# On the geometry of mixed states and the Fisher information tensor

## Abstract

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

- Authors:

- Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801 (United States)
- Dipartimento di Fisica e Astronomia, Università di Bologna and INFN, V. Irnerio 46, 40127 Bologna (Italy)
- Institut für Mathematik, Winterthurerstrasse 190, 8057 Zürich (Switzerland)

- Publication Date:

- OSTI Identifier:
- 22596851

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GEOMETRY; MIXED STATES; ORBITS; QUANTUM INFORMATION; QUANTUM MECHANICS; SYMMETRY; TENSORS

### Citation Formats

```
Contreras, I., E-mail: icontrer@illinois.edu, Ercolessi, E., E-mail: ercolessi@bo.infn.it, and Schiavina, M., E-mail: michele.schiavina@math.uzh.ch.
```*On the geometry of mixed states and the Fisher information tensor*. United States: N. p., 2016.
Web. doi:10.1063/1.4954328.

```
Contreras, I., E-mail: icontrer@illinois.edu, Ercolessi, E., E-mail: ercolessi@bo.infn.it, & Schiavina, M., E-mail: michele.schiavina@math.uzh.ch.
```*On the geometry of mixed states and the Fisher information tensor*. United States. doi:10.1063/1.4954328.

```
Contreras, I., E-mail: icontrer@illinois.edu, Ercolessi, E., E-mail: ercolessi@bo.infn.it, and Schiavina, M., E-mail: michele.schiavina@math.uzh.ch. Wed .
"On the geometry of mixed states and the Fisher information tensor". United States.
doi:10.1063/1.4954328.
```

```
@article{osti_22596851,
```

title = {On the geometry of mixed states and the Fisher information tensor},

author = {Contreras, I., E-mail: icontrer@illinois.edu and Ercolessi, E., E-mail: ercolessi@bo.infn.it and Schiavina, M., E-mail: michele.schiavina@math.uzh.ch},

abstractNote = {In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.},

doi = {10.1063/1.4954328},

journal = {Journal of Mathematical Physics},

number = 6,

volume = 57,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}

DOI: 10.1063/1.4954328

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