The Exercise

A channel $\mathcal{E}$ acts on a quantum system $S$ in one of two ways. With probability $p$ it discards the input and prepares the system in a fixed state $\rho_0$ ; with probability $1-p$ it applies another channel $\mathcal{E}'$ . An input state $\tau$ therefore leaves as $\mathcal{E}(\tau) = p\,\rho_0 + (1-p)\,\mathcal{E}'(\tau), \qquad 0 < p \le 1.$ (1) The exercise asks to show that $\mathcal{E}$ is strictly contractive in the trace distance, and that it has exactly one fixed point, a state it leaves unchanged.

A Quantum System

A quantum system $S$ is a physical system 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 $S$ .^✦

For a weight $p$ with $0 \le p \le 1$ , the mixture of two states $\rho$ and $\sigma$ of $S$ is the state $p\,\rho + (1-p)\,\sigma .$ (2)^✦ The output (1) is the mixture of $\rho_0$ and $\mathcal{E}'(\tau)$ with weight $p$ .

A quantum channel $\mathcal{E}$ on $S$ is a map from the states of $S$ to the states of $S$ that agrees on states with a completely positive, trace preserving linear map $\Phi$ on the operators on the state space of $S$ : $\mathcal{E}(\rho) = \Phi(\rho) \quad \text{for every state } \rho, \qquad \operatorname{tr} \Phi(A) = \operatorname{tr} A \quad \text{for every operator } A.$ (3)^✦

Distance Between States

The trace distance between two states $\rho$ and $\sigma$ of $S$ is half the trace norm of their difference, $D(\rho,\sigma) = \tfrac{1}{2} \operatorname{tr} \lvert \rho - \sigma \rvert .$ (4)^✦

A map $\mathcal{E}$ on the states of $S$ is strictly contractive when, for any two distinct states, the trace distance between their images is strictly smaller than the trace distance between them.^✦

The Claim

The library states the two parts of the exercise as two theorems.

Theorem 1. Let $S$ be a quantum system, $\rho_0$ a state of $S$ , $p$ a real number with $0 < p \le 1$ and $\mathcal{E}'$ a channel on $S$ , and let $\mathcal{E}$ be the map on the states of $S$ given by (1). Then $\mathcal{E}$ is strictly contractive: for any two distinct states $\rho$ and $\sigma$ of $S$ , $D\bigl(\mathcal{E}(\rho), \mathcal{E}(\sigma)\bigr) < D(\rho, \sigma).$ (5)^✦

Theorem 2. Let $S$ , $\rho_0$ and $p$ be as before, and let $\mathcal{E}$ and $\mathcal{E}'$ be channels on $S$ related by (1) for every state $\tau$ . Then $\mathcal{E}$ has exactly one fixed point: there is one and only one state $\rho$ of $S$ with $\mathcal{E}(\rho) = \rho .$ (6)^✦

What Is Checked

Both theorems are stated over a quantum system $S$ and its states, the channels $\mathcal{E}$ and $\mathcal{E}'$ , the mixture (2) of two states, and the trace distance (4), in which strict contractivity is measured.

The library proves that the mixture (2) of two states is again a state, so that (1) gives a state for every input $\tau$ .