Permutation symmetry and entanglement in multipartite quantum states of unequal subsystem dimensions

symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by

\sigma

. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation

\sigma

. Eigenvectors of this matrix have the required symmetry. We then examine entanglement of states in the igenspaces of these matrices. It is found that all nonsymmetric eigenspaces of such matrices are completely entangled subspaces, with states being equally entangled in both the given tensor product space and the permuted tensor product space.

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k

subsystems (of arbitrary dimensions) and a permutation operation

\sigma

over

k

\sigma

. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation

\sigma

k

subsystems (of arbitrary dimensions) and a permutation operation

σ

over

k

σ

. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation

σ

j

with a spin

1∕2

. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.

k

-positive map witness, optimal Schmidt-number witnesses, and entanglement measures.