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Let's say I encounter particles and I want to know, with certainty, if they are entangled. Can entanglement witness help me determine this?

More specifically: 1. Can I prepare a witness at any time to determine this? Even after the entanglement event occurred? 2. Will the entanglement witness tell me if the particles are entangled with 100% certainty? And not give me a false positive?

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  • $\begingroup$ The particles are always entangled. The problem is that most of the time they are entangled with lots of unrelated objects, so the few particles we are interested in can't be considered entangled only within the couple. If they get entangled with an observer, that's called measurement (from the observer's POV). So you'd better define what you mean by "entangled with 100% certainty". $\endgroup$ – Ruslan Jun 25 at 16:16
  • $\begingroup$ It'd be with 100% certainty. Check this out: en.m.wikipedia.org/wiki/Delayed-choice_quantum_eraser Do note that in this particular experiment, we use two entangled photons and get those results, of course you can do the same experiment with non entangled photons, coincidence counter will be your witness, and it will show you other results. $\endgroup$ – Paradoxy Jun 28 at 19:25
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The notion of a "witness" is that it is a measure which will not give a false positive. That is:

  • If witness reports there is entanglement, then there is (assuming the experiment has not gone wrong in some way).

  • If witness does not report there is entanglement, then there may or may not be entanglement present.

The study of entanglement witnesses is itself quite complex, so it's hard to answer your further questions. It depends on the scenario. In the standard Bell state case, you have a pair of particles and you want to know whether or not they are in a product state. I think that if you only have access to one copy of a pair of particles in some given state, it won't be possible to determine whether or not they are entangled, because for each entangled state $|\psi\rangle$, there are product states $| u \rangle$ such that $\langle u | \psi \rangle \ne 0$, so there is always a product state which can partially 'mimic' an entangled state.

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