Mutually unbiased bases: common states, completion, and extension obstructions
The research investigates pure states that yield uniform outcomes across multiple mutually unbiased measurements (MUBs). It proves methods to complete these common states into full orthonormal bases, especially for Fourier pairs. Crucially, it identifies specific mathematical obstructions—like amplitude-profile issues in dimension six or Hadamard matrix neighborhoods—that prevent the extension of a set of MUBs to a larger set.