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arXiv:1007.3920 · Quantum Information and Computation

Symmetric states: local unitary equivalence via stabilizers

Curt D. Cenci, David W. Lyons, Laura M. Snyder, Scott N. Walck

Published Nov 1, 2010• 14 citations
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Abstract

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group $SO(3)$, and give examples of states with discrete stabilizers.

DOI
10.26421/qic10.11-12-12

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We classify, up to local unitary equivalence, the set of nnn-qubit states that is stabilized by the diagonal subgroup of the local unitary group. We exhibit a basis for this set, parametrized by diagrams of nonintersecting chords connecting pairs of points on a circle, and give a criterion for when the stabilizer is precisely the diagonal subgroup and not larger. This investigation is part of a larger program to partially classify entanglement type (local unitary equivalence class) via analysis of stabilizer structure.
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.
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In order to study a large number of generalized Bell states (GBSs) efficiently, it is important to classify them by local unitary (LU) equivalence. For example, there are C254(=12650) sets of four GBSs in C5⊗C5, which greatly exceed the C164(=1820) sets of four GBSs in C4⊗C4, and it is inconceivable to explore these sets directly without prior classification. In this paper, we consider the classification of GBS sets in C5⊗C5. First, two simple and useful characterizations of LU-equivalence of GBS sets are given. Then, one characterization is used to divide all 12 650 sets of four GBSs into eight equivalence classes including six locally distinguishable classes and two classes that cannot be discriminated by one-way local operations and classical communication. The second characterization is powerful and can be used to write a program, by using which all C255(=53130) sets of five GBSs in C5⊗C5 are divided into 21 equivalence classes including 9 locally distinguishable classes and 12 locally indistinguishable classes.