User:Tb519

Entropic Uncertainty Relations and MUBs
There is an alternative characterisation 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 $$B_1 = \{ |a_{i}\rangle{i=1}^d \} $$ and  $$B_2 = \{ | b_{j} \rangle {j=1}^{d} \}$$:


 * $$ H_{B_1} + H_{B_2} \geq -2\log c.$$

where $$c = max | \langle a_j | b_k \rangle |$$ and $$ H_{B_1}$$ and $$H_{B_2}$$ is the respective entropy of the bases $$B_1$$ and $$B_2$$, when measuring a given state. The fact that entropic uncertainty relations do not depend on this state often makes this a more useful uncertainty relation than the Heisenberg uncertainty principle.

The best possible lower bound for the uncertainty of two bases occurs when those bases are mutually unbaised. This is because mutually unbaised 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 we have :


 * $$ H_{B_1} + H_{B_2} \geq \log (N)$$

for any pair of mutualy unbiased bases $$B_1$$ and $$B_2$$.

If the dimension, $$d$$, of the space is a prime power, we can construct $$d+1$$ MUBs, and then :


 * $$ \sum_{k=1}^{d+1} H_{B_k} \geq (d+1) \frac{1}{2} \log(d+1)$$

which bigger than the one we would get from pairing up the sets and then using the Maassen and Uffink equation.

Thus we have a characterisation of $$d+1$$ mutually unbiased bases as those for which the bounds of the uncertainty relations are highest. If we consider uncertainty realtions to be the limit on how much we can know about a system, MUBs provide...

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.