# Exercise 1

## from Revision class solutions

### Exercise 1.

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 $$\mathcal{B}(\mathcal{H})$$.
2. Using the above identity, prove that $D(\rho,\sigma) = \max_P \operatorname{tr}(P(\rho -\sigma))$ where the maximisation is over all projection operators $$P\in \mathcal{B}(\mathcal{H})$$.
3. 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.
4. 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 $$\Lambda(\rho)$$.