Correction to: npj Quantum Information https://doi.org/10.1038/s41534-025-01071-4, published online 08 July 2025

The original version of this Article contained several errors:

  1. 1.

    In the paragraph below equation (10), an extra comma was removed and the expression now reads \(\alpha \in (0,\infty )\).

  2. 2.

    On page 3, in the paragraph under equation (13), the phrase ‘Notably, the right-hand side is proportional to’ has been amended to ‘Notably, the right-hand side is inversely proportional to’.

  3. 3.

    In equation (18), certain operational symbols were missing due to author oversight. The corrected equation now reads: \(\sqrt{n}\cdot {\left[\frac{{(1-F)}^{2}}{F}+\mathop{\sum }\limits_{i=1}^{N-1}{q}_{i}\right]}^{\frac{1}{2}}\ge \alpha\).

These errors have now been corrected in both the PDF and HTML versions of the Article.