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If I have some state $|\psi> = \alpha |0> + \beta|1>$, I know that the probability of observing $|0>$ is $p_1 = |\alpha|^2$. Is it possible to change the probability of observing $|0>$ by some $\Delta p$ without knowing the initial state?

I've been thinking about it in terms of the Bloch sphere representation $$ |\psi> = \cos\frac{\theta}{2}|0> + e^{i\phi}\sin\frac{\theta}{2}|1>, $$ so the probability of observation is just a function of $\theta$. The problem is that the change in $\theta$ is a nonlinear function of the initial and final probability $$ \Delta \theta = 2\arccos(\sqrt{1-p_2}) - 2\arccos(\sqrt{1-p_1}). $$ There doesn't seem to be a way to effect a given $\Delta p$ unless you know the initial probability $p_1$.

I've been wondering if there's some way to "encode" the information in $\phi$ since that doesn't affect the observation probability.

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This cannot be possible. First, take $\Delta p>0$. Then this process will not work on an initial state with $p_1=1$. Then, take $\Delta p<0$; this process will not work for an initial state with $p_1=0$. It is therefore impossible to find a transformation that simply shifts the probability by a constant amount regardless of the input state.

As to the latter question, information can be encoded in $\phi$ because other measurements are sensitive to its value. It is true that measuring $|0\rangle$ and $|1\rangle$ does not give any information about $\phi$, but this information can be gleaned from a measurement in a basis $(|0\rangle\pm |1\rangle)/\sqrt{2}$, or $(|0\rangle\pm i|1\rangle)/\sqrt{2}$, etc.

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