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Isn't the dot product already the squared magnitude?pic.twitter.com/1briCwprrz
hmm no for instance a*b can be negative or positive |a*b|^2 is real positive definite
Ah, so |𝜙⟩ and |-𝜙⟩ are indistinguishable. Reminds me of quaternion double cover of 3D orientations. Square root of scaling factor makes sense there, since you need sandwich product to apply rotation which has side effect of applying any scaling twice.
Actually, for normalized states |𝜙⟩ and Exp(i θ) |𝜙⟩ are indistinguishable for any phase θ. These equivalence classes are often called rays, and the space of such rays correspond to the space of pure quantum states. This is very similar to Projective spaces.
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