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While writing this answer I was wondering whether the analogy of the nature of entanglement in the GHZ state with Borromean rings is more than a mere analogy (cf. Aaronson's lecture).

The question in my mind basically is: can quantum entanglement, at least for the finite-dimensional cases, be expressed in terms of knot theory? Has there been any approach in this direction? If yes, then I suppose it would be a nice way to visualize qubit entanglements in various scenarios.

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    $\begingroup$ Could the well-known GHZ:Borromean Rings analogy somehow be related to the BQP-completeness of approximating the Jones polynomial of a knot? $\endgroup$
    – Mark Spinelli
    Commented Dec 4, 2019 at 3:59
  • $\begingroup$ Kauffman et al.'s papers appear relevant: 1, 2. Note: Jones polynomials and the ER=EPR hypothesis do seem to arise in this context, but I haven't read through the arguments yet. $\endgroup$
    – Sanchayan Dutta
    Commented Dec 4, 2019 at 8:22
  • $\begingroup$ Heh, there's another paper that says Braiding Operators are Universal Quantum Gates. $\endgroup$
    – Sanchayan Dutta
    Commented Dec 4, 2019 at 8:31

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In this paper, the authors used Knot theory to define what they call 'Path Model Representation'. In a later section they convert this representation to qubits by switching to binary.

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