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Imagine a following thought experiment.

Suppose we have a large amount of entangled particle pairs, several million or billion. Now suppose there are two observers, each carrying one member of particle pair - a slower-then-light spaceship and mission control on Earth, for example.

Before departure, both spaceship crew and Earth crew agreed to order their entangled pairs by into sets of 100 (or 1000,10000, ...), ordered by the time of particle pair creation or any other way, as long as the order is the same on both Earth and spaceship.

Now, spaceship departs Earth with v << c, so relativistic effects are negligible and the clocks on the ship and Earth are more-or-less in sync for the foreseeable future. Each day mission control on Earth is using the their particles in the pre-determined pair set to construct a quantum computer (can entangled particle in a pair be used as qubit? used for quantum gate?) and runs a specific quantum algorithm that should give some defined answer (Shor's, or Grover's).

Afterwards results are measured, breaking the entanglement and then compared against the expected one. Each day next set of particle is used and discarded. The spaceship crew does nothing, except noting which particle set has been used up by Earth team.

Now, suppose after many hundreds years of flight spaceship is several light-years away from Earth, but there are still entangled particle sets available for both crew and Earth team and both team know which sets have been used and what set is going to be the next one for the next day.

Now, imagine our spaceship encounter hostile aliens. On that day and all the subsequent ones as long as the spaceship crew survives, they observe all remaining particle pair sets, breaking the entanglement with the matching ones on Earth. Earth team is not aware of the encounter and they still attempt to construct quantum computer from their matching half of the particle pair sets every day.

But each day as they try to run the quantum algorithm on their half of particle pair set, their computation FAILS as their particles are no longer in a quantum superposition state and can no longer be used for quantum computation. So, after a few days, Earth team learns that spaceship crew sent them a 1-bit distress signal, even though a ship is several light-years away, receiving this information faster then light.

Now question: where is the mistake in my chain of thought?

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2 Answers 2

The mistake is here:

But each day as they try to run the quantum algorithm on their half of particle pair set, their computation FAILS as their particles are no longer in a quantum superposition state and can no longer be used for quantum computation.

Whether or not measurements are performed by the spaceship crew has no effect on the reduced density matrix for the system on Earth. Hence the ground crew have no way of knowing anything is going on. The number of particles involved and the timescale is irrelevant, so long as no classical information is transmitted between the parties. See here and links therein.

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Because all entanglement pair require an extra information through a classical channel, which travel slower than light speed, $v\le c$. Therefore, there is is no faster than light communication. Without that information, the qubit on Earth is essentially random in any statistical measurement.

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