

author = {Cappellini, Valerio and Sommers, Hans-Juergen and Zyczkowski, Karol and Fachbereich Physik, Universitaet Duisburg-Essen, Campus Duisburg, 47048 Duisburg and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnikow 32/44, 02-668 Warsaw},

abstractNote = {We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M{sub N} of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (HS) (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M{sub N} induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in M{sub N{sup 2}}.},

doi = {10.1063/1.2738359},

journal = {Journal of Mathematical Physics},

number = 5,

volume = 48,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}

DOI: 10.1063/1.2738359

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