arXiv:0709.1105 · Physical Review A

Classification of nonproduct states with maximum stabilizer dimension

David W. Lyons, Scott N. Walck, Stephanie A. Blanda

Published Feb 11, 2008• 11 citations

Abstract

Nonproduct $n$-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized $n$-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 $n=4$, 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.

DOI: 10.1103/physreva.77.022309

Related papers

Maximum stabilizer dimension for nonproduct states

Scott N. Walck, David W. Lyons

arXiv:0706.1785Physical Review AAug 3, 200710 citations

Symmetric mixed states of n qubits: Local unitary stabilizers and entanglement classes

David W. Lyons, Scott N. Walck

arXiv:1107.1372Physical Review AOct 31, 201111 citations

Only n -Qubit Greenberger-Horne-Zeilinger States Are Undetermined by Their Reduced Density Matrices

Scott N. Walck, David W. Lyons

The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.

arXiv:0707.4428Physical Review LettersFeb 4, 200837 citations

Local unitary group stabilizers and entanglement for multiqubit symmetric states

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

arXiv:1011.5229Nov 23, 20104 citations

Symmetric states: local unitary equivalence via stabilizers

Curt D. Cenci, David W. Lyons, Laura M. Snyder +1 more

arXiv:1007.3920Quantum Information and ComputationNov 1, 201014 citations

Invariance of reduced density matrices under Local Unitary operations

Ana María Guerra Martins

arXiv:1307.0948Jul 3, 20130 citations

arXiv:0709.1105 · Physical Review A

Classification of nonproduct states with maximum stabilizer dimension

David W. Lyons, Scott N. Walck, Stephanie A. Blanda

Published Feb 11, 2008• 11 citations

arXiv HTML PDF

Abstract

Nonproduct

n

-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized

n

-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=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.

DOI: 10.1103/physreva.77.022309

Related papers

Maximum stabilizer dimension for nonproduct states

Scott N. Walck, David W. Lyons

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.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.

arXiv:0706.1785Physical Review AAug 3, 200710 citations

Symmetric mixed states of n qubits: Local unitary stabilizers and entanglement classes

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.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.

arXiv:1107.1372Physical Review AOct 31, 201111 citations

Only n -Qubit Greenberger-Horne-Zeilinger States Are Undetermined by Their Reduced Density Matrices

Scott N. Walck, David W. Lyons

arXiv:0707.4428Physical Review LettersFeb 4, 200837 citations

Local unitary group stabilizers and entanglement for multiqubit symmetric states

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.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.

arXiv:1011.5229Nov 23, 20104 citations

Symmetric states: local unitary equivalence via stabilizers

Curt D. Cenci, David W. Lyons, Laura M. Snyder +1 more

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.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.

arXiv:1007.3920Quantum Information and ComputationNov 1, 201014 citations

Invariance of reduced density matrices under Local Unitary operations

Ana María Guerra Martins

We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of {\it cyclic local} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$ . 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 derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of {\it cyclic local} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$. 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.

arXiv:1307.0948Jul 3, 20130 citations

Nonproduct

n

-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized

n

-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=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.

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.

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.

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.

SO(3)

, and give examples of states with discrete stabilizers.

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.