Elitzur-Dolev Quantum Liar Paradox

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Quantum of Solace
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I've seen Elitzur's brief presentations on this. Two excited atoms pointed at a detector. The detector goes off, it isn't known which particle fired the photon, so you interrogate one, it gives a definite answer but violates bell's inequality? I'm sorry but I'm lost at such a vague description and this was the best I could find. What is the logic behind this test? I appreciate any references you could offer.
 
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Quantum of Solace said:
I've seen Elitzur's brief presentations on this...
Do you have a reference? Without that, it's going to be hard for anyone to comment sensibly.
 
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A quick google search found, well, this very thread. But also these slides:

Two slides: top slide titled The Elitzur-Dolev Quantum Liar Paradox with bullets on two excited atoms A1 and A2 in cavities facing a beam-splitter and the state (1/√2)(|1>_A|0>_B - |0>_A|1>_B); bottom slide titled The Quantum Liar Paradox.


Basically it's describing this quantum circuit, where a parity measurement creates entanglement between two qubits that start off separable:

Quantum circuit with three qubit lines starting in |0>: two qubits pass through Hadamard gates and CZ-style controlled operations shown as Bloch-sphere entanglers, then CNOT gates targeting a third line, a dotted post-select line, a measurement meter, and a post-select output of |0>|0>.


This is an example of "performing a measurement forces the system into the state indicated by the measurement outcome, even if that state is entangled and you feel like that's weird".

This particular setup works on inputs that are in X-axis states (e.g. On+Off), but not Z-axis states (e.g. just On). Doing an X-axis or Y-axis parity measurement in addition to the Z-axis parity measurement would fix that, but kinda breaks the analogy with the simple physical system. With both parity measurements it's a full fledged Bell basis measurement, which must force the system into one of the Bell basis states (and tell you which one). All of those basis states are fully entangled, so the output is entangled.
 
It took a while, but I finally found the reference:

http://a-c-elitzur.co.il/uploads/articlesdocs/MultipleIFM.pdf

Section 10, Page 13 "The Quantum Liar Paradox"

Still trying to understand this properly, but it seems to me until the Z measurement (which box?) is performed, each particle Z exists in the blocking(or non blocking) box only as part of a superposed state, and is thus not forced to interact with the photon, leaving it to interfere with itself.

If two entangled particles have a 50% chance of being in the "blocking" box, (mutually exclusive), does that mean one of them MUST interact with a traversing photon? or does quantum mechanics allow a possibility for photons to traverse both boxes? If so, then I'm not sure I get why this is supposed to be a paradox.