Broken symmetries in the entanglement of formation

We compare some recent computations of the entanglement of formation in quantum information theory and of the entropy of a subalgebra in quantum ergodic theory. Both notions require optimization over decompositions of quantum states. We show that both functionals are strongly related for some highly symmetric density matrices. Indeed, for certain interesting regions the entanglement of formation can be expressed by the entropy of a commuting subalgebra, and the corresponding optimal decompositions can be obtained one from the other. We discuss the presence of broken symmetries in relation with the structure of the optimal decompositions.

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