Existence Problem
Let denote the maximal number of mutually unbiased bases in the d-dimensional Hilbert space Cd. It is an open question how many mutually unbiased bases, one can find in Cd, for arbitrary d.
In general, if
is the prime number decomposition of d, where
then the maximal number of mutually unbiased bases which can be constructed satisfies
It follows that if the dimension of a Hilbert space d is an integer power of a prime number, then it is possible to find d + 1 mutually unbiased bases. This can be seen in the previous equation, as the prime number decomposition of d simply is . Therefore,
Though the maximal number of mutually unbiased bases is known when d is an integer power of a prime number, it is not known for arbitrary d.
Read more about this topic: Mutually Unbiased Bases
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