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Suppose we have a qubit P in an unknown quantum state. Unknown as in we didn't prepare it, and don't know how it was prepared either. Without measuring the qubit in any way, we encode it into two qubits Q and R as follows:

$$\alpha \left| 0 \right\rangle_P +\beta\left|1\right\rangle_P \to \frac{\alpha}{\sqrt 2} \left[ |00\rangle_{QR} + |11\rangle_{QR} \right] + \frac{\beta}{\sqrt 2} \left[ |01\rangle_{QR} - |10\rangle_{QR} \right]$$

Qubit P is "destroyed" along the way due to no-cloning. Then, we encode Q in a photon and beam it off into space in one direction, while we beam R off in the opposite direction. Both photons eventually cross the cosmological horizon.

Is the quantum information originally encoded within P lost for good? We can suppose there might be an observer outside the cosmological horizon, but he can only observe at most either Q or R, but not both. And just looking at one qubit leaves a maximally mixed qubit.

Or maybe the causal patch conjecture is true. Then, both qubits are absorbed by the stretched horizon and their information are eventually reradiated back to us in a delocalized manner across cosmological scales with no information loss.

Suppose there are two detectors lying outside the cosmological horizon measuring Q and R respectively in the $\left\{ |0\rangle, |1\rangle \right\}$ basis. Both detectors can never ever be in causal contact with each other. Better still, replace the detectors with two obsevers. So, can they cause a delayed choice retrocausal collapse of P? Do both observers even "commute" with each other?

PS: actually, I could have just assumed only qubit Q gets beamed out, while qubit R remains with us in our reference frame. This doesn't change the essence of the question, but it does simplify details somewhat.

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  • $\begingroup$ Isn't it obvious? $\endgroup$ – CuriousOne Jan 4 '15 at 15:41
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    $\begingroup$ is everything obvious to you? $\endgroup$ – Just a lil kid Jan 4 '15 at 15:53
  • $\begingroup$ "Questions" so blatantly obvious are. $\endgroup$ – CuriousOne Jan 4 '15 at 15:56
  • $\begingroup$ The way I see it cosmological horizons are just "constraints" on a localized observer i.e the information exists as it is, the question whether it is physically possible to patch them up will depend on how much time you have and has nothing to do with information theory. $\endgroup$ – Avrham Aton Jan 4 '15 at 18:05

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