  Fluctuation theorems impose fundamental bounds in the statistics of the
entropy production, with the second law of thermodynamics being the most
famous. Using information theory, we quantify the information of entropy
production and find an upper tight bound as a function of its mean from the
strong detailed fluctuation theorem. The bound is given in terms of a maximal
distribution, a member of the exponential family with nonlinear argument. We
show that the entropy produced by heat transfer using a bosonic mode at weak
coupling reproduces the maximal distribution in a limiting case. The upper
bound is extended to the continuous domain and verified for the heat transfer
using a levitated nanoparticle. Finally, we show that a composition of qubit
swap engines satisfies a particular case of the maximal distribution regardless
of its size.
