# Rank penalized estimation of a quantum system

Abstract : We introduce a new method to reconstruct the density matrix $\rho$ of a system of $n$-qubits and estimate its rank $d$ from data obtained by quantum state tomography measurements repeated $m$ times. The procedure consists in minimizing the risk of a linear estimator $\hat{\rho}$ of $\rho$ penalized by given rank (from 1 to $2^n$), where $\hat{\rho}$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order $dn(4/3)^n /m$ and consistency for the estimator of the rank. The proposed methodology is computationaly efficient and is illustrated with some example states and real experimental data sets.
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Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-00705755
Contributor : Pierre Alquier <>
Submitted on : Wednesday, September 25, 2013 - 11:38:27 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM
Document(s) archivé(s) le : Friday, April 7, 2017 - 2:43:22 AM

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### Identifiers

• HAL Id : hal-00705755, version 3
• ARXIV : 1206.1711

### Citation

Pierre Alquier, Cristina Butucea, Mohamed Hebiri, Katia Meziani, Tomoyuki Morimae. Rank penalized estimation of a quantum system. 2013. ⟨hal-00705755v3⟩

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