Corrigendum: Simple sufficient condition for subspace to be completely or genuinely entangled (2021 New J. Phys.23 103016)

Proof. Consider a superposition of k pure mutually orthogonal states |ϕ_i ⟩

$$\begin{equation}\vert {\Psi}\rangle =\sum\limits _{i=1}^{k}{\alpha }_{i}\vert {\phi }_{i}\rangle ,\end{equation} \tag{ 1 }$$

and recall that all the considered entanglement quantifiers can be wrapped up in a single formula

$$\begin{equation}\mathcal{E}(\vert \psi \rangle )=1-\underset{\vert \varphi \rangle \in \mathcal{S}}{\mathrm{max}}\vert \langle \varphi \vert \psi \rangle {\vert }^{2},\end{equation} \tag{ 2 }$$

where $\mathcal{S}$ is any set considered in the main text.

Due to the triangle inequality |x + y| ⩽ |x| + |y|, the expression under the maximum on the right-hand side of the above for the superposition |Ψ⟩ can be upper bounded as

$$\begin{align}\hfill \vert \langle \varphi \vert {\Psi}\rangle {\vert }^{2}& \leqslant {\left(\sum\limits _{i=1}^{k}\vert {\alpha }_{i}{\Vert}\langle \varphi \vert {\phi }_{i}\rangle \vert \right)}^{2}\hfill \\ \hfill & =\sum\limits _{i=1}^{k}\vert {\alpha }_{i}{\vert }^{2}\vert \langle \varphi \vert {\phi }_{i}\rangle {\vert }^{2}+2\sum\limits _{i< j}\vert {\alpha }_{i}{\alpha }_{j}{\Vert}\langle \varphi \vert {\phi }_{i}\rangle \langle \varphi \vert {\phi }_{j}\rangle \vert ,\hfill \end{align} \tag{ 3 }$$

which holds for any φ. Plugging this into equation (2) and using the fact that ∑_i |α_i |² = 1, we obtain

$$\begin{align}\hfill \mathcal{E}(\vert {\Psi}\rangle )& \geqslant \sum\limits _{i=1}^{k}\vert {\alpha }_{i}{\vert }^{2}\mathcal{E}(\vert {\phi }_{i}\rangle )-2\enspace \underset{\vert \varphi \rangle \in \mathcal{S}}{\mathrm{max}}\sum\limits _{i< j}\vert {\alpha }_{i}{\alpha }_{j}{\Vert}\langle \varphi \vert {\phi }_{i}\rangle \langle \varphi \vert {\phi }_{j}\rangle \vert \hfill \\ \hfill & \geqslant \sum\limits _{i=1}^{k}\vert {\alpha }_{i}{\vert }^{2}\mathcal{E}(\vert {\phi }_{i}\rangle )-2\sum\limits _{i< j}\vert {\alpha }_{i}{\alpha }_{j}\vert \underset{\vert \varphi \rangle \in \mathcal{S}}{\mathrm{max}}\vert \langle \varphi \vert {\phi }_{i}\rangle \vert \underset{\vert \varphi \rangle \in \mathcal{S}}{\mathrm{max}}\vert \langle \varphi \vert {\phi }_{j}\rangle \vert ,\hfill \end{align} \tag{ 4 }$$

where in the second inequality we have first exploited the fact that the maximum of the sum is upper bounded by the sum of maxima, and then bounded from above each maximum of products by the product of maxima. With the aid of the fact that $\underset{\vert \varphi \rangle \in \mathcal{S}}{\mathrm{max}}\vert \langle \varphi \vert {\phi }_{i}\rangle \vert =\sqrt{1-\mathcal{E}(\vert {\phi }_{i}\rangle )}$ this gives the claimed inequality. □