Classifying Quantum Entanglement through Topological Links

Abstract

We propose a new classification scheme for quantum entanglement based on topological links. This is done by identifying a non-rigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for a given number of rings. To determine all different possibilities, we develop a formalism which associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. In order to demonstrate the use of this classification scheme, we choose qubit quantum states as our example of physical system. A possible procedure to obtain qubit states from the polynomials is also introduced, providing an example state for each link class. We apply the formalism for the quantum systems of three and four qubits, and demonstrate the potential of these new tools in a context of qubit networks.

arXiv comment13 pages, 12 figures

References in corpus (11)

• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Three qubits can be entangled in two inequivalent ways
• Four qubits can be entangled in nine different ways
• Generalized Schmidt decomposition and classification of three-quantum-bit states
• Comparing Quantum Entanglement and Topological Entanglement
• Inductive Entanglement Classification of Four Qubits under SLOCC
• Four-qubit entanglement from string theory
• Polynomial invariants for discrimination and classification of four-qubit entanglement
• Classification of Multipartite Entanglement via Negativity Fonts
• Borromean Entanglement Revisited

Cited by in corpus (10)

• Cut-resistant links and multipartite entanglement resistant to particle loss
• Braiding quantum gates from partition algebras
• Entanglement Classification from a Topological Perspective
• Local invariants of braiding quantum gates -- associated link polynomials and entangling power
• Coarse-grained entanglement classification through orthogonal arrays
• Geometric representations of braid and Yang-Baxter gates
• Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation
• Majorana fermions solve the tetrahedron equations as well as higher simplex equations
• Generating W states with braiding operators
• Graph approach to quantum teleportation dynamics