Derivation of Equation $8.7$ in Nielsen Chuang

Question

Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358)

$$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\dagger}\rbrack). \tag{8.6} \label{eq:sp1}$$

And then they say in page 359 that initially $\rho_{env} = |0\rangle\langle0|$ and then we apply $U$ to the combined state.(here $U$ is CNOT). The equation \eqref{eq:sp1} becomes (after plugging these values)

$$ \mathcal{E}(\rho) = P_{0}\rho P_{0} + P_{1}\rho P_{1} \tag{8.7} \label{eq:sp2}$$ where $P_{m}=|m\rangle\langle m|$.

How are they arriving at \eqref{eq:sp2}?

Answer 1

Just plug in all of the relevant stuff you state in the question, i.e. $$ U = |0\rangle \langle 0 | \otimes I + |1 \rangle \langle 1 | \otimes X $$ and $$ \rho_{\mathrm{env}} = |0\rangle \langle 0 |. $$ Then expand and simplify $$ \begin{aligned} \mathcal{E}(\rho) &= \mathrm{Tr}_{\mathrm{env}}[(P_0 \otimes I + P_1 \otimes X)(\rho \otimes |0\rangle \langle 0 |) (P_0 \otimes I + P_1 \otimes X)] \\ &= \mathrm{Tr}_{\mathrm{env}}[(P_0 \otimes I + P_1 \otimes X)(\rho P_0 \otimes |0\rangle \langle 0| + \rho P_1 \otimes |1\rangle \langle 0|)] \\ &= \mathrm{Tr}_{\mathrm{env}}[(P_0\rho P_0 \otimes |0\rangle \langle 0| + P_0\rho P_1 \otimes |1\rangle \langle 0| + P_1\rho P_0 \otimes |1\rangle\langle 0| + P_1 \rho P_1 \otimes |1\rangle\langle1|] \\ &= P_0 \rho P_0 + P_1 \rho P_1. \end{aligned} $$