# How to change the probability of observation by some set amount when initial probability is unknown?

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.

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.