Information Theory on Wave Function Collapse

Time_

I was trying to understand wave function collapse in terms of superposition, but I ran into some problems when relating back to information theory/entropy. It is given in the definition of information in terms of entropy energy is needed to transfer information. That is something we have always been taught, but if that means information is associated with different eigenstates, what happens to the information associated with different collapse forms of a particle? Is it that there is no energy associated with the information of the superpositions of a particle, but if that were the case, how could those states exist in the first place? Perhaps the answer is in Schrödinger's equation. Any help is welcomed! Thanks.

kith

Eigenstates of an observable and superpositions of such eigenstates are not different regarding their (von Neumann) entropy: both are so-called pure states and have zero entropy. When you think about superpositions you should keep in mind that every state can be a superposition of states. This is just a question of which basis you chose. Physically, this corresponds to which observable you are going to measure.

Collapse only occurs during measurements, where additional degrees of freedom are introduced. The interaction with the measurement apparatus leads the initially pure state to a classical probabilistic mixture of states, which has a higher entropy (this is called "decoherence"). However, there is no universally accepted mechanism how a single outcome is chosen from this mixture. So I'm not sure, if there is a satisfying answer to your question.

Time_

Alas I feel the same way, I cannot find definitive answer, however, could you answer this, mathematically, or logically? Is there energy associated with the different superpositions? If so, where does that energy go? I have a thought, but I'd like to hear others opinions first.

kith

What do you mean by the first question? Arbitrary states |ψ> can be decomposed in a superposition of energy eigenstates. If you have such a superposition, the energy of the state is not well-defined. The best you can do is to talk about average values. During a measurement, this average value changes due to the interaction with the measurement apparatus. So the apparatus is where the "energy goes".