Mutually unbiased bases

Remarks

Example: in $\C^2$, take the standard basis $(\lvert 0 \rangle, \lvert 1 \rangle)$, the basis $(\lvert 0 \rangle \pm \lvert 1 \rangle)/\sqrt2$ and the basis $(\lvert 0 \rangle \pm i\lvert 1 \rangle)/\sqrt2$. These are the eigenbases of the Pauli matrices $Z$, $X$ and $Y$. Each inner product between vectors of different bases has modulus $1/\sqrt2$, so its square is $1/2 = 1/d$.

Non-example: the standard basis and the basis $(\cos t\, \lvert 0 \rangle + \sin t\, \lvert 1 \rangle, -\sin t\, \lvert 0 \rangle + \cos t\, \lvert 1 \rangle)$ with $t = \pi/6$. Here $\abs{\langle 0 \vert (\cos t\, \lvert 0 \rangle + \sin t\, \lvert 1 \rangle)}^2 = \cos^2 t = 3/4 \ne 1/2$.

In $\C^d$ the standard basis and the Fourier basis $f_k = d^{-1/2}\sum_j e^{2\pi i jk/d} e_j$ are mutually unbiased. A family of mutually unbiased bases of $\C^d$ has at most $d + 1$ members. When $d$ is a prime power, a family of $d + 1$ members exists. For other $d$, such as $d = 6$, the largest size is unknown.