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The question is really one question that leads to the final one:

  1. Is it possible to realize a qubit that naturally flips between two quantum states on a definite and fixed period without any ongoing external stimulation?

  2. If so, if such a qubit were forced into entanglement with another such oscillating qubit and then one of the qubits accelerated up to very near light-speed, would relativistic time dilation lead to disentanglement/desynchronization of these two "qubit clocks"?

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  • $\begingroup$ why was this question downvoted? $\endgroup$ – Oke Uwechue Aug 27 at 15:11
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(1) Yes. A simple example is when the two states are not eigenstates of the energy operator. Imagine that the two lowest energy eigenstates are $|1\rangle$ and $|2\rangle$, with energy $E_1$ and $E_2$ respectively, and form a superposition state from them.

A good example are the two lowest energy states of a 1D double well, where $|1\rangle$ would be the lowest energy symmetric and $|2\rangle$ the anti-symmetric state. The states $|+\rangle=(|1\rangle+|2\rangle)/\sqrt{2}$ and $|-\rangle=(|1\rangle-|2\rangle)/\sqrt{2}$ would then correspond to having the system in the left or right well respectively. But if placed in state $|+\rangle$ at $t=0$ the state will evolve as $ (|1\rangle e^{-iE_1t/\hbar}+|2\rangle e^{-iE_2t/\hbar})/\sqrt{2}$, wobbling back and forth between the two wells at a frequency set by the difference in energy between the two states.

(2) Entanglement is not generally a Lorenz boost invariant property. There are papers discussing how they entangle spin and momentum degrees of freedom, and this changes under the Lorenz transformation. Momentum-momentum entagnlement does not change. In general, how entangled different observables are depends on what they are.

So the answer seems to be that if one of the qubit clocks are boosted, if the degree of freedom is something like spin, then entanglement can remain although they would no longer be in perfect sync. Remember that entanglement does not mean the two systems are perfectly the same or unchanging, but rather that describing one system will necessarily imply things about the other system in a nontrivial way.

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