# What is a maximal entangled multipartite state?

We know the four Bell states are the maximal entangled states for two-qubit states, and we know if a state cannot be written as the tensor product by its subsets, then it is a entangled state, so is there a definition of maximal entangled state?

– glS
Sep 22 '21 at 14:13
• Yes, some description about maximal entanglement for multipartite states. Sep 22 '21 at 14:15

A pure state is said to be maximally entangled if the Von Neumann Entropy $$S(\rho_{A})$$, where $$\rho_{A}=Tr_{B}(\rho_{AB})$$ is the maximum value. ie $$S(\rho_{A})=log(d)$$ where $$d$$ is the dimension of the subsystem $$A$$
Edit: Just gonna add in here that, in the case of multipartite states, the entropy of the marginals in any bipartition of said state should be log(d), where $$d$$ in this case is the minimum over the dimensions of the systems involved in said bipartition.
• you might want to specify that this applies to pure states. Otherwise, the bipartite state $I\otimes I/dd'$ (with $d,d'$ dimensions of two spaces) also has "maximal entropy".