As I understand, in physics, 'information' is closely tied to thermodynamic entropy. Does this relationship imply that if the Universe expands and ends in 'heat death' (maximum entropy?) that it reaches a state of maximum information (so at the Big Bang it had very low entropy and information content)? I sense this is incorrect given the principle of conservation of information, so what does an increase in entropy in the Universe imply for its information?
I think using entropy when you are inside a system is meaningless, entropy describe the number of possible state that a system can have giving some "global information"
That is to say, if you have an isolated box, and only know that you have 10 particles inside and that the total energy inside the box is 100 joules, you can't know what are the exact configuration of what is inside the box, you just know that some configuration are allowed and some are not, for example you know that all the particle cannot have zero velocity since the box contain more than zero energy.
But inside the box particle are in a define configuration, on a single deterministic trajectory, so I don't think talking about the entropy of universe is meaningful, since we are inside and entropy describe system from an exterior point of view.
EDIT: Another thing is that it is often believe that an increasing entropy means that the situation is becoming more and more "a mess", that everything is going toward a kind of big shapeless soup without any structure, but it is only from this "global" point of view : the number of possible configuration increases, but it does not mean that the current configuration of the system is not structured and look like a mess ...