# Number of representations providing noiseless subsystems

## Abstract

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function f{sub d}(n) which is the fraction of all d-dimensional quantum systems which preserve n bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.

- Authors:

- Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138 (United States)

- Publication Date:

- OSTI Identifier:
- 20786291

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevA.72.062328; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; ALGEBRA; DISTRIBUTION; ERRORS; HILBERT SPACE; LIE GROUPS; QUANTUM COMPUTERS; QUANTUM DECOHERENCE

### Citation Formats

```
Ritter, William Gordon.
```*Number of representations providing noiseless subsystems*. United States: N. p., 2005.
Web. doi:10.1103/PHYSREVA.72.0.

```
Ritter, William Gordon.
```*Number of representations providing noiseless subsystems*. United States. doi:10.1103/PHYSREVA.72.0.

```
Ritter, William Gordon. Thu .
"Number of representations providing noiseless subsystems". United States.
doi:10.1103/PHYSREVA.72.0.
```

```
@article{osti_20786291,
```

title = {Number of representations providing noiseless subsystems},

author = {Ritter, William Gordon},

abstractNote = {This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function f{sub d}(n) which is the fraction of all d-dimensional quantum systems which preserve n bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.},

doi = {10.1103/PHYSREVA.72.0},

journal = {Physical Review. A},

number = 6,

volume = 72,

place = {United States},

year = {Thu Dec 15 00:00:00 EST 2005},

month = {Thu Dec 15 00:00:00 EST 2005}

}