# Mutually unbiased bases

Two orthonormal bases B and Bʹ of a d-dimensional complex inner-product space are called mutually unbiased if and only if Klappenecker03constructions

\forall {x \in \mathcal{B}}\ \forall{ y\in\mathcal{B'}} |\langle x|y\rangle|^2=\frac{1}{d}

### An example for d = 2

A simple example of a set of mutually unbiased bases in a 2 dimensional Hilbert space consists of the three bases composed of the eigenvectors of the Pauli matrices σx, σz and their product σxσz. The three bases are

{∣0⟩, ∣1⟩}

$$\left\{ \frac{| 0 \rangle+| 1 \rangle}{\sqrt{2}},\frac{| 0 \rangle-| 1 \rangle}{\sqrt{2}} \right\}$$

$$\left\{ \frac{| 0 \rangle+i | 1 \rangle}{\sqrt{2}},\frac{| 0 \rangle-i| 1 \rangle}{\sqrt{2}} \right\}$$

which form a set of mutually unbiased bases.