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Permutation symmetries of multipartite quantum states are defined only whenthe constituent subsystems are of equal dimensions. In this work we extend thisnotion of permutation symmetry to heterogeneous systems, that is, systemscomposed of subsystems having unequal dimensions. Given a tensor product spaceof $k$ subsystems (of arbitrary dimensions) and a permutation operation$\sigma$ over $k$ symbols, these states are such that they have identicaldecompositions (up to an overall phase) in the given tensor product space andthe tensor product space obtained by the permuting the subsystems by $\sigma$.Towards this, we construct a matrix whose action is to simultaneously permutethe subsystem label and subsystem dimension of a given state according topermutation $\sigma$. Eigenvectors of this matrix have the required symmetry.We then examine entanglement of states in the igenspaces of these matrices. Itis found that all nonsymmetric eigenspaces of such matrices are completelyentangled subspaces, with states being equally entangled in both the giventensor product space and the permuted tensor product space.

Permutation symmetry and entanglement in quantum states of heterogeneous systems