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I'm just starting to study quantum mechanics. Please explain the error in this thinking:

You set up decay of two $\pi$ mesons and get $2\mathrm{e}^-$ on Mars and $2\mathrm{e}^+$ on Earth.

On Earth you may or may not measure the spin of those positrons, with 50% probability that they are the same spin.

On Mars, your buddy "immediately afterwards" takes a $\mathrm{He}^{2+}$ ion and adds the electrons.

If you made the measurement, the $\mathrm{He}$ has a 50%+ chance of being in a higher-then-base energy level due to Pauli exclusion.

If you didn't make the measurement, the $\mathrm{He}$ has a much lower chance of being in a higher-than-base energy level.

Your buddy measures the energy state.

Result: You just superluminally transmitted $\approx 0.8$ bits of information.

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Talk us through your jump from how making the measurement affects the He state. If you make the measurement, what are the possible outcomes and their probabilities, and what are the outcomes for He in each of those cases? If you don't make the measurement, what are the outcomes for He and their probabilities. Also, in your last sentence, do you really mean "subluminally"? Transmission of information at subluminal speeds is A-OK in physics, and happens literally all the time. – Anonymous Coward Aug 23 '11 at 19:07
I approved the edit to "superluminally" because as AC pointed out, there's nothing special about subluminal motion. But could you clarify what you really meant there, Full Decent? – David Z Aug 23 '11 at 19:18
I don't understand your sentence beginning "If you made the measurement ..." Whether you made the measurement or not, he has a 50% chance that his two electrons have the same spin, and a 50% chance that they have opposite spin. – Ted Bunn Aug 23 '11 at 20:06
up vote 4 down vote accepted

If you didn't make the measurement, the He has a much lower chance of being in a higher-than-base energy level.

I think this is the incorrect step in your reasoning. The $He$ atom would be in a 50-50 superposition of the ground and an excited state if the electrons on earth were not measured, and should collapse to one state or the other with equal probablility if they are. In either case, a mesurement of the $He$ atom to determine if it were in the ground or an excited state has exactly the same distribution of results.

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