3天前

Do you believe in free will?

There is an argument that free will doesn't exist because there is an unbroken chain of causality we are riding on that dates back to the beginning of time. Meaning that every time you fart, scratch your nose, blink, or make lifechanging decisions there is a pre existing reason. These reasons might be anything from the sensory enviornment you were in the past minute, the hormone levels in your bloodstream at the time, hormones you were exposed to as a baby, or how you were parented growing up. No thought you have is really original and is more like a domino affect of neurons firing off in reaction to what you have experienced. What are your thoughts on this?

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I am not that good with abstract language. It helps to put it into more logical terms.
It sounds like what you are saying is that you begin with something a superposition of states like (1/√2)(|0⟩ + |1⟩) which we could achieve with the H operator applied to |0⟩ and then you make that be the cause of something else which we would achieve with the CX operator and would give us (1/√2)(|00⟩ + |11⟩) and then measure it. We can call these t=0 starting in the |00⟩ state, then t=1 we apply H operator to the least significant, and then t=2 is the CX operator with the control on the least significant.
I can't answer it for the two cats literally because they are made up it a gorillion particles and computing it for all of them would be computationally impossible. But in this simple case you would just compute the weak values which requires you to also condition on the final state which in this case the final states could be |00⟩ or |11⟩. For each observable, let's say we're interested in the one at t=x, you construct your final state vector by starting on this final state, specifically its Hermitian transpose, and multiplying it by the reversed unitary evolution from t=2 to t=x and multiply that by the observable then multiply that by the forwards-in-time evolution from t=0 to t=x multiplied by the initial state, and then normalize the whole thing by dividing it by the Hermitian transpose of the final state times the whole reverse time evolution from t=2 to t=0 and then by the final state.
In the case where the measured state at t=3 is |00⟩ we get for the observables (most significant followed by least significant)...
t=0: (0,0,+1);(+1,+i,+1)
t=1: (0,0,+1);(+1,-i,+1)
t=2: (0,0,+1);(0,0,+1)
In the case where the measured state at t=3 is |11⟩ we get for the observables....
t=0: (0,0,+1);(-1,-i,+1)
t=1: (0,0,+1);(+1,+i,-1)
t=2: (0,0,-1);(0,0,-1)
The values |0⟩ and |1⟩ just mean that the Z observable has a value of +1 or -1, so if we just look at the values of the Z observables we can rewrite this in something a bit more readable.
|00⟩ → |00⟩ → |00⟩
|00⟩ → |01⟩ → |11⟩
Even though the initial conditions both began at |00⟩ they have different values on their other observables which then plays a role in subsequent interactions. The least significant qubit in the case where the final state is |00⟩ begins with a different signage on its Y observable than in the case when the outcome is |11⟩. That causes the H opreator to have a different impact, in one case it flips the least significant qubit and in another case it does not. If it gets flipped then, since it is the control for the CX operator, it will flip the most significant qubit as well, but if it's not then it won't flip it.
Notice how there is also no t=3, because t=3 is when we measure, and the algorithm guarantees that the values are always in the state you will measure before you measure them. So your measurement does reveal what is really there.
If we say |0⟩ = no sleepy gas is released and the cat is awake, and |1⟩ = sleepy gas is released and the cat go sleepy time, then in the case where both cats are observed to be awake when you opened the box, at t=1: |00⟩ meaning the first one's sleepy gas didn't get released, and so at t=2: |00⟩ it doesn't cause the other one's to get released. In the case where both cats are observed to be asleep when you open the box, then t=1: |01⟩ meaning the first one's did get released, and at t=2: |11⟩ that causes the second's to be released.
When you compute this algorithm you find that the values of the observables are always set locally. Whenever two particles interact such that they become entangled, then they will form correlations for their observables in that moment and not later when you measure them, and you can even figure out what those values specifically are.
To borrow an analogy I heard from the physicist Emily Adlam, causality in quantum mechanics is akin to filling out a Sudoku puzzle. The global rules and some "known" values constrains the puzzle so that you are only capable of filling in very specific values, and so the "known" values plus the rules determine the rest of the values. If you are given the initial and final conditions as your "known" values plus the laws of quantum mechanics as the global rules constraining the system, then there is only one way you can fill in these numbers, those being the values for the observables.
- Sorry, it's been a long time since I last looked at the mathematical side of quantum mechanics, so most of your comment flew over my head. Let me put it in as simple terms as I can:
  If there are multiple paths a system can take to reach a final state, how can you accurately determine which path was taken if you only know the initial & final state? IMO this shouldn't be possible.
  
  Let's say the initial state is at time t=x, the final state is at time t=z, and the state we're interested in is at time t=y where x < y < z.
  In classical mechanics you condition on the initial known state at t=x and evolve it up to the state you're interested in at t=y. This works because the initial state is a sufficient constraint in order to guarantee only one possible outcome in classical mechanics, and so you don't need to know the final state ahead of time at t=z.
  This does not work in quantum mechanics because evolving time in a single direction gives you ambiguities due to the uncertainty principle. In quantum mechanics you have to condition on the known initial state at t=x and the known final state at t=z, and then evolve the initial state forwards in time from t=x to t=y and the final state backwards in time from t=z to t=y where they meet.
  Both directions together provide sufficient constraints to give you a value for the observable.
  I can't explain it in more detail than that without giving you the mathematics. What you are asking is ultimately a mathematical question and so it demands a mathematical answer.
  
  But that's ignoring that there are multiple paths that can lead to state z, isn't it?
  I'll try to design the simplest possible experiment: you have to radioactive atoms, each connected to a detector, and the detectors are connected to a counter. You leave the room and come back - the counter shows 2. How do you determine which atom decayed first?
  
  That's a classical ambiguity, not a quantum ambiguity. It would be like if I placed a camera that recorded when cars arrived but I only gave you information on when it detected a car and at what time and no other information, not even providing you with the footage, and asked you to derive which car came first. You can't because that's not enough information.
  The issue here isn't a quantum mechanical one but due to the resolution of your detector. In principle if it was precise enough, because the radiation emanates from different points, you could figure out which one is first because there would be non-overlapping differences. This is just a practical issue due to the low resolution of the measuring device, and not a quantum mechanical ambiguity that couldn't be resolved with a more precise measuring apparatus.
  A more quantum mechanical example is something like if you apply the H operator twice in a row and then measure it, and then ask the value of the qubit after the first application. It would be in a superposition of states which describes both possibilities symmetrically so the wavefunction you derive from its forwards-in-time evolution is not enough to tell you anything about its observables at all, and if you try to measure it at the midpoint then you also alter the outcome at the final point, no matter how precise the measuring device is.
  
  I see what you're trying to get at. It's not that we can definitely know the state, it's that we could build the experiment in such a way that we can definitely know the state - and by not building it this way we're essentially deliberately "throwing away" information about the final state.
  Thanks for the explanation!

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