  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 {\it 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,
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, 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 $1\leq p\leq 2$ 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 previous papers. When the objective function of DE
is replaced by the negative von Neumann entropy, we obtain sharp convergence
rate in Kullback-Leibler divergence.
