The Exercise

A quantum system is prepared in one of two states, $\rho$ or $\sigma$, and then evolves as a closed system under a unitary $U$, the same whichever state it started in. The exercise is to show that the fidelity of the two states, which measures how close they are, is the same after the evolution as before it.^✦

A Quantum System

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$; pure and mixed states are both states in this sense.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $$U^\dagger U = U U^\dagger = 1,$$(1)^✦ and it describes a closed evolution of $S$. The evolution of a state $\rho$ of $S$ under $U$ is the state $$\rho \mapsto U \rho \, U^\dagger .$$(2)^✦

Fidelity

The fidelity of two states $\rho$ and $\sigma$ of $S$ is the nonnegative real number $$F(\rho,\sigma) = \operatorname{tr} \sqrt{\sqrt{\rho}\,\sigma \sqrt{\rho}} ,$$(3)^✦ the trace taken over the state space of $S$. Here the square root of a positive operator is taken eigenvalue by eigenvalue: it has the same eigenvectors, and each eigenvalue is replaced by its nonnegative square root.

The Claim

What Is Checked

The claim is stated over the fidelity (3) of two states of $S$ after their evolution (2) under one unitary of $S$.

The library proves that conjugating a density operator by a unitary gives a density operator, so that $U \rho \, U^\dagger$ and $U \sigma U^\dagger$ are states of $S$ and the left side of (4) is the fidelity of two states.