The Exercise

Two states $\rho$ and $\sigma$ of a quantum system $S$ are sent through the same quantum channel, a physical process acting on $S$, noisy or not. The fidelity measures how close two states are.

A Quantum System

A quantum system $S$ is described by its state space, a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a density operator on the state space of $S$: a positive operator of unit trace.^✦

The Fidelity

The fidelity of two states $\rho$ and $\sigma$ of $S$ is the nonnegative real number $$F(\rho,\sigma) = \operatorname{tr}\sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}},$$(1)^✦ where the square root of a positive operator is the positive operator with the same eigenvectors and the square roots of its eigenvalues.

Quantum Channels

A quantum channel $f$ on $S$ is a map taking states of $S$ to states of $S$ that agrees on states with a completely positive, trace preserving linear map $\Phi$ on the operators of the state space of $S$: $$f(\rho) = \Phi(\rho) \ \text{for every state } \rho, \qquad \operatorname{tr}\Phi(A) = \operatorname{tr}A \ \text{for every operator } A.$$(2)^✦

The Claim

What Is Checked