Possible label mismatch in Box 8.5 of Nielsen & Chuang — are ρ₂′ and ρ₃′ swapped?

Question

I’ve been working through Box 8.5 of Nielsen & Chuang’s textbook (Quantum Computation and Quantum Information) related to single-qubit quantum process tomography.

Specifically, in equations (8.175) and (8.176), the output density matrices $\rho_2'$ and $\rho_3'$ are constructed from linear combinations of $\mathcal{E(|+\rangle\langle+|)}$ and $\mathcal{E(|-\rangle\langle-|)}$. However, when I compare the results of those expressions to the definitions given in earlier equations—namely:

$$\rho_2=\rho_1X, \quad \rho_3=X\rho_1$$

I notice that the results from (8.175) and (8.176) actually yield $\rho_3'$ and $\rho_2'$, respectively. In other words, the labels seem to be reversed.

So my question is: Is this a labelling error in the book (i.e., $\rho_2'$ and $\rho_3'$ should be swapped in (8.175)-(8.176))?

Or is there another subtlety in the definitions or assumptions that I’m missing?

Answer 1

Indeed, it seems like the version of the book I have also bears this mistake. We can express $\rho_i$ in terms of the basis components $|0\rangle\langle0|$, $|1\rangle\langle1|$, $|+\rangle\langle+|$ and $|-\rangle\langle-|$: $$|0\rangle\langle0| = \rho_1 $$ $$|1\rangle\langle1| = \rho_4 $$ $$\begin{align} |+\rangle\langle+| &= 1/2(|0\rangle\langle0| + |1\rangle\langle1| + |0\rangle\langle1| + |1\rangle\langle0|) \\ &= 1/2(\rho_1+\rho_2+\rho_3+\rho_4) \end{align}$$ $$\begin{align} |-\rangle\langle-| &= 1/2(|0\rangle\langle0| + |1\rangle\langle1| + |0\rangle\langle1| + |1\rangle\langle0|) \\ &= 1/2(\rho_1+\rho_4-i\rho_2+i\rho_3) \end{align}$$

from these equalities we can isolate $\rho_i$: $$\rho_3 = |+\rangle\langle+| -i|-\rangle\langle-| - \frac{1-i}{2}(\rho_1+\rho_4)$$ $$\rho_2 = |+\rangle\langle+| +i|-\rangle\langle-| - \frac{1+i}{2}(\rho_1+\rho_4)$$

Giving you the equations 8.175 and 8.176 with swapped labels.

Note: there is a website that collect errors in Nielsen and Chuang but it is not up to date (see this post for more information).