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I like path integrals. I prefer to try to understand quantum phenomena in terms of path integrals rather than Hamiltonian mechanics. However, most of the standard texts on quantum mechanics start from Hamiltonian mechanics and mention the path integrals as a side note. This makes it difficult for me to gain insight into common phenomena (with a few exceptions like certain areas of optics) from the perspective of path integrals formulations. I am aware of some exceptions to this rule (books on path integrals in particular), but I have only seen entanglement explained from the perspective of eigenstates of a Hamiltonian.

I was wondering how entanglement would be interpreted if you started from a path integral formulation. I would be more than happy to hear from a quantum field theoretic perspective as well, but the usual quantum mechanical path integral would certainly satisfy my curiosity.

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Path integrals have been used to discuss entanglement of quantum fields. The path integral is used to trace out part of the system, so that the entanglement is given in terms of Von Neumann entropy, in terms of the trace of the partial density matrix. You can find an example here for instance:

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