Entropic Uncertainty Relations and MUBs
There is an alternative characterization of mutually unbiased bases that considers them in terms of uncertainty relations.
Entropic uncertainty relations are analogous to the Heisenberg uncertainty principle, and Maassen and Uffink found that for any two bases and :
where and and is the respective entropy of the bases and, when measuring a given state.
Entropic uncertainty relations are often preferable to the Heisenberg uncertainty principle, as they are not phrased in terms of the state to be measured, but in terms of c.
In scenarios such as quantum key distribution, we aim for measurement bases such that full knowledge of a state with respect to one basis implies minimal knowledge of the state with respect to the other bases. This implies a high entropy of measurement outcomes, and thus we call these strong entropic uncertainty relations.
For two bases, the lower bound of the uncertainty relation is maximized when the measurement bases are mutually unbiased, since mutually unbiased bases are maximally incompatible: the outcome of a measurement made in a basis unbiased to that in which the state is prepared in is completely random. In fact, for a d-dimensional space, we have:
for any pair of mutually unbiased bases and . This bound is optimal: If we measure a state from one of the bases then the outcome has entropy 0 in that basis and an entropy of in the other.
If the dimension of the space is a prime power, we can construct d + 1 MUBs, and then it has been found that
which is stronger than the relation we would get from pairing up the sets and then using the Maassen and Uffink equation. Thus we have a characterization of d + 1 mutually unbiased bases as those for which the uncertainty relations are strongest.
Although the case for two bases, and for d + 1 bases is well studied, very little is known about uncertainty relations for mutually unbiased bases in other circumstances.
When considering more than two, and less than bases it is known that large sets of mutually unbiased bases exist which exhibit very little uncertainty. This means merely being mutually unbiased does not lead to high uncertainty, except when considering measurements in only two bases. Yet there do exist other measurements that are very uncertain.
Read more about this topic: Mutually Unbiased Bases
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