Tuesday, July 18, 2017

How is quantum superposition different from mixed state?


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+|ψ22,

which is a superposition of the states |ψ1 and |ψ2. Let |ψi be eigenstates of the Hamiltonian operator. Then measurements of energy will give 50% chance of it being E1 and 50% of being E2. But this then corresponds to the definition above of mixed state! However, superposition is defined to be a pure state.


So, what is the mistake here? What is the real difference between mixed state and superposition of pure states?



Answer



The state


|Ψ=12(|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=12(|ϕ1+|ϕ2)

|ψ2=12(|ϕ1|ϕ2)
Then I can ask for the probability of measuring ϕ and having the system collapse to the state |ϕ1, given that the state is |Ψ, it's 100%. So I have predictions for the two experiments, one measuring ψ and the other ϕ, given knowledge that the state is Ψ.


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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