I EPR revisited

[anuttarasammyak]

In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state
$$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$
At point A, a measurement is made. The state then collapses to
$$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$

We assume that A has the state $|a>_A$ and B has $|b>_B$ simultaneously, i.e., when their synchronized clocks both read time T

However, in other inertial frames, due to the relativity of simultaneity, the moment when B has $|b>_B$ occurs when B’s clock reads a time before or after T

Therefore, may we conclude that the statement “B has $|b>_B$" is not invariant, but frame-dependent?

 

[PeterDonis]

may we conclude that the statement “B has $|b>_B$" is not invariant, but frame-dependent?

You may conclude that that statement, as you make it, is not even well-defined. You didn't specify a time in any frame at which B has that state.

A true statement would be that the time ordering of spacelike separated events is not invariant, but frame-dependent. But specifying an event requires specifying a place and a time in some frame--an event is a point in spacetime.

You are also sweeping under the rug complications that involve quantum mechanics as well as relativity. For example, suppose we say that the measurement on A happens at a particular event--a particular place and time in some chosen frame. You are assuming that the state of B collapses when the measurement on A is made. But at what event does the collapse of B happen? There is no invariant way to specify that--which should be a clue that the intuitive picture of QM you are using is not, in fact, correct--more precisely, it is not relativistic. In relativistic QM, i.e., quantum field theory, the whole idea of "collapse" is much more problematic, because there are no invariant answers to questions like the one I posed just now.

In short, you unfortunately have chosen a bad example to illustrate the relativity of simultaneity.

 

[anuttarasammyak]

Thank you @PeterDonis. I should be somewhat satisfied to know that studies are going on.

 

[PeterDonis]

I should be somewhat satisfied to know that studies are going on.

I'm not sure what "studies" you're referring to, but it's certainly true that there have been plenty of experiments testing both relativity of simultaneity and the QM predictions for measurements on entangled particles.