2. Shifting certainty
The state of a single qubit is characterized by three numbers, \langle \sigma^x \rangle, \langle \sigma^y \rangle and \langle \sigma^z \rangle,
defined as
\langle \sigma^\alpha \rangle = p^\alpha_0 - p^\alpha_1,
where p^z_0 is the probability of the outcome \texttt{0} for a Z measurement, and so on.
For any single qubit superposition |\psi\rangle = c_0 |0\rangle + c_1 |1\rangle,
$
\langle \sigma^x \rangle^2 + \langle \sigma^y \rangle^2 + \langle \sigma^z \rangle^2 = 1
$
Verify this for the following states.
- (a)
|\psi\rangle = \cos \theta \, |0\rangle + \sin \theta \, |1\rangle
- (b)
|\psi\rangle = \frac{1}{\sqrt{2}} \, \left( |0\rangle \,+\, e^{i \phi} |1\rangle \right)
3. A useful matrix
Find the 2\times2 matrix M such that
p^z_0 - p^z_1 = \langle \psi | M | \psi \rangle \,\,\, \forall |\psi\rangle
