cost Distillable entanglement Entanglement of formation Concurrence Relative entropy of entanglement Squashed entanglement Logarithmic negativity Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (NP-hard) to compute. Quantum field theory The Reeh-Schlieder theorem of quantum field theory is sometimes seen as an analogue of quantum entanglement. Applications Entanglement has many applications in quantum information theory. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the best-known applications of entanglement are superdense coding and quantum teleportation. Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some). Entanglement is used in some protocols of quantum cryptography, but to prove the security of QKD under standard assumptions does not require entanglement. However, the device independent security of QKD is shown exploiting entanglement between the communication partners. Entangled states There are several canonical entangled states that appear often in theory and experiments. For two qubits, the Bell states are These four pure states are all maximally entangled (according to the entropy of entanglement) and form an orthonormal basis (linear algebra) of the Hilbert space of the two qubits. They play a fundamental role in Bell's theorem. For M>2 qubits, the GHZ state is which reduces to the Bell state for . The traditional GHZ state was defined for . GHZ states are occasionally extended to qudits, i.e., systems of d rather than 2 dimensions. Also for M>2 qubits, there are spin squeezed states, a class of squeezed coherent states satisfying certain restrictions on the uncertainty of spin measurements, which are necessarily entangled. Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement. For two bosonic modes, a NOON state is This is like the Bell state except the basis kets 0 and 1 have been replaced with "the N photons are in one mode" and "the N photons are in the other mode". Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam splitter. They are the sum of multiple of NOON states, and can be used to achieve the Heisenberg limit. For the appropriately chosen measures of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to