Mutually Unbiased Bases - Methods For Finding Mutually Unbiased Bases - Hadamard Matrix Method

Hadamard Matrix Method

Given that one basis in a Hilbert space is the standard basis, then all bases which are unbiased with respect to this basis can be represented by the columns of a complex Hadamard matrix multiplied by a normalization factor. For d = 3 these matrices would have the form

U = \frac{1}{\sqrt{d}} \begin{bmatrix} 1 & 1 & 1 \\ e^{i \phi_{10}} & e^{i \phi_{11}} & e^{i \phi_{12}} \\ e^{i \phi_{20}} & e^{i \phi_{21}} & e^{i \phi_{22}} \end{bmatrix}

The problem of finding a set of k+1 mutually unbiased bases therefore corresponds to finding k mutually unbiased complex Hadamard matrices.

An example of a one parameter family of Hadamard matrices in a 4 dimensional Hilbert space is

H_4(\phi) = \frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & e^{i\phi} & -1 & -e^{i \phi} \\ 1 & -1 & 1 & -1 \\ 1 & -e^{i\phi} & -1 & e^{i\phi} \end{bmatrix}

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