Absolutely maximally entangled state

The absolutely maximally entangled (AME) state is a concept in quantum information science, which has many applications in quantum error-correcting code, discrete AdS/CFT correspondence, AdS/CMT correspondence, and more. It is the multipartite generalization of the bipartite maximally entangled state.

Definition
The bipartite maximally entangled state $$|\psi\rangle_{AB}$$ is the one for which the reduced density operators are maximally mixed, i.e., $$\rho_A=\rho_B=I/d$$. Typical examples are Bell states.

A multipartite state $$|\psi\rangle $$ of a system $$S$$ is called absolutely maximally entangled if for any bipartition $$A|B$$ of $$S$$, the reduced density operator is maximally mixed $$\rho_A=\rho_B=I/d$$, where $$d=\min\{d_A,d_B\}$$.

Property
The AME state does not always exist; in some given local dimension and number of parties, there is no AME state. There is a list of AME states in low dimensions created by Huber and Wyderka.

The existence of the AME state can be transformed into the existence of the solution for a specific quantum marginal problem.

The AME can also be used to build a kind of quantum error-correcting code called holographic error-correcting code.