Daily Archives: February 6, 2008

Qutrit Mutually Unbiased Bases (MUBs)

Mutually Unbiased Bases are sets of bases for a Hilbert space that are “unbiased:” the transition probabilities between any two states from different bases are equal. For a Hilbert space of dimension 3 (i.e. qutrits), the transition probability is 1/3. The operator space of a Hilbert space of dimension n is n^2, in this case the operator space has 9 dimensions. Each base consists of 3 quantum states. It turns out that a base uses up 3-1 = 2 degrees of freedom of the operator space, and the scalar part of the operator space is shared by all. So for a 3-dimensional Hilbert space, there are at most four mutually unbiased bases. In this post I will derive a set of four such bases.
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