Difference between Quantum Superposition and Mixed States

[Trixie Mattel]

Hello, sorry, I do realize that this question has been asked before but there are just a few things I would like to figure out.

So, in my mind the differences lie in knowing the states that the system could end up in, and also the difference in the probabilities.

Is this thinking correct:

For a mixed state, there exists classical randomness. So for example some machine prepares a state
|ψ1> and that is the desired state I would like to get. However the machine is faulty and also prepares some state |ψ2>. The probability of getting |ψ1> is P1 and the probability of state |ψ2> being prepare is 1-P1.
The probabilities here are classical. Also |ψ2> is some state, made by the faulty machine, that I am not really aware of. I am not sure of its characteristics. Its prepared due to classical randomness. And so I get a mixed state of P1|ψ1> + P2|ψ2>Then there's for example a particle. Which can be in spin up or spin down, |0> and |1> respectively.
Here I am aware of the possible state the particle may end up in and the characteristics of those states.
However I can not be sure of which state the particle is in until i make a measurement (the wavefunction collapses) and then it will be in one state or the other. Here in QM we can think of the particle, before measurement as existing in a quantum superposition of both states. α|0> + β|1>
where α and β are not classical probabilities, they are quantum probability amplitudes. And also we can think of this quantum superposition of pure states as just another pure state before measurement.Is that a correct way of thinking of the difference between mixed states and quantum superposition?

Also, to be in quantum superposition, is it a necessity that the superimposed states are eigenstates of some hermitian operator??Thank you very much

 

[DrClaude]

A couple of comments:

A mixed state occurs because of a lack of information. For instance, if a particle is entailed entangled (darn autocorrect) with another but the state of that particle is unknown (or it is not even known that the particle has an entanglement partner), then its state is essentially a mixed state.

Quantum systems are always in superpositions. If they are in an eigenstate of a given observable, then they are in a superposition of states of any observable that does not not commute with the first one.