# What are dimensions/states in a quantum system?

Since I am just starting out in quantum physics, posing meaningful questions is still hard for me, so please keep that in mind.

First of all, are the dimensions in a quantum system the same as states? For example a qubit is a quantum system with two states, so if we represent this quantum system as a Hilbert space, will it have two dimensions? If so, doesn't it have more than two states, since any two dimensional vector within this Hilbert space represents a state, no?

Secondly, what exactly are states? What does it mean if a quantum system has only two states? How many states does a "real" quantum system have, i.e. the quantum system we live in?

The possible states of a quantum system are modelled as vectors in a Hilbert space. In finite dimension, this is the same as just a vector space. It follows that the system has as many linearly independent states as the dimension of the vector space, just by definition. You can always orthonormalize a basis of linearly independent vectors, so usually you would say that a system has as many orthogonal states as the dimension of the vector space.

So by "two states system" people usually mean that the system that can be in the state

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$

where $$|0\rangle$$ and $$|1\rangle$$ are orthogonal, hence it lives on a two dimensional vector space spanned by these two vectors.

Why do we care about orthogonality? Orthogonal states are how we encode the fact that they're "different" states, i.e. they're perfectly distinguishable. Consider the observable

$$S_z=|0\rangle\langle0|-|1\rangle\langle 1|$$

and suppose you know somehow that your system is somehow either in the state $$|0\rangle$$ or in the state $$|1\rangle$$, but you don't know which. Call the state $$|\psi\rangle$$. If you measure $$S_z$$, you will observe $$1$$ (the first eigenvalue) with probability $$\langle 0|\psi\rangle$$ and $$-1$$ with probability $$\langle 1|\psi\rangle$$. That is, if the system is in the state $$|0\rangle$$, you will observe $$1$$ with certainty, and viceversa if it is in the state $$|1\rangle$$, you will observe $$-1$$ with certainty.

But what if now you know that your system is either in the state $$|0\rangle$$ or in the state $$1/\sqrt{2}(|0\rangle+|1\rangle)$$? Now if you measure the same observable, and the system is actually in the second state, you might observe the result $$1$$ with probability $$1/2$$, hence upon observing the result $$1$$, you cannot be sure of what the state of the system is. These two states are not orthogonal, and hence they cannot be perfectly distinguishable. The second state has a component along the first.

So orthogonal states are a good way to keep track of what possible distinguishable states your system can be in.

Usually two states systems are idealizations of real systems, you might have for example an atom where the state $$|0\rangle$$ is the ground state, and the state $$|1\rangle$$ is an excited state, and you're reasonably certain the atom won't go to higher excited states. You can find all sorts of dimensionalities in real systems, for example you can think of photons in some cavity as an infinite dimensional space, where $$|n\rangle$$ corresponds to a state with $$n$$ photons in the cavity.

Dimension is not really the same as the number of states available. It merely means the minimum number of states you need for the given system to describe all possible states. For example: Consider a particle in a $$2D$$ state space, that means that all possible states of the particle can be described as a linear combination of $$2$$ linearly independent states from that space: $$|\Psi\rangle_{gen}=c_1|\Psi\rangle_1+c_2|\Psi\rangle_2$$ where $$c_1$$ and $$c_2$$ are complex numbers and are different for different states.

In quantum mechanics, knowing a quantum state means knowing all that you can know about the system. When you study potential wells later, you will discover that for the ground state in an infinite potential well, the probability of the particle being found at the middle of the well is higher than any other point along with certain other interesting things. This is all that you can know about the particle. You can't question where it is if it is in the ground state, you can only question the probabilities. The point to emphasize is that the probabilistic nature is (as per the most accepted interpretation of QM) not something because of your lack of knowledge, but that is just how nature works at the microscopic level.