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To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the use of a "circle" (ignoring the Bloch sphere) to represent a combination of |0> and |1> states rather than, for example, a system with the condition that alpha+beta=1 (a line, in terms of its geometry). This still preserves the Complementary Rule if we consider the probability of |0> to be its coefficient alpha and |1> beta respectively. Is it just easier to use a circle, because the state can be represented in polar form in terms of some angle? If you had, for example, alpha+beta=1 or alpha^3+beta^3=1 the magnitude would also change but I don't necessarily see an issue with that because the magnitude doesn't need to represent the total probability of either event occurring (1).

So what is the fundamental issue with representing a quantum state using other conditions which still preserve the probability conditions? Does it have something to do with the magnitude? Linearity? Is it just convenient?

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    $\begingroup$ Welcome to QCSE. To write an equations you can use mathjax. It is as simple as just adding dollar signs, for example $\alpha^2$ produces $\alpha^2$ $\endgroup$
    – Mauricio
    Mar 11 at 0:25
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    $\begingroup$ "So what is the fundamental issue with representing a quantum state using other conditions which still preserve the probability conditions?" The fundamental issue is that the natural world doesn't work that way. Quantum mechanics describes how nature is, not how we would like it to be. $\endgroup$
    – hft
    Mar 11 at 2:26

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The issue is that coefficients $\alpha$ and $\beta$ are complex numbers representing both probability of measuring certain basis state and so-called quantum phase. Note that the phase can be used for encoding information to it. So, it is not ephemeral artifact of quantum mechanics but real feature of Nature. It is hard to imagine the phase in classical day-to-day world, but if you imagine a particle like a wave, you can thin about the quantum phase like a phase of the wave.

It follows from mathematical foundations upon the quantum mechanics is built that probability of measuring certain state is $|\alpha|^2$ (or $|\beta|^2$). If we have two states we easily come to $|\alpha|^2+|\beta|^2 =1$. This can be generalized for any number of basis states you want, i.e. if we have a general quantum state $|\psi\rangle = \sum_{i=1}^n \alpha_i |b_i\rangle$, where $|b_i\rangle$ are basis states, we have condition $\sum_{i=1}^n|\alpha_i|^2=1$.

If you want to understand why this is true in details, please consult any textbook on quantum mechanics. What I wanted to say is that we cannot use any metric or relation we want but need to dealt with those that correspond to reality.

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The fact that you use a circle instead of a line is the fundamental defining difference between quantum mechanics and statistics. See Lecture 9 of Scott Aaronson's lecture notes.

One of his slides:

enter image description here

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  • $\begingroup$ His one slide is a bit too pithy for my taste... $\endgroup$
    – hft
    Mar 10 at 23:16

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