Non-ideal coin tossing

Can someone please check if the following makes sense?

We have a non-ideal coin-tossing scheme as follows. Alice and Bob know what $|0rangle,|1rangle$ are. Bob wins when the coin is 1.

1. Honest Alice commits a random bit $ain {0,1}$. The state of their joint system with Bob is $|arangle ,$ while Bob has $rho_a^B=Tr{|aranglelangle a|}$
2. Bob chooses a bit $a’in{0,1}$
3. Alice reveals her guess.
4. The coin is $c=aoplus a’$.

Probability that $c=0$ is $p_0$ and similarly, the probability that $c=1$ is $p_1$. I need to determine bias Bob can achieve, i.e. bounds on $max|p_0-p_1|$ if Bob is dishonest.

Now suppose, Bob is acting dishonestly and he wants the coin to be 1 in the end. He will try to guess what bit Alice has commited so he can choose the opposite. All he has is his reduced density matrix. He will try to distinguish $rho_a^B$ from $rho_0^B$ and from $rho_1^B$.
By Holevo-Helstrom his probability of guessing correctly is at most $$frac{1}{2}+frac{1}{4}||rho_0^B-rho_1^B||_1.$$ Suppose, we are given that $F(rho_0^B,rho_1^B)ge 1-delta.$

We have the inequalities:

$$2-2F(rho_0^B,rho_1^B)le ||rho_0-rho_1||_1le 2sqrt{1-F(rho_0^B,rho_1^B)^2}$$

Therefore,

$$1-frac{F(rho_0^B,rho_1^B)}{2}le frac{1}{2}+frac{1}{4}||rho_0^B-rho_1^B||_1le frac{1}{2}+frac{sqrt{2delta-delta^2}}{2}$$

Any feedback, any additional comments, references and suggestions are greatly appreciated.