1 Correction to: Quantum Information Processing (2019), 18:318 https://doi.org/10.1007/s11128-019-2430-3

The original version of this article unfortunately contained an error in the proof of Theorem 13. The correct transformation of the proof of Theorem 13 is as follows,

$$ \begin{aligned} (1 - \alpha )^{ - 1} { \log }\mathop \sum \limits_{n,k} (\lambda_{n} \mu_{k}^{\left( n \right)} )^{\alpha } & = (1 - \alpha )^{ - 1} { \log }\mathop \sum \limits_{n} (\lambda_{n} )^{\alpha } \mathop \sum \limits_{k} (\mu_{k}^{\left( n \right)} )^{\alpha } \\ & \ge (1 - \alpha )^{ - 1} { \log }\mathop \sum \limits_{n} (\lambda_{n} )^{\alpha } \\ & = {\text{S}}_{\alpha }^{K\left( \theta \right)} \left( \varphi \right) \\ \end{aligned} $$