Reply to Comment on ‘Product states and Schmidt rank of mutually unbiased bases in dimension six’

Abstract McNulty and Weigert (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908–25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>×</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> complex Hadamard matrix whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> complex Hadamard submatrices of any H 2 -reducible matrix is not <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>10</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mn>16</mml:mn> <mml:mo>,</mml:mo> <mml:mn>18</mml:mn> </mml:mrow> </mml:math> . We also put forward some possible directions for further development.