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,^[1] discrete AdS/CFT correspondence,^[2] AdS/CMT correspondence,^[2] and more. It is the multipartite generalization of the bipartite maximally entangled state.

Definition

The bipartite maximally entangled state ${\displaystyle |\psi \rangle _{AB}}$ is the one for which the reduced density operators are maximally mixed, i.e., ${\displaystyle \rho _{A}=\rho _{B}=I/d}$. Typical examples are Bell states.

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

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.^[3][4]

The existence of the AME state can be transformed into the existence of the solution for a specific quantum marginal problem.^[5]

The AME can also be used to build a kind of quantum error-correcting code called holographic error-correcting code.^[2][6][7]