States in Quantum Mechanics

The states aren't really important, and they aren't really physical. The fundamental thing is the operator algebra
My Physics Prof

When you're first learning quantum, you learn to think of states as "things", and operators/observables as "measurements we do to states". Given a state $\ket \psi$ and a Hermitian operator $\mathcal{O}$, we get the "expected value of measuring $\mathcal{O}$ on $\ket \psi$" by computing $\mathbb{E}_{\ket\psi}[\mathcal{O}] := \bra \psi \mathcal{O} \ket \psi$.

But it turns out we can also look at the problem from a different perspective. When you learn intro quantum, you don't actually spend that long learning about properties of states. Instead, you learn a lot about the properties of the observables. You study their commutation relations, and that sort of thing. Really, the basic objects that we study are these observables. And in fact, we can take observables to be our fundamental "things". Then, you can think of states as "ways of measuring observables".

And we can make this formal. We can say that a state is any positive, linear functional of norm 1 on the space of observables. By linear functional, I mean that it's a function that takes in operators and spits out real numbers. By positive, I mean that the functions are nonnegative on positive semidefinite operators. And by norm 1, I mean that these functions are 1 on the identity. It's pretty simple to check that the expected value function $\mathbb{E}_{\ket\psi}$ that I defined above has these properties.

Amazingly, the GNS construction tells us that ever positive linear function of norm 1 can be represented as a vector state (i.e. it is an expected value for some state vector). So the reasonable measurements that we can do to operators look like measuring state vectors! We can think of state vectors as being a convenient representation of the measurements we can do to our operators.

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