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.

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.

“Free will” usually refers to the belief that your decisions cannot be reduced to the laws of physics (e.g. people who say “do you really think your thoughts are just a bunch of chemical reactions in the brain???”), either because they can’t be reduced at all or that they operate according to their own independent logic. I see no reason to believe that and no evidence for it.

Some people try to bring up randomness but even if the universe is random that doesn’t get you to free will. Imagine if the state forced you to accept a job for life they choose when you turn 18, and they pick it with a random number generator. Is that free will? Of course not. Randomness is not relevant to free will. I think the confusion comes from the fact that we have two parallel debates of “free will vs determinism” and “randomness vs determinism” and people think they’re related, but in reality the term “determinism” means something different in both contexts.

In the “free will vs determinism” debate we are talking about nomological determinism, which is the idea that reality is reducible to the laws of physics and nothing more. Even if those laws may be random, it would still be incompatible with the philosophical notion of “free will” because it would still be ultimately the probabilistic mathematical laws that govern the chemical reactions in your brain that cause you to make decisions.

In the “randomness vs determinism” debate we are instead talking about absolute determinism, sometimes also called Laplacian determinism, which is the idea that if you fully know the initial state of the universe you could predict the future with absolute certainty.

These are two separate discussions and shouldn’t be confused with one another.

In a sense it is deterministic. It’s just when most people think of determinism, they think of conditioning on the initial state, and that this provides sufficient constraints to predict all future states. In quantum mechanics, conditioning on the initial state does not provide sufficient constraints to predict all future states and leads to ambiguities. However, if you condition on both the initial state and the final state, you appear to get determinstic values for all of the observables. It seems to be deterministic, just not forwards-in-time deterministic, but “all-at-once” deterministic. Laplace’s demon would just need to know the very initial conditions of the universe and the very final conditions.

Many Worlds is an incredibly bizarre point of view.

Quantum mechanics has two fundamental postulates, that being the Schrodinger equation and the Born rule. It’s impossible to get rid of the Born rule in quantum mechanics as shown by Gleason’s Theorem, it’s an inevitable consequence of the structure of the theory. But Schrodinger’s equation implies that systems can undergo unitary evolution in certain contexts, whereas the Born rule implies systems can undergo non-unitary evolution in other contexts.

If we just take this as true at face value, then it means the wave function is not fundamental because it can only model unitary evolution, hence why you need the measurement update hack to skip over non-unitary transformations. It is only a convenient shorthand for when you are solely dealing with unitary evolution. The density matrix is then more fundamental because it is a complete description which can model both unitary and non-unitary transformations without the need for measurement update, “collapse,” and does so continuously and linearly.

However, MWI proponents have a weird unexplained bias against the Born rule and love for unitary evolution, so they insist the Born rule must actually just be due to some error in measurement, and that everything actually evolves unitarily. This is trivially false if you just take quantum mechanics at face value. The mathematics at face value unequivocally tells you that both kinds of evolution can occur under different contexts.

MWI tries to escape this by pointing out that because it’s contextual, i.e. “perspectival,” you can imagine a kind of universal perspective where everything is unitary. For example, in the Wigner’s friend scenario, for his friend, he would describe the particle undergoing non-unitary evolution, but for Wigner, he would describe the system as still unitary from his “outside” perspective. Hence, you can imagine a cosmic, godlike perspective outside of everything, and from it, everything would always remain unitary.

The problem with this is Hilbert space isn’t a background space like Minkowski space where you can apply a perspective transformation to something independent of any physical object, which is possible with background spaces because they are defined independently of the relevant objects. Hilbert space is a constructed space which is defined dependently upon the relevant objects. Two different objects described with two different wave functions would be elements of different Hilbert spaces.

That means perspective transformations are only possible to the perspective of other objects within your defined Hilbert space, you cannot adopt a “view from nowhere” like you can with a background space, so there is just nothing in the mathematics of quantum mechanics that could ever allow you to mathematically derive this cosmic perspective of the universal wave function. You could not even define it, because, again, a Hilbert space is defined in terms of the objects it contains, and so a Hilbert space containing the whole universe would require knowing the whole universe to even define it.

The issue is that this “universal wave function” is neither mathematically definable nor derivable, so it only has to be postulated, as well as its mathematical properties postulates, as a matter of fiat. Every single paper on MWI ever just postulates it entirely by fiat and defines by fiat what its mathematical properties are. Because the Born rule is inevitable form the logical structure of quantum theory, these mathematical properties always include something basically just the same as the Born rule but in a more roundabout fashion.

None of this plays any empirical role in the real world. The only point of the universal wave function is so that whenever you perceive non-unitary evolution, you can clasp your hands together and pray, “I know from the viewpoint of the great universal wave function above that is watching over us all, it is still unitary!” If you believe this, it still doesn’t play any role in how you would carry out quantum mechanics, because you don’t have access to it, so you still have to treat it as if from your perspective it’s non-unitary.