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Operations on many qubits at the same time is the same as operation of qubits with many states.

For example the CNOT gate matrix with 1 control qubit will be the same of a "gate" acting on a qubit with 4 possible states , no?

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    $\begingroup$ what does "4 possible states" mean in this sentence? I don't know if that's what you're trying to ask, but an $n$-qubit state is essentially a single $2^n$-dimensional qudit, yes. With some underlying assumption on the type of operations and questions you're going to ask it. $\endgroup$
    – glS
    Dec 1, 2023 at 11:54

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By definition, qubit is a quantum system which has two basis states. If more states are allowed, this is called a qudit.

It is true that 2 qubits can be seen as one system with 4 basis states $\left|00\right>$, $\left|01\right>$, $\left|10\right>$, $\left|11\right>$. In general it is not $n+2$, but $2n$ — the system containing a qubit and a $n$-dimensional qudit has $2n$ basis states (of the form $\left|0k\right>$ and $\left|1k\right>$).

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  • $\begingroup$ you probably meant $2^n$ basis states, not $2n$ $\endgroup$
    – glS
    Dec 1, 2023 at 11:55
  • $\begingroup$ For $n$ qubits yes, the dimension would be $2^n$. I meant 1 qubit and 1 $n$-dimensional qudit, then it is $2n$. $\endgroup$ Dec 1, 2023 at 14:36

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