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​ 1 ​ ( ∣ ↑ ⟩ + i ∣ ↓ ⟩) . Then $\langle$ f|$\psi$$\rangle$ = $\tfrac12(1-i)$ and $\langle$ f|$\sigma_z$|$\psi$$\rangle$ = $\tfrac12(1+i)$ , so the weak value of $\sigma_z$ is $\langle \sigma_z \rangle_w = \frac{\tfrac12(1+i)}{\tfrac12(1-i)} = i$. $\sigma_z$ has eigenvalues $\pm$1 only, yet its weak value here is purely imaginary, not merely outside the eigenvalue range but off the real line entirely. This is an exact, closed-form evaluation of the defining ratio, not a measured or simulated number, stated here only to make concrete what “anomalous weak value” means before the same word is applied to a spatial coordinate rather than a spin component.

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