For mixed quantum states, entanglement can be characterized by a quotient group of local unitary stabilizers of the density matrix, and any nontriviality for separable states must come from multipartite entanglement with the purifying system.
Multiparticle singlet states cannot be maximally entangled for the bipartitions
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One way to explore multiparticle entanglement is to ask for maximal entanglement with respect to different bipartitions, leading to the notion of absolutely maximally entangled states or perfect tensors. A different path uses unitary invariance and symmetries, resulting in the concept of multiparticle singlet states. We show that these two concepts are incompatible in the sense that the space of pure multiparticle singlet states does not contain any state for which all partitions of two particles versus the rest are maximally entangled. This puts restrictions on the construction of quantum codes and contributes to discussions in the context of the AdS/CFT correspondence and quantum gravity.
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Entanglement groups for mixed states
For mixed quantum states, entanglement can be characterized by a quotient group of local unitary stabilizers of the density matrix, and any nontriviality for separable states must come from multipartite entanglement with the purifying system.