The Exercise

A qubit, the principal system, interacts with a second qubit, its environment, which starts in the state $|0\rangle$. The interaction is a controlled flip with the system as the control: the environment is flipped when the system is in $|1\rangle$ and left alone when the system is in $|0\rangle$. The environment is then discarded. The exercise asks for the quantum operation this process induces on the system, in the operator sum representation.

The library defines that operation from the interaction and the environment’s initial state, and states its operator sum form.

A Qubit and Its Environment

Let $H$ be the state space of a qubit and $|0\rangle$, $|1\rangle$ its computational basis. The system and the environment each have state space $H$, and together they have $H \otimes H$, with the system’s factor first.

For $i$ in $\{0,1\}$, the projector $P_i$ onto $|i\rangle$ is the rank one operator $|i\rangle\langle i|$ on $H$.^✦ The bit flip $X$ is the operator on $H$ that exchanges the two basis vectors: $$X = |0\rangle\langle 1| + |1\rangle\langle 0|.$$(1)^✦

The controlled flip $U$ is the operator on $H \otimes H$ given by $$U = P_0 \otimes I + P_1 \otimes X,$$(2)^✦ with $I$ the identity on $H$.

The dilation $V$ of the controlled flip is the linear map from $H$ to $H \otimes H$ that adjoins the environment in $|0\rangle$ and then applies $U$: $$V|\psi\rangle = U\bigl(|\psi\rangle \otimes |0\rangle\bigr)$$(3)^✦ for every vector $|\psi\rangle$ in $H$.

The dephasing operation $\mathcal{E}$ is the map that sends each operator $\rho$ on $H$ to the partial trace $\operatorname{tr}_{\mathrm{env}}$ over the environment, the second factor, of $V \rho V^\dagger$: $$\mathcal{E}(\rho) = \operatorname{tr}_{\mathrm{env}}\bigl(V \rho V^\dagger\bigr).$$(4)^✦

The Claim

What Is Checked

The claim is stated over the dephasing operation $\mathcal{E}$ of (4), which rests on the dilation $V$ of (3) and through it on the controlled flip $U$ of (2), built from the bit flip $X$ and the projectors $P_0$ and $P_1$.

What the library proves about these constructions is the identity (5), for every operator $\rho$ on $H$ and not only for density operators.