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There have been similar questions asked here [1] [2], but I'm asking something a bit more specific.

I've ofter heard the argument that quantum mechanics might somehow make free will possible because it incorporates probabilities into its predictions. For example, this quote by philosopher of science Henry Margenau sums up the viewpoint nicely:

Our thesis is that quantum mechanics leaves our body, our brain, at any moment in a state with numerous (because of its complexity we might say innumerable) possible futures, each with a predetermined probability. Freedom involves two components: chance (existence of a genuine set of alternatives) and choice. Quantum mechanics provides the chance, and we shall argue that only the mind can make the choice by selecting (not energetically enforcing) among the possible future courses.

What are some common counter points against this argument?

Most modern philosophers I know of (Daniel Dennett, Douglas Hofstadter, Sean Carroll) don't take this viewpoint seriously, from what I gather. And I'm inclined to agree since random chance (quantum mechanics) is not the same as having control over an outcome (free will). But at the same time, I can't think of an obvious experiment that could disprove the idea, and I also believe that the meaning of "random chance" is not fully understood or agreed upon by philosophers. So even though I'm inclined to disagree, I'd like to know what the best counter points are.

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Quantum mechanics only introduces randomness, it is to say unpredictability.

That this randomness can be extended to macroscopic systems like a human brain has yet to be demonstrated, but even if we grant it this is only a counter argument to determinism, or the idea that, if one knew the state of a system one could possibly predict its future states with 100% accuracy.

However, non-determinism is not the same thing as free will. Even if quantum probabilities made it so that my brain can unpredictably take different decisions given the same stimuli, it does not mean that I made the decision.

To link free will and quantum determination, one would need to demonstrate how my will could influence the random outcome of quantum particles interactions in my brain, which is a very bold and, so far, unsubstantiated statement.

In one sentence: the unpredictability of quantum mechanics can give you the "free", but not the "will".

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I will be making a stronger counter argument: There is no system permitted by standard quantum mechanics which can have true free-will (different from that of a dice roll). An intriguing discussion and links to support the same can be seen from here:

So in quantum mechanics everything usually evolves deterministically(/unitarily). However, if a measurement is done where your measuring device instrument measures the state of a system then that is given by the Born Rule and one observes a real number. The Born Rule is where probability is introduced.

What is the origin of the Born Rule? If one thinks of the measurement, it sounds dodgy that system 1 can interact with system 2 and one does not take into account the Hamiltonian of system 1 to tell the time evolution (of system 2), where system 1 is the measuring device.

So, probability appeared when I introduced a macroscopic apparatus of a special sort: one with emergent classical behavior (the pointer) specially designed to behave in a certain way when presented with position eigenstates. This makes me tempted to say that probability has no fundamental role in quantum theory, it’s a subtle feature of the emergence of classical behavior from the more fundamental quantum behavior, that will appear in certain circumstances, governed by the Born rule.

So where does probability come from?

A macroscopic experimental apparatus never has a known pure state. If I want to carefully analyze such a setup, I need to describe it by quantum statistical mechanics, using a mixed state. Balian and collaborators claim that if they do this for a specific realistic model of an experimental apparatus, they get as output not the problematic superposition of states of the measurement problem, but definite outcomes, with probabilities given by the Born rule.

It comes from the measuring device being in a mixed state. Or one can think of it as ignorance of the what is the state of the measuring device.

Hopeful Claim(/ Ongoing Research): If one models system 1 and system 2 unitarily. One will get superpositions of states. However, in each superposition the measuring device returns a number.

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