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arXiv:0706.1785 · Physical Review A

Maximum stabilizer dimension for nonproduct states

Scott N. Walck, David W. Lyons

Published Aug 3, 2007• 10 citations
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Abstract

Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for $n$-qubit pure states, and find its maximum dimension to be $n\ensuremath{-}1$ for nonproduct states of three qubits and higher. The $n$-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: We show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the $n$-qubit GHZ state.

DOI
10.1103/physreva.76.022303

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Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for nnn-qubit pure states, and find its maximum dimension to be n\ensuremath−1n\ensuremath{-}1n\ensuremath−1 for nonproduct states of three qubits and higher. The nnn-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: We show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the nnn-qubit GHZ state.
Nonproduct nnn-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized nnn-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for $n\ensuremath{\geqslant}3$, $n\ensuremath{\ne}4$. We characterize the Lie algebra of the stabilizer subgroup for these states. For n=4n=4n=4, there is an additional maximal stabilizer subalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well.
We classify, up to local unitary equivalence, local unitary stabilizer Lie algebras for symmetric mixed states of nnn 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.
The entanglement properties of a multiparty pure state are invariant under local unitary transformations. The stabilizer dimension of a multiparty pure state characterizes how many types of such local unitary transformations exist for the state. We find that the stabilizer dimension of an nnn-qubit $(n\ensuremath{\ge}2)$ graph state is associated with three specific configurations in its graph. We further show that the stabilizer dimension of an nnn-qubit $(n\ensuremath{\ge}3)$ graph state is equal to the degree of irreducible two-qubit correlations in the state.
We refine recent local unitary entanglement classification for symmetric pure states of nnn 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.