we can extend a given POVM by an appropriate number of zero operators $A_j = 0_d \in {\mathbb C}^{d \times d}$ where $0_d$ denotes the zero matrix of size $d$. This extension does not change the probability distribution of the POVM since $p_j = {\rm tr}(\rho 0_d) = 0$ for a zero operator $A_j=0_d$. In our example above we add the zero operator $A_4 = 0_2$ to the three POVM operators. We obtain a POVM that can be implemented on a register of two qubits.

The basic idea of Neumark’s theorem is to implement an orthogonal measurement $({\tilde A}_j)$ on the extended system with $n$ dimensions that corresponds to the POVM $(A_j)$ in the sense that it reproduces the correct probabilities $p_j$. We now consider the construction of the orthogonal measurement $({\tilde A_j})$. Let $\rho
\in {\mathbb C}^{d \times d}$ be the density matrix of the state to be measured. Then the state of the extended system with $n$ dimensions can be written as ${\tilde \rho} = \rho \oplus 0_{n-d} \in {\mathbb
C}^{n \times n}$. When we write the vectors ${|\Psi_j\rangle}$ as columns of the matrix $$M= \left( {|\Psi_1\rangle} \ldots {|\Psi_n\rangle} \right) \in
{\mathbb C}^{d \times n},$$ the operators ${\tilde 