Multipartite entanglement: combinatorics, topology, and … astronomy.

A brief introduction to entanglement of multipartite pure quantum states will be given. As the Bell states are known to be maximally entangled among all two-qubit quantum states, a natural question arises: What is the most entangled state for the quantum system consisting of N sub-systems with d levels each? The answer depends on the entanglement measure selected, but already for four-qubit system there is no state, which displays maximal entanglement with respect to all three possible splittings of the systems into two pairs of qubits.
To construct strongly entangled multipartite quantum states one can use various mathematical techniques involving combinatorial designs, topological methods related to knot theory or the Majorana (stellar) representation of permutation symmetric quantum states. Absolutely maximally entangled (AME) states of 2n subsystems, being maximally entangled with respect to all possible symmetric splitting of the system, find their applications for information processing tasks. For instance, the standard |GHZ_4^3> state of four qutrits allows one to teleport a single qutrit between any two parties, while the ‘more entangled’ AME state of four qutrits enables us to teleport two qutrits from any selected pair of users to the remaining two parties.