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In the simple example that we measure the spin of two entangled particles, we measure one to have spin up so we know the other has spin down.
If we could (theoretically) measure both particles at the same exact time what happens?

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    $\begingroup$ It would be the same as when they are measured at different times. The measurement would show one spin is up and the other down. $\endgroup$ – mmesser314 Jul 14 '15 at 13:05
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First an important clarification about simultaneity you need to be aware of:

In special relativity we learn that there is no such thing as two spatially separated events A and B happening 'at the same time', at least not in any absolute sense. If one inertial observer sees the events as simultaneous, another perfectly legitimate inertial observer sees A occur before B, while another can see B happen before A. None of them are correct in any ultimate objective sense, since simultaneity is relative.

Second, in relation to your specific question, the measured spins will be anti-correlated (one up, one down) no matter what the time difference between the measurements in the frame chosen. Entangled systems do not 'care' about the magnitude of their spatial or temporal separation - the correlation is there to stay so long as no further interactions occur.

Entangled systems behave as though they are joined at the hip, even when they are arbitrarily separated, and that's what makes the behavior so unusual.

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You seem to think that when you measure one entangled particle, the other is affected even if you have not interacted with it. If that were true, then the result of the measurement on the first particle would determine the result of the measurement on the second. And then if you measure them at the same time, there is a problem of how to determine the tie break.

But this is a result of a misconception in the way entanglement is usually described. It is common for people to say that Bell's theorem shows quantum mechanics is non-local, i.e. - measurement on one system determines what happens to a distant system. This leads to problems like the one you described above. However, Bell's theorem doesn't show that quantum mechanics is non-local.

Bell's theorem is a mathematical result. It states that if the outcomes of measurements are described by stochastic variables then those variables have to be non-local to reproduce the correlations predicted by quantum mechanics. But quantum mechanics isn't a theory about classical stochastic variables. Rather, the physical quantities that describe the evolution of a quantum system are Hermitian operators that evolve entirely locally:

http://xxx.lanl.gov/abs/quant-ph/9906007

http://arxiv.org/abs/1109.6223

These operators describe physical reality as being a more complex structure than the universe as described by classical physics that, in some approximations, resembles multiple non-interacting versions of the world as described by classical physics.

For each measurement there will be two versions of the measuring apparatus after the measurement. One of the versions of the measuring apparatus will record spin up, the other will record spin down. When a joint measurement is done on records of each result they then become correlated.

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