Symmetric mixed states: local unitary stabilizers and entanglement classes

Curt D. Cenci, David W. Lyons, Scott N. Walck

Luca Candelori, Vladimir Chernyak, John R. Klein +1 more

Z. Gedik

David W. Lyons, Scott N. Walck

We classify, up to local unitary equivalence, local unitary stabilizer Lie algebras for symmetric mixed states of $n$ qubits into six classes. These include the stabilizer types of the Werner states, the Greenberger-Horne-Zeilinger state and its generalizations, and Dicke states. For all but the zero algebra, we classify entanglement types (local unitary equivalence classes) of symmetric mixed states that have those stabilizers. We make use of the identification of symmetric density matrices with polynomials in three variables with real coefficients and apply the representation theory of SO(3) on this space of polynomials.

Curt D. Cenci, David W. Lyons, Scott N. Walck

We refine recent local unitary entanglement classification for symmetric pure states of $n$ qubits (that is, states invariant under permutations of qubits) using local unitary stabilizer subgroups and Majorana configurations. Stabilizer subgroups carry more entanglement distinguishing power than do the stabilizer subalgebras used in our previous work. We extend to mixed states recent results about local operations on pure symmetric states by showing that if two symmetric density operators are equivalent by a local unitary operation, then they are equivalent via a local unitary operation that is the {\em same} in each qubit. A geometric consequence, used in our entanglement classification, is that two symmetric pure states are local unitary equivalent if and only if their Majorana configurations can be interchanged by a rotation of the Bloch sphere.

Thierry Bastin, Stéphanie Krins, Pierre Mathonet +3 more