arXiv:quant-ph/0506241 · Journal of Physics A Mathematical and General

Classification ofn-qubit states with minimum orbit dimension

David W. Lyons, Scott N. Walck

Published Feb 22, 2006• 5 citations

Abstract

The group of local unitary transformations acts on the space of n-qubit pure states, decomposing it into orbits. In a previous paper we proved that a product of singlet states (together with an unentangled qubit for a system with an odd number of qubits) achieves the smallest possible orbit dimension, equal to 3n/2 for n even and (3n + 1)/2 for n odd, where n is the number of qubits. In this paper we show that any state with minimum orbit dimension must be of this form, and furthermore, such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets.

DOI: 10.1088/0305-4470/39/10/013

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arXiv:quant-ph/0506241 · Journal of Physics A Mathematical and General

Classification ofn-qubit states with minimum orbit dimension

David W. Lyons, Scott N. Walck

Published Feb 22, 2006• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/39/10/013

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SU(2)^{\otimes n}

. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.

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.