According to Wikipedia, if a system has 50% chance to be in state |ψ1⟩ and 50% to be in state |ψ2⟩, then this is a mixed state.
Now, consider the state |Ψ⟩=|ψ1⟩+|ψ2⟩√2,
So, what is the mistake here? What is the real difference between mixed state and superposition of pure states?
Answer
The state
|Ψ⟩=1√2(|ψ1⟩+|ψ2⟩)
is a pure state. Meaning, there's not a 50% chance the system is in the state |ψ1⟩ and a 50% it is in the state |ψ2⟩. There is a 0% chance that the system is in either of those states, and a 100% chance the system is in the state |Ψ⟩.
The point is that these statements are all made before I make any measurements.
It is true that if I measure the observable corresponding to ψ (ψ-gular momentum :)), then there is a 50% chance after collapse the system will end up in the state |ψ1⟩.
However, let's say I choose to measure a different observable. Let's say the observable is called ϕ, and let's say that ϕ and ψ are incompatible observables in the sense that as operators [ˆψ,ˆϕ]≠0. (I realize I'm using ψ in a sense you didn't originally intend but hopefully you know what I mean). The incompatibliity means that |ψ1⟩ is not just proportional to |ϕ1⟩, it is a superposition of |ϕ1⟩ and |ϕ2⟩ (the two operators are not simulatenously diagonalized).
Then we want to re-express |Ψ⟩ in the ϕ basis. Let's say that we find |Ψ⟩=|ϕ1⟩
For example, this would happen if |ψ1⟩=1√2(|ϕ1⟩+|ϕ2⟩)
But now let's say that there's a 50% chance that the system is in the pure state |ψ1⟩, and a 50% chance the system is in the pure state |ψ2⟩. Not a superposition, a genuine uncertainty as to what the state of the system is. If the state is |ψ1⟩, then there is a 50% chance that measuring ϕ will collapse the system into the state |ϕ1⟩. Meanwhile, if the state is |ψ2⟩, I get a 50% chance of finding the system in |ϕ1⟩ after measuring. So the probability of measuring the system in the state |ϕ1⟩ after measuring ϕ, is (50% being in ψ1)(50% measuring ϕ1) + (50% being in ψ2)(50% measuring ϕ1)=50%. This is different than the pure state case.
So the difference between a 'density matrix' type uncertainty and a 'quantum superposition' of a pure state lies in the ability of quantum amplitudes to interfere, which you can measure by preparing many copies of the same state and then measuring incompatible observables.
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