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 \(\Lambda(\rho)\).