Mutually unbiased bases

Two orthonormal bases \mathcal{B} and \mathcal{B}' of a d-dimensional complex inner-product space are called mutually unbiased if and only if [1]

math

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

\left\{ | 0 \rangle,| 1 \rangle \right\}
\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.

See also

  • See the paper by Bengtsson[2] for a review.

References

  1. ↑ Klappenecker, A. & Roetteler, M.(2003). Bib
  2. ↑ Bengtsson, I.(2006) Three ways to look at mutually unbiased bases. IN ArXiv, . Bib
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