I've been reading about the Bernoulli-factory problem and I'm particularly interested in deriving the results using the density matrix formalism, i.e., given required numbers of copies of the initial state
\begin{pmatrix} p & \sqrt{p(1-p)} \\ \sqrt{p(1-p)} & 1-p \end{pmatrix}
I'm interested in finding the set of achievable densitry matrices after quantum processing- applying quantum gates and making measurements.
I've tried finding the orbit of the state using the conjugation map but it doesn't lead to the result found in the paper by Jian, Zhang and Sun.
A few observations that I can make are:
- The eigenvalues are same for all elements of the set
- If we start with a tensor product of
xcopies of the intial state, then the final state has functions of degreexon its diagonals, which are linear compination of p, 1-p and $\sqrt{p(1-p)}$
Now, as a final step I want to prove that the ratio of any two elements in the resultant matrix is of the form
$\frac{a_{ii}}{a_{jj}} = (\frac{g_1(p)}{g_2(p)}\sqrt{\frac{p}{1-p}} + \frac{g_3(p)}{g_4(p)})^2$, where $g_i(p)$ are polynomials in p. Note that we have an extra constraint that the ratio being square of the term. Any ideas on how to prove this result?
