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How many dimensions does an $n$-qubit system have?

What is definition of dimension for a quantum state? Is it related to the number of rows and columns of a density matrix?

My guess is that it has $2^n$ dimensions, but I am not sure about that.

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Yes, $n$ qubits are represented by a vector in $2^n$ dimensional Hilbert space (which is, in finite dimensions, just the same as a vector space).

So for $n = 1$, the system is described by a two dimensional vector space spanned by two vectors, which are $|0\rangle$ and $|1\rangle$ when in the computational basis. However, for a general quantum state it's important to note that dimensionality doesn't just equate to the number of available states. Instead, it signifies the minimal number of states required to represent all potential states within the given system.

In terms of density matrices, in an $N$ dimensional space, the density matrix has $N^2$ real parameters. So, you need $2^{2n}$ parameters to describe an $n$-qubit density matrix (you can find an explanation for this on the Physics SE).

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