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arXiv:2011.14507

Constraints on Maximal Entanglement Under Groups of Permutations

Alexander Meill, Jayden Butts, Elijah Sanderson

Published Nov 30, 2020• 0 citations
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Abstract

We provide a simplified characterization of entanglement in physical systems which are symmetric under the action of subgroups of the symmetric group acting on the party labels. Sets of entanglements are inherently equal, lying in the same orbit under the group action, which we demonstrate for cyclic, dihedral, and polyhedral groups. We then introduce new, generalized relationships for the maxima of those entanglement by exploiting the normalizer and normal subgroups of the physical symmetry group.

DOI
10.48550/arxiv.2011.14507

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We provide a simplified characterization of entanglement in physical systems which are symmetric under the action of subgroups of the symmetric group acting on the party labels. Sets of entanglements are inherently equal, lying in the same orbit under the group action, which we demonstrate for cyclic, dihedral, and polyhedral groups. We then introduce new, generalized relationships for the maxima of those entanglement by exploiting the normalizer and normal subgroups of the physical symmetry group.
We provide an initial characterization of pairwise concurrence in quantum states which are invariant under cyclic permutations of party labeling. We prove that maximal entanglement can be entirely described by adjacent pairs, then give explicit descriptions of those states in specific subsets of four- and five-qubit states (X states). We also construct a monogamy bound on shared concurrences in the same subsets in four and five qubits, finding that above nonmaximal entanglement thresholds, no other entanglements are possible.
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.
Quantum entanglement rapidly becomes unwieldy to calculate as the number of particlesand the dimension of the spaces associated to those particles increase. One meaningful approachwhich simplifies that analysis is the restriction to subsets of states which obey some physicallyrelevant symmetry. In this thesis, entanglement properties of totally permutation-symmetric,translationally invariant, and party-site symmetric states are examined, as well as those of smallbond-dimensional matrix product states.
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.
We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using the Kostant-Sternberg theorem, we show that separable states form a unique symplectic orbit, whereas orbits of entangled states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement. The above statements remain true for systems with an arbitrary number of components, moreover the presented method is general and can be applied also under different additional symmetry conditions stemming, e.g., from the indistinguishability of particles. We show how to calculate the degeneracy for various multiparticle systems providing also simple criteria of separability.
Permutation symmetries of multipartite quantum states are defined only when the constituent subsystems are of equal dimensions. In this work we extend this notion of permutation symmetry to heterogeneous systems, that is, systems composed of subsystems having unequal dimensions. Given a tensor product space of kkk subsystems (of arbitrary dimensions) and a permutation operation σσσ over kkk symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by σσσ. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation σσσ. Eigenvectors of this matrix have the required symmetry. We then examine entanglement of states in the igenspaces of these matrices. It is found that all nonsymmetric eigenspaces of such matrices are completely entangled subspaces, with states being equally entangled in both the given tensor product space and the permuted tensor product space.