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I'm relatively qualified. Studied physics all through college and spent a couple years working in quantum computing. I'll chime in here because Schrodinger's cat jokes are a pet peeve.

You are correct that, as far as we understand, it is literally impossible. There has been a competing theory for decades, but I'm not really up on the specifics https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory. The reason it is generally rejected is that it appears to violate relativity.

Anyway.. the cat thought experiment is such a fun thought experiment to me because it specifically makes us think about a very practical issue with respect to quantum computing: decoherence. If you take his thought experiment to an extreme, it actually should be theoretically possible to create a state in which a macroscopic object (the cat) and a quantum object (the radioactive source) are indeed entangled. But that is absurd according to everything we've ever seen. So what's up? The missing concept here is decoherence -- while this state may theoretically exist, it'd decohere on timescales so small we can't even imagine. The fun connection here is that decoherence is the exact thing we're trying to fight in quantum computing. Essentially we're trying to make this thought experiment a reality for a much less complex system.

Some more on decoherence: https://en.wikipedia.org/wiki/Quantum_decoherence