### Exercise 1.

- Define the trace distance \(D(\rho,\sigma)\) between two states \(\rho,\sigma\in \mathcal{D}(\mathcal{H})\) and prove that it can be expressed in the form: \[
D(\rho,\sigma) = \frac{1}{2}(\operatorname{tr}Q + \operatorname{tr}R)
\] where \(Q\) and \(R\) are suitably defined positive semi-definite operators in \(\mathcal{B}(\mathcal{H})\).
- Using the above identity, prove that \[
D(\rho,\sigma) = \max_P \operatorname{tr}(P(\rho -\sigma))
\] where the maximisation is over all projection operators \(P\in \mathcal{B}(\mathcal{H})\).
- Further, prove that \[
D(,\rho,\sigma) = \max_T \operatorname{tr}(T(\rho-\sigma))
\] where the maximisation is over all positive semi-definite operators \(T\in \mathcal{B}(\mathcal{H})\) with eigenvalues less than or equal to unity.
- Let \(\rho\) be a quanutm state and \(\Lambda\) be a linear completely positive trace-preserving map. Prove that \[
F_e (\rho,\Lambda) \leq ( F(\rho,\Lambda(\rho)))^2
\] where \(F_e(\rho,\Lambda)\) denotes the entanglement fidelity, and \(F(\rho,\Lambda(\rho))\) denotes the fidelity of the states \(\rho\) and \(\Lambda(\rho)\).