Quantum Information for Quantum Cats (14)

Entanglement of Two Quantum Arrows

The counter-intuitive feature of quantum mechanics is not limited to the effect of the observer to the object observed. The state of an object can be changed by an observer residing far away who observes something other than the object in question! That would be VERY strage indeed. But probably the reader who have followed to this point are ready for that surprize.

Suppose there are two quantum arrows. We place one of them upward in state | u >, and the other one downward in state | d >. Writing the state of the first arrow | >1 , and the second arrow | >2 , we can have twoo possibilities

These are two different states, which is evident from their orthogonality < A | B > = 0. Assuming that | u > and | d > are normalized, < u | u > = 1, < d | d > = 1, | A > nd | B > are normalized too, < A | A > = 1 and <B|B> = 1. As we have stressed before, the superposition of two quantum states has to be a quantum state. So the state | S > defined by

is a legitimate quantum state describing the system of two arrows. The coefficients \sqrt2 are ther to guantee the normalization <S|S> = 1.

We now ask a question: Which way is the first arrow in the state |S> pointing? In the state |A >, it is upward, and in | B > it is downward, and the probability of finding |A > and | B> in sthe state | S> is given by

So, is it that the first arrow in | S > is upward with 50 % chance and downward with 50 %? The answer is not that simple. To see that, we take the product of | S > with the states of the second arrow only:

The meaning of these equation is that, given the condition that the second arrow is found upward (downward), the first arrow is always downward (upward). So the observation of the second arrow changes the direction of the first arrow. The coefficient 1/\sqrt2 is a reminder that each even occurs with the condition of having the second particle in up or down state, which itself could occur in 1/2 - 1/2 chance.