in distant regions. Let Alice’s physical system be described by the Hilbert space $\mathcal{H}_A$ and that of Bob by $\mathcal{H}_B$. Then the joint physical system of Alice and Bob is described by the tensor product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_{B}$.

**Def. 1** *Product and entangled pure states:*\
*A pure state, i.e. a projector $|\psi_{AB}\rangle \langle \psi_{AB}|$ on a vector $|\psi_{AB}\rangle \in {\cal H}_A \otimes {\cal H}_B $, is a product state if the states of local subsystems are also pure states, that is, if $|\psi_{AB}\rangle= |\psi_A\rangle \otimes |\psi_B\rangle$. However, there are states that cannot be written in this form. These states are called entangled states.*

An example of an entangled state is the well-known singlet state ${\ensuremath{|\Psi^-\rangle}}=(\left|01\right\rangle - \left|10\right\rangle)/\sqrt{2}$, where $|0\rangle$ and $|1\rangle$ are two orthonormal states. Operationally, product states correspond to those states, that can be locally prepared by Alice and Bob at two separate locations. Entangled states can, however, be prepared only after the particles of Alice and Bob have interacted either directly or by means of an ancillary system. The second option is necessary due to the existence of the phenomenon of entanglement swapping [@ZZHE]. A very useful representation, only valid for pure bipartite states, is the, 