Direct sum in Eq.(25) of *A Grand Unification of Quantum Algorithms* should be changed into $\sum_\lambda$?

Question

I have a problem when I read this paper, is there someone who has occasionally read this paper and can help me solve the problem?

The problem is just about the notation used in the paper, the authors stated that $U=\left[\begin{array}{cc}\mathcal{H} & \sqrt{I-\mathcal{H}^{2}} \\ \sqrt{I-\mathcal{H}^{2}} & -\mathcal{H}\end{array}\right]$, where $\mathcal{H}=\sum_{\lambda} \lambda|\lambda\rangle\langle\lambda|$. But later, they stated again in eq.(25): $$ \begin{aligned} U &=\bigoplus_{\lambda}\left[\begin{array}{cc} \lambda & \sqrt{1-\lambda^{2}} \\ \sqrt{1-\lambda^{2}} & -\lambda \end{array}\right] \otimes|\lambda\rangle\langle\lambda| \\ &=\bigoplus_{\lambda}\left[\sqrt{1-\lambda^{2}} X+\lambda Z\right] \otimes|\lambda\rangle\langle\lambda| \\ &=: \bigoplus_{\lambda} R(\lambda) \otimes|\lambda\rangle\langle\lambda|, \end{aligned} $$ My question is, should the direct sum be changed into $\sum_\lambda$? Because the dimension in the direct sum form seems not the same as the original $U$.

Answer 1

Yes, it seems that two possible notations, either $$ \sum_\lambda R(\lambda)\otimes |\lambda\rangle\langle\lambda| $$ or $$ \bigoplus_\lambda R(\lambda) $$ have been merged (the second needs to be interpreted as 'in the basis $\{|\lambda\rangle\}$'). As you say, I think you're better off with the normal sum.