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This question refers to the following point made in Susskind's book Quantum Mechanics- The Theoretical Minimum:
In the classical world, the relationship between the state of a system and the result of a measurement on that system is very straightforward. In fact, it’s trivial. The labels that describe a state (the position and momentum of a particle, for example) are the same labels that characterize measurements of that state. To put it another way, one can perform an experiment to determine the state of a system. In the quantum world, this is not true. States and measurements are two different things, and the relationship between them is subtle and nonintuitive.
I'm not sure I understand the last line since it seems to imply that states and measurements are not "different things" in the classical realm. Are both of them the "same" in the sense that they both refer to a point in the system's phase space?
Of course, any particular state would uniquely specify a measurement in classical mechanics and conversely a set of measurements would uniquely specify a state. Such a correspondence doesn't exist in quantum mechanics. So is that what the author means by "states and measurements are two different things"?
Finally, what does the author mean by "labels"? Do they simply refer to the values of the various degrees of freedom of the system?
Now, coming to the next part:
Attached to the electron is an extra degree of freedom called its spin. [...] We can and will abstract the idea of a spin, and forget that it is attached to an electron. The quantum spin is a system that can be studied in its own right.
Why is the spin being called a "system"? Isn't a system supposed to be something physical instead of a mathematical abstraction? (And it's defined as a degree of freedom in the first place- from what I understand, a degree of freedom is meant to characterize a physical system.)