Hilbert space

OK, lay it out for me.

Well, the framework of quantum theory operates on strict, non-random mathematical structures, like vectors rotating in a multidimensional mathematical arena called Hilbert space.

Sounds very precise.

Highly precise.

So no probability at all?

None.

Instead, she took the mathematical structure of quantum mechanics, those state vectors and Hilbert space we mentioned, and simply assigned a generic mathematical function, just a neutral number, to each possible measurement outcome.

OK, let me make sure I'm following the mathematical translation here.

But the math suggests extensions of this phenomenon where an entire region of possibilities remains mathematically favored over time.

To understand the quantum birthplace, we have to introduce Hilbert space.

Oh, boy.

More space.

I know.

But earlier we talked about phase space, which maps position and momentum.

Hilbert space is an even more abstract mathematical arena where quantum states live.

So what does it map?

I always thought of covariance as just, you know, a statistical formula you plug into Excel to see if two stocks move together.

Right, but the text reframes random variables with a mean of zero as literal vectors in a mathematical space, specifically a Hilbert space.

Let me make sure I'm visualizing this correctly.

If the random variables are vectors, the origin point, zero, is their mean.

Wait, a projection?

Yes.

Remember the Hilbert space of random variables we discussed earlier? Imagine you have a random variable y whose outcome you want to guess.

It's a vector pointing off into space.

My, uh, cup is either on the table or it's off the table.

Um, however, a qubit init- is technically mathematically a ray in the Hilbert space, a complex two to the n-dimensional Hilbert space.

Um, in other words, it's a state vector who can exist between two levels or any combination between those two levels, two computational basis states.

So essentially a qubit is an object, mathematical object that eventually becomes a physical object instantiated through superconductors, the same way a classical bit gets instantiated, same way that its mathematical object being instantiated through Boolean logic, through, uh, regular semiconductors.

It is the smallest quantum mechanical system you can have, which is two level quantum system.

Uh, and so the math connection though, this is super interesting, um, since we're talking about that.

Um, I mentioned that it's a ray in the Hilbert space, two to the n-dimensional Hilbert space.

The Hilbert space was discovered by David Hilbert.

Um, so at this point, if I remember correctly, the Organians are trying to find a way to break through the thought screen because their godlike abilities are currently trapped on their planet.

Um, the anti-thought shield is a problem for them, and the energy they use to break through comes directly from Hilbert space, which is a, uh, thing that Blish talked about earlier in the novel.

It's a physics concept.

Yeah, it's a mathematical model.