## Electronic Journal of Statistics

### Estimation of low rank density matrices by Pauli measurements

Dong Xia

#### Abstract

Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in quantum state tomography (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten $p$-norms with $1\leq p\leq+\infty$ for low rank density matrices estimation by Pauli measurements. In our previous paper [14], these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST.

Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers [14] and [25], we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten $p$-norms with $1\leq p\leq2$ and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten $p$-norms with $1\leq p\leq+\infty$. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in [14] and [25]. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 50-77.

Dates
First available in Project Euclid: 5 January 2017

https://projecteuclid.org/euclid.ejs/1483585973

Digital Object Identifier
doi:10.1214/16-EJS1222

Mathematical Reviews number (MathSciNet)
MR3592698

Zentralblatt MATH identifier
1366.62156

#### Citation

Xia, Dong. Estimation of low rank density matrices by Pauli measurements. Electron. J. Statist. 11 (2017), no. 1, 50--77. doi:10.1214/16-EJS1222. https://projecteuclid.org/euclid.ejs/1483585973

#### References

• [1] P. Alquier, C. Butucea, M. Hebiri, K. Meziani, and T. Morimae. Rank-penalized estimation of a quantum system., Physical Review A, 88(3) :032113, 2013.
• [2] G. Aubrun. On almost randomizing channels with a short Kraus decomposition., Communications in Mathematical Physics, 288(3) :1103–1116, 2009.
• [3] T. Cai, D. Kim, Y. Wang, M. Yuan, and H. H. Zhou. Optimal large-scale quantum state tomography with Pauli measurements., The Annals of Statistics, 44(2):682–712, 2016.
• [4] E. J. Candés and Y. Plan. Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements., IEEE Transactions on Information Theory, 57(4) :2342–2359, 2011.
• [5] S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators., New Journal of Physics, 14(9) :095022, 2012.
• [6] D. Gross. Recovering low-rank matrices from few coefficients in any basis., IEEE Transactions on Information Theory, 57(3) :1548–1566, 2011.
• [7] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert. Quantum state tomography via compressed sensing., Physical Review Letters, 105(15) :150401, 2010.
• [8] O. Guédon, S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann. Majorizing measures and proportional subsets of bounded orthonormal systems., Revista Matemática Iberoamericana, 24(3) :1075–1095, 2008.
• [9] V. Koltchinskii. The Dantzig selector and sparsity oracle inequalities., Bernoulli, 15(3):799–828, 2009.
• [10] V. Koltchinskii., Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems: École d’Été de Probabilités de Saint-Flour XXXVIII -2008. Springer, 2011.
• [11] V. Koltchinskii. von Neumann entropy penalization and low-rank matrix estimation., The Annals of Statistics, 39(6) :2936–2973, 2011.
• [12] V. Koltchinskii. Sharp oracle inequalities in low rank estimation. In, Empirical Inference, pages 217–230. Springer, 2013.
• [13] V. Koltchinskii, K. Lounici, and A. B. Tsybakov. Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion., The Annals of Statistics, 39(5) :2302–2329, 2011.
• [14] V. Koltchinskii and D. Xia. Optimal estimation of low rank density matrices., Journal of Machine Learning Research, 16 :1757–1792, 2015.
• [15] Y.-K. Liu. Universal low-rank matrix recovery from Pauli measurements. In, Advances in Neural Information Processing Systems, pages 1638–1646, 2011.
• [16] S. Mendelson. Upper bounds on product and multiplier empirical processes., Stochastic Processes and Their Applications, 2016.
• [17] S. Negahban and M. J. Wainwright. Restricted strong convexity and weighted matrix completion: Optimal bounds with noise., Journal of Machine Learning Research, 13(May) :1665–1697, 2012.
• [18] M. Nielsen and I. Chuang., Quantum Computation and Quantum Information. Cambridge University Press, 2000.
• [19] A. Pajor. Metric entropy of the Grassmann manifold., Convex Geometric Analysis, 34:181–188, 1998.
• [20] A. Rohde and A. B. Tsybakov. Estimation of high-dimensional low-rank matrices., The Annals of Statistics, 39(2):887–930, 2011.
• [21] J. A. Tropp. User-friendly tail bounds for sums of random matrices., Foundations of Computational Mathematics, 12(4):389–434, 2012.
• [22] A. B. Tsybakov., Introduction to Nonparametric Estimation. Springer, 2008.
• [23] Y. Wang. Asymptotic equivalence of quantum state tomography and noisy matrix completion., The Annals of Statistics, 41(5) :2462–2504, 2013.
• [24] D. Xia. Optimal schatten-q and ky-fan-k norm rate of low rank matrix estimation., arXiv preprint arXiv:1403.6499, 2014.
• [25] D. Xia and V. Koltchinskii. Estimation of low rank density matrices: Bounds in schatten norms and other distances., Electron. J. Statist., 10(2) :2717–2745, 2016.