Quantumania 1 - Say No to Blurry Arrows

I want to make some notes about QM, and in regard to how useful they are likely to be to anyone else, I’ve decided to name them after a widely hated Marvel film.

Away With Wishy Washy Wobbliness

Popular accounts of quantum mechanics are plagued by a need to get across how weird everything is, and the language and imagery leans particularly on the idea that everything is fundamentally blurry and uncertain, so nothing can be pinned down.

So it is perhaps surprising that, at the most fundamental level, QM actually says nothing of the sort. The first postulate of QM is that the state of any system can fully represented by a unit vector, though not in physical space, rather in a specially-defined “state space”, each vector representing one state of all the possible states.

The circle is important: it’s the unit circle, radius 1, and all state vectors reach from the origin to that circle, like the needle of an old fashion dial meter. Only the direction matters. The other thing that’s important is the sharpness and definiteness of the arrow. Because this is QM, can the arrow be blurry? Or can the state of the system be “in two states at once?”

No! Absolutely not. There are other situations considered by QM in which representations of blurriness of this ilk can be appropriate, but they are not the fundamental representation we’re talking about here. The state is singular; to say “the state of the system is that it is in multiple states at once” is just self-contradiction. The state of the system is always one precise, sharply defined vector. Yes, whether you’re looking at it or not.

This is still true with entanglement involved, because if two systems are entangled then they are really one isolated system described by its own state space, and that system must be in exactly one state from that space. The only problem is that if we follow this reasoning to its ultimate conclusion, the only truly isolated system is the entire universe. So we’ll have to figure out what the state of a compound system tells us about the subsystems of which it is composed, and that is going to somewhat blurry.