The Theorem

Theorem 9.2 of the book concerns two states $\rho$ and $\sigma$ of one quantum system, both sent through the same quantum channel $\mathcal{E}$: the noise of a transmission line, say, acting in the same way on whichever state enters it. After the channel, the trace distance between $\mathcal{E}(\rho)$ and $\mathcal{E}(\sigma)$ is at most the trace distance between $\rho$ and $\sigma$.^✦

A Quantum System

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

Trace Distance

The trace distance $D(\rho,\sigma)$ between two states $\rho$ and $\sigma$ of $S$ is half the trace norm of their difference, $$D(\rho,\sigma) = \frac{1}{2}\,\|\rho-\sigma\|_1 = \frac{1}{2}\sum_i |\lambda_i| ,$$(1)^✦ where $\lambda_i$ runs over the eigenvalues of the operator $\rho-\sigma$.

Quantum Channels

A quantum channel $\mathcal{E}$ on $S$ is a map on the states of $S$ that is the restriction to states of a linear map $\Phi$ on the operators on the state space of $S$, where $\Phi$ is completely positive and preserves the trace: $$\mathcal{E}(\rho) = \Phi(\rho) \quad \text{for every state } \rho, \qquad \mathrm{tr}\,\Phi(A) = \mathrm{tr}\,A \quad \text{for every operator } A .$$(2)^✦

The Claim

What Is Checked

The claim is stated over a quantum channel $\mathcal{E}$ on a quantum system $S$, in the sense of (2), two states $\rho$ and $\sigma$ of $S$, and the trace distance (1).