# Confusion over HSW theorem depicted in Nielsen and Chuang

On page 560, it states that

$$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}\varepsilon(|{\psi}\rangle\langle{\psi}|)-\frac{1}{2}\varepsilon(|{\varphi}\rangle\langle{\varphi}|)).$$

However, shouldn't this be

$$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}S(\varepsilon(|{\psi}\rangle\langle{\psi}|))-\frac{1}{2}S(\varepsilon(|{\varphi}\rangle\langle{\varphi}|)).$$

as on the same page it states that if $$\varepsilon(|{\psi_{j}}\rangle\langle{\psi_{j}}|) = p|{\psi_{j}}\rangle\langle{\psi_{j}}|+(1-p)\frac{I}{2}$$ then $$S(\varepsilon(|{\psi_{j}}\rangle\langle{\psi_{j}}|)) = H(\frac{1+p}{2})$$, so $$C(\varepsilon)=1-H(\frac{1-p}{2})$$, which can't come about unless it's the entropy of the two states, not just the channel acting on them, or am I misreading this completely?

• I think it has to be a typo, the term with $S(\cdot)$ is a scalar while the output of $\mathcal{E}(\cdot)$ is a state! So I think you're right, there is a $S(\cdot)$ missing. Jun 20 '20 at 0:28

## 1 Answer

Adding my comment as an answer: I think this is a typo. The equation contains a scalar, namely $$S(\cdot)$$ and an operator, the output of the channel $$\mathcal{E}(\cdot)$$ in linear combination (their dimensions do not match). Therefore, this has to be a typo (there are no hidden Identity operators in the scalar terms as is easy to check).

• Alright, thanks, I wasn't sure if I was completely misinterpreting the meaning of the theorem. Jun 20 '20 at 14:39