All possible permutational symmetries of a quantum system

TL;DR

This paper explores intermediate permutational symmetries in qubit systems, showing that random states are nearly antisymmetric and that antisymmetric subspaces host highly entangled states with maximally mixed single-qubit reductions.

Contribution

It characterizes intermediate permutational symmetries in qubits and links antisymmetry to increased multipartite entanglement and maximal mixedness of reduced states.

Findings

Random pure states are nearly as antisymmetric as possible.

More antisymmetric subspaces exhibit higher multipartite entanglement.

States in the most antisymmetric subspace have maximally mixed single-qubit states.

Abstract

We investigate the intermediate permutational symmetries of a system of qubits, that lie in between the perfect symmetric and antisymmetric cases. We prove that, on average, pure states of qubits picked at random with respect to the uniform measure on the unit sphere of the Hilbert space are almost as antisymmetric as they are allowed to be. We then observe that multipartite entanglement, measured by the generalized Meyer- Wallach measure, tends to be larger in subspaces that are more antisymmetric than the complete symmetric one. Eventually, we prove that all states contained in the most antisymmetric subspace are relevant multipartite entangled states in the sense that their 1-qubit reduced states are all maximally mixed.