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arXiv:2109.06921 · Physical Review A

Local unitary classes of states invariant under permutation subgroups

David W. Lyons, Jesse R. Arnold, Ashley F. Swogger

Published Mar 25, 2022• 10 citations
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Abstract

The study of entanglement properties of multiqubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.

DOI
10.1103/physreva.105.032442

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The study of entanglement properties of multiqubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.
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.
Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.
This paper introduces the Entanglement Dimension, a novel group invariant derived from the application of quantum information theory concepts to algebraic structures. Traditional group invariants, such as order, nilpotency class, and solvability length, offer crucial insights but often fail to distinguish between non-isomorphic groups sharing many common properties. By mapping group elements or their representations to quantum states and subsequently quantifying the entanglement present in superpositions formed from these states, we propose a new measure that captures nuanced aspects of group structure. We rigorously define the Entanglement Dimension and demonstrate its computation for various finite groups, including cyclic, dihedral, and symmetric groups. Our results indicate that this invariant can differentiate between groups previously indistinguishable by classical methods, providing a finer characterization of their internal symmetries and interactions. The Entanglement Dimension offers a fresh perspective on group theory, bridging algebraic structures with quantum mechanical principles and opening new avenues for research in both fields, particularly in areas like quantum computing and fundamental physics.
Abstract The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group SU(2), whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group U q ( su ( 2 ) ) . We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.
Local unitary stabilizer subgroups constitute powerful invariants for distinguishing various types of multipartite entanglement. In this paper, we show how stabilizers can be used as a basis for entanglement verification protocols on distributed quantum networks using minimal resources. As an example, we develop and analyze the performance of a protocol to verify membership in the space of Werner states, that is, multiqubit states that are invariant under the action of any one-qubit unitary applied to all the qubits.
We discuss the entanglement properties of symmetric states of n qubits. The Majorana representation maps a generic such state into a system of n points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an overcomplete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well-documented three-qubits case. In the SLOCC case, a cross ratio of related Möbius transformations are shown to play a central role, exemplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.