basic principles of quantum mechanics (Born’s rule), its probability distribution is given by $$\label{prob.fomala}
\forall \mathbf{s} \in \mathcal{R}^n,
p_{\mathbf{a},\mathbf{s}} := \mathbb{P}
 (R^\mathbf{a}= \mathbf{s})
  = {\rm Trace} \left(\rho \cdot
 P_{\mathbf{s}}^{\mathbf{a}} \right),$$ where $ P_{\mathbf{s}}^{\mathbf{a}} :=
P_{s_1}^{a_{1}}\otimes \dots \otimes P_{s_n}^{a_n}$ and $P_{s_i}^{a_i}$ is the orthogonal projection associated to the eigenvalue $s_i$ in the diagonalization of $ \sigma_{a_i} $ for $a_i\in \{x,y,z\}$ and $ s_i\in \{-1,1\} $ – that is $ \sigma_{a_i} = -1P^{a_i}_{-1}+1P^{a_i}_{+1}$.

The quantum state tomography problem is as follows: a physicist has access to an experimental device that produces $n$-qubits in a state $\rho^0$, and $\rho^0$ is assumed to be unknown. He/she can produce a large number of replications of the $n$-qubits and wants to infer $\rho^0$ from this.

In the complete measurement case, for [*each*]{} experiment setting $ \mathbf{a}\in\mathcal{E}^n$, the experimenter repeats $ m $ times the experiment corresponding to $\mathbf{a}$ and thus collects $m$ independent random copies of $R^\mathbf{a}$, say $R^\mathbf{a}_1,\dots,R^\mathbf{a}_m$. As there are $3^n$ possible experiment settings $ \mathbf{a}$, we define the **quantum sample** size as $ N:=m\cdot 3^n $. We will refer to $(R^\mathbf{a}_i)_{i\in\{1,\dots,m\},\mathbf{a}\in\mathcal{E}^n}$ as $\mathcal{D}$ (for data).

Note that the case where we would only have access to experiments $ \mathbf{a}\in\mathcal{A}$ where $\mathcal{A}$ is some 