Classifying quantum entanglement through topological links

We propose an alternative classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid 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 that associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. 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 tools in a context of qubit networks.

Figure 1

The Hopf link, an element of the class $2^1$.

Figure 2

An element of the link class $3^3$.

Figure 3

Diagram of one-cuts for the link class $3^3$. The arrows correspond to the link after the cut, and the numbers above each arrow represent the number of possible cuts which lead to a link of that class.

Figure 4

The one-cut diagram of the link $3^1$.

Figure 5

The Borromean link, an element of the link class $3^1$.

Figure 6

The one-cut diagram of the link class $3^2$.

Figure 7

An element of the link class $3^2$.

Figure 8

The one-cut diagram of the link class $3^4$.

Figure 9

An element of the link class $3^4$.

Figure 10

An element of the link class $4^{20}$.

Figure 11

Basic building blocks of four-ring links: on the top left, the two-variable term block; on the top right, the four-variable term block; and on the bottom, the three-variable term block.

Figure 12

All one-cut diagrams corresponding to each four-ring link class.