# Optimal full estimation of qubit mixed states

## Abstract

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.

- Authors:

- Grup de Fisica Teorica and IFAE, Facultat de Ciencies, Edifici Cn, Universitat Autonoma de Barcelona, 08193 Bellaterra (Barcelona) (Spain)
- Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht (Netherlands)
- (Netherlands)

- Publication Date:

- OSTI Identifier:
- 20786850

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.73.032301; (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; DISTRIBUTION; ENERGY LEVELS; MIXED STATE; PROBABILITY; QUANTUM COMPUTERS; QUBITS

### Citation Formats

```
Bagan, E., Monras, A., Munoz-Tapia, R., Ballester, M. A., Gill, R. D., and EURANDOM, P.O. Box 513, 5600 MB Eindhoven.
```*Optimal full estimation of qubit mixed states*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Bagan, E., Monras, A., Munoz-Tapia, R., Ballester, M. A., Gill, R. D., & EURANDOM, P.O. Box 513, 5600 MB Eindhoven.
```*Optimal full estimation of qubit mixed states*. United States. doi:10.1103/PHYSREVA.73.0.

```
Bagan, E., Monras, A., Munoz-Tapia, R., Ballester, M. A., Gill, R. D., and EURANDOM, P.O. Box 513, 5600 MB Eindhoven. Wed .
"Optimal full estimation of qubit mixed states". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20786850,
```

title = {Optimal full estimation of qubit mixed states},

author = {Bagan, E. and Monras, A. and Munoz-Tapia, R. and Ballester, M. A. and Gill, R. D. and EURANDOM, P.O. Box 513, 5600 MB Eindhoven},

abstractNote = {We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 3,

volume = 73,

place = {United States},

year = {Wed Mar 15 00:00:00 EST 2006},

month = {Wed Mar 15 00:00:00 EST 2006}

}