be any integer between $1$ and $2^n$, but physicists are especially interested in pure states and a pure state $ \rho $ can be defined by ${\rm rank}(\rho)=1$.

The objective of quantum tomography is to estimate $\rho$ on the basis of experimental observations of many independent and identically systems prepared in the state $\rho$ by the same experimental device.

For each particle (qubit), one can measure one of the three Pauli observables $\sigma_x, \, \sigma_y, \, \sigma_z$. The outcome for each will be $ 1 $, or $ -1 $, randomly (the corresponding probability depends on the state $\rho$ and will be given in  below). Thus for a $ n $-qubits system, we consider $3^n$ possible experimental observables. The set of all possible performed observables is $$\begin{aligned}
\{\sigma_{\mathbf{a}} = \sigma_{{a}_1} \otimes
\ldots \otimes \sigma_{{a}_n}; \,  \mathbf{a} =
(a_1,\ldots,a_n) \in \mathcal{E}^n := \{x,y,z\}^{n}\},\end{aligned}$$ where vector $\mathbf{a} $ identifies the experiment. The outcome for each fixed observable setting will be a random vector $ \mathbf{s} = (s_1, \ldots , s_n)   \in \mathcal{R}^n :=\{-1,1\}^{n}$, thus there are $ 2^n $ outcomes in total.

Let us denote $R^{\mathbf{a}}$ a $\mathcal{R}^n$-valued random vector that is the outcome of an experiment indexed by $\mathbf{a}$. From the 