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Not a duplicate. Question and answers given elsewhere do not answer this one.

Context. Scott Aaronson, lecture 2, course 6.896 Quantum Complexity Theory, september 2008 says that a matrix such as $$\begin{bmatrix}1&0\\0&-1\end{bmatrix}$$ maps the state $\alpha|0\rangle + \beta|1\rangle$ to $\alpha|0\rangle - \beta|1\rangle$.

He then asks

What is in between these two states? To have a continuous unitary transformation between these two states requires the use of complex numbers.

Question. Could you elaborate this conclusion with an example? I can't quite see the deduction. I can see why we want it to be continuous: the Schrödinger equation is a differential equation with respect to time, which therefore requires time to be continuous. So if that matrix maps a state into another one, we may ask what happened in between the change of state. And how does that require complex numbers?

Source: enter image description here

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    $\begingroup$ Thank you @FDGod for showing me how to write matrices in LaTeX and how to use a proper right-angled bracket. I appreciate it. $\endgroup$ Nov 11, 2023 at 19:04

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We are told that there is an operation $$ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} \alpha \\ \beta \end{bmatrix}=\begin{bmatrix} \alpha \\ -\beta \end{bmatrix}. $$ We are now asked to speculate about whether there are intermediate operations, what that means for intermediate states, and whether we are forced to include complex numbers.

If operation is continuous, I can stop somewhere and later restart. So, my evolution would be described by two matrices $M_1$ and $M_2$. $$ M_2M_1=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $$ I'm imposing that $M_1$ and $M_2$ are neither identity or the target. To simplify the maths, consider the specific possibility that we stopped half way, which would suggest we can set $M_1=M_2$. So, no you're asking what $2\times 2$ matrices $M_1$ there are such that $$ M_1^2=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. $$ Try solving it. You'll find the only solutions contain complex numbers. (If you're not so happy with the $M_2=M_1$ special case, you'll probably need to introduce the extra restriction that the matrices are unitary, so if you want them to be real, they'll look something like $$ M_1=\begin{bmatrix} \sqrt{p} & \sqrt{1-p} \\ \pm\sqrt{1-p} & \mp\sqrt{p} \end{bmatrix}, $$ and you'll still find there's no solution.

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  • $\begingroup$ Omg, thanks a lot. I can finally see why we need them with the clarity that we should. Eternally grateful --- sincerely. $\endgroup$ Nov 10, 2023 at 21:03
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    $\begingroup$ Thank you for the example! It turns out there's also a theory why, in an even more general sense, you can't describe quantum computing/physics without complex numbers even if you use another systems e.g. quaternions etc. The authors start off by stating some assumptions (which if violated have other strange consequences, so they seem reasonable) and design a Bell-type thought experiment with 3 parties. They then go on to show that the results of this experiment cannot be described with without complex numbers nature.com/articles/s41586-021-04160-4 $\endgroup$ Nov 11, 2023 at 9:27

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