\section*{Mutually Unbiased Bases}

Mutually unbiased bases (MUBs) play an important role in quantum information theory, quantum cryptography, and the foundations of quantum mechanics. Let \(\mathcal{H}=\mathbb{C}^d\) be a \(d\)-dimensional Hilbert space. Two orthonormal bases
\[
\mathcal{B}_1=\{ |e_1\rangle,\ldots,|e_d\rangle \}, \qquad
\mathcal{B}_2=\{ |f_1\rangle,\ldots,|f_d\rangle \}
\]
are called \textbf{mutually unbiased} if for all \(i,j\)
\[
|\langle e_i , f_j \rangle|^2=\frac{1}{d}.
\]

Intuitively, if a quantum system is prepared in a state belonging to one basis, then a measurement in the other basis produces completely random outcomes. A collection of orthonormal bases is called mutually unbiased if every pair of bases in the collection satisfies the above condition.

\section*{Known Results}

In a \(d\)-dimensional Hilbert space the number of mutually unbiased bases is at most
\(d+1\). It is known that a maximal family of \(d+1\) mutually unbiased bases exists whenever
\[
d=p^n
\]
is a power of a prime. This result was established in the works of
\begin{itemize}
\item I.~D.~Ivanovic (1981) \cite{Ivanovic1981},
\item W.~K.~Wootters and B.~D.~Fields (1989) \cite{WoottersFields1989},
\end{itemize}
using constructions based on finite fields and discrete Fourier transforms over the finite field \(\mathbb{F}_{p^n}\). For dimensions that are not prime powers the problem remains open. In particular, the existence of \(d+1\) mutually unbiased bases in dimension
\[
d=6
\]
is a well-known open problem in quantum information theory. For a comprehensive overview of these open existence problems and various construction methods, see the extensive review by Durt, Englert, Bengtsson, and Zyczkowski \cite{Durt2010}.

\begin{thebibliography}{9}

\bibitem{Ivanovic1981}
I.~D.~Ivanovic,
``Geometrical description of quantal state determination,''
\textit{Journal of Physics A: Mathematical and General},
14(12), 3241--3245 (1981).
doi:10.1088/0305-4470/14/12/019

\bibitem{WoottersFields1989}
W.~K.~Wootters and B.~D.~Fields,
``Optimal state-determination by mutually unbiased measurements,''
\textit{Annals of Physics},
191(2), 363--381 (1989).
doi:10.1016/0003-4916(89)90322-9

\bibitem{Durt2010}
T.~Durt, B.-G.~Englert, I.~Bengtsson, and K.~Zyczkowski,
``On mutually unbiased bases,''
\textit{International Journal of Quantum Information},
8, 535--640 (2010).
doi:10.1142/s0219749910006502

\end{thebibliography}