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I am an amateur physics enthusiast and a science fiction writer who also sometimes writes factual articles. I like to inspire lay people to get interested in the big scientific problems and am currently writing a sort of primer on the question of time. Getting across the basic ideas is the most important thing and I add disclaimers saying that what I'm writing is at best only part of the truth.

I'm thinking about the question is there really an arrow of time? So far all the arrows turn out to be illusory. Physical processes are time symmetrical, even entropy may be a double headed arrow. Quantum mechanics also involves time reversible processes and maths. I believe there are several proposals to explain the collapse of a wave function and some of them are time symmetrical, some aren't. I'd be happy to read more thoughts on the time symmetry of collapse, in layman's terms, but I'm not trying to get the full story here, I want to follow a particular train of though, without saying anything that's wrong.

It seems to me that a wave function contains a lot of information because it tells you the probability of a particle being in any location, while after the collapse you only have binary information for all possible locations - the particle is there or not there. Therefore information is lost and the process appears to be non-reversible.

I realise that the wave function can be represented by a curve and that curves can be represented by formulas – so maybe the wave function is effectively compressed information and does not really contain more data?

I'll be grateful for any thoughts.

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This question is extremely open ended, so I will just try to provide some intuition about a small part.

A central problem in your question seems to concern the idea of "information" and quantifying it in the first place. The question seems to be, if a wavefunction with some uncertainty (for example a particle in a big box) seems to have more information than a collapsed wavefunction (a particle in the left side of the box vs. right side). As a function, it seems to be more complex. Another example, a probabilistic mixture $\rho_0 + \rho_1 + ... + \rho_n$ seems to be harder to write down than a single one of those states $\rho_3$. But, we are a little bit unsure when the wavefunction and its collapsed state are both very complex (and maybe more information?) on their own.

How can we characterize information content? Well for one, we can look at all the possible states that the wavefunction can collapse to. This is your intuition. We can say that a dice roll is more information (and therefore more entropic) than a coin flip, since it has more possible outcomes. What about a weighted vs. unweighted dice? Or, a better analogy, what if we had a 26 sided dice, vs. the English language. Surely they can't have the same information content, since the letters like Q and J and Z are used much less likely. It's almost like a 23 letter alphabet with some random stragglers. We need to take into account the fact that some letters are more probable than others, just like how Heads in an unfair coin might be more probable than Tails.

The measure that takes all of this into account is the Shannon Entropy, and a generalization for quantum systems is the Von Neumann Entropy. It defines entropy $S$ as something like

$$S = - \sum_i p_i \ln p_i$$

If we had a dice roll, it's entropy is $\ln(6)$, and a coin flip is $\ln(2)$. An unfair coin would have something smaller, and a continuous function (like a continuous wavefunction, which is a curve)

may be defined as

$$S = -\int p(x) \ln p(x) dx$$

For some coordinate $x$. This captures the essense of the ideas that we are interested in.

As a side note, the derivation of Shannon entropy is very similar to our intuition. We want to take into account not just the number of possible states, but also their relative probability. It might be helpful to look at a derivation of Shannon entropy in the first place, and see how incredibly intuitive it is as a measure of information content.

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