Rank-penalized estimation of a quantum system

We introduce a 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 of minimizing the risk of a linear estimator $\widehat{\rho }$ of $\rho$ penalized by a given rank (from 1 to $2^n$), where $\widehat{\rho }$ 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 the penalized estimator, evaluated with the Frobenius norm, which is of order $dn{(4/3)}^n/m$ and consistent for the estimator of the rank. The proposed methodology is computationally efficient and is illustrated with some example states and real experimental data sets.

Figure 1

Frequency of good selection of the true rank $d$, based on Eq. (10) with $\nu ={\nu }_n^{\left(1\right)}$ (green squares) and with $\nu ={\nu }_n^{\left(2\right)}$ (blue stars). The results are established for 20 repetitions. A value equal to 1 in the $y$ axis means that the method always selects the good rank, whereas 0 means that it always fails. Left, $m=50$ measurements; right, $m=100$ measurements.

Figure 2

Evaluation of the operator norm $\sqrt{{\nu }_n^{\left(1\right)}}=\parallel \widehat{\rho }-\rho \parallel$. The results are established for 20 repetitions. Top, $n=4$, $m=50$ repetitions of the measurements; we compare the errors when $d$ takes values between 1 and 6. Middle, $n=5$, $m=100$; we compare the errors when $d$ takes values between 1 and 6. Bottom, the rank equals $d=4$ and we compare the error for $m=50$ and 100.

Figure 3

Eigenvalues of the linear estimator in increasing order and the penalty choice; $m=100$ and $n=4$, 5, or 6, respectively.