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Suppose Alice and Bob are arrested for killing Eve, and are taken to two different interrogation rooms. The police quiz Alice and separately Bob, asking them a bunch of different questions along the way. If Alice and Bob can get their stories straight they'll be set free, but if Alice is inconsistent with Bob they'll both be sent to the event horizon of the nearest black hole, to be shredded and spaghettified to smithereens.

But the cops know that Alice and Bob are quantum experimentalists and could have pre-shared some entanglement prior to their arrests. So, the coppers suspect that Alice could be engaging in a CHSH-game with Bob.

Is there anything the cops can do to assure themselves that Alice and Bob did not share any entanglement before being arrested? Is there a set of questions that the cops can ask of Alice and Bob such that they can be assured that Alice and Bob are not engaged in CHSH-like trickery?

At least information-theoretically, I don't think so. I think this is precisely what was ruled out by the no-go theorem on quantum bit commitment. Perhaps there is some post-quantum protocol that can classically bind Alice and Bob, but otherwise I think the cops are stuck assuming Alice and Bob have such entanglement.

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    $\begingroup$ I don't imagine you'll ever manage a proof that they don't have entanglement (if they know you're testing that, they just don't use their entanglement). But for the scenario you specify, you could reframe it: can the questions they ask (that need to be correlated) be embedded into some larger protocol that does not have any advantage to using entanglement? $\endgroup$
    – DaftWullie
    Feb 14 at 13:55
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    $\begingroup$ On a practical note, you'll want to decohere Alice and Bob's entanglement. So you will think about all of the ways in which they might retain coherence, then try to couple as many of those degrees of freedom as possible to large environments. This answer doesn't give "questions" for the cops to ask, but the cops could be pretty confident if they put Alice and Bob in contact with large warm thermal baths... $\endgroup$ Feb 14 at 14:11
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    $\begingroup$ 1. Cops could confiscate all quantum systems in Alice's & Bob's possession. 2. Cops could estimate an upper bound on entanglement that Alice & Bob could have prepared or may possess and then could lie to them, e.g. that consistently winning in a GHZ game will set them free, so that they end up using it all up. 3. If the cops suspect a specific entangled resource state, then they could measure Alice & Bob in the appropriate basis to obtain evidence of entanglement. 4. If Alice & Bob are actually rebel factions and cops caught sufficiently many members of each, then they could perform joint QST. $\endgroup$ Feb 14 at 14:20
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    $\begingroup$ Under some assumptions you could try to argue that Alice and Bob are not entangled because they are entangled with other parties and use monogamy of entanglement results. E.g., Suppose Alice's system maximally violates CHSH with Cop1 and so should be maximally entangled. Then her state cannot be entangled with Bob by monogamy. $\endgroup$
    – Rammus
    Feb 14 at 14:56
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    $\begingroup$ @AdamZalcman How do the cops distinguish a quantum system from any other (e.g. atoms in their body)? $\endgroup$
    – DaftWullie
    Feb 15 at 8:44

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