The Information Conservation Principle

Nobody asks where energy goes. We bookkeep it. Energy is conserved, so every apparent loss is a transfer, and the entire discipline of thermodynamics is what you get from taking that bookkeeping seriously. The Information Conservation Principle says: do the same for information. Treat conserved information the way we treat conserved energy, and two things in quantum foundations that look like separate mysteries turn out to be the same bookkeeping move.

The principle

Stated plainly: for the closed system, information is neither created nor destroyed. Apparent loss is redistribution. When a measurement seems to destroy information, the principle says look harder, and the harder look finds it spread across system, apparatus and environment, which is exactly where decoherence studies say it goes.

So far this is just unitarity said in a thermodynamic accent, and it would be fair to ask what the accent buys. It buys a discipline about which quantities are allowed to appear in your accounting. If information is conserved, then any rule for weighting outcomes has to be expressed as a redistribution of a conserved quantity, and that constraint is surprisingly tight.

Where the Born rule comes from

Quantum mechanics uses the Born rule to weight outcomes: the probability of a result is the squared magnitude of its amplitude. Textbooks postulate this. Generations of physicists have found the postulate unsatisfying, because a rule about squares arrives with no explanation of why not cubes, or absolute values, or anything else.

The ICP route derives it. If the conserved quantity is information, and you require that branch bookkeeping be consistent when systems are composed, independent of basis, and continuous in the amplitudes, then the unique measure that does the job is the squared-amplitude measure. Anything else either double-counts on composition or breaks the conservation story. The squared in the Born rule is not an aesthetic choice. It is what conservation forces.

I want to be precise about the strength of that claim, because it is easy to overstate. This is a derivation given the axioms: conservation, composition, continuity, and the rest of the stated list. Different axiom sets in the literature derive different measures, and some derive no unique measure. The honest statement is that within this bookkeeping, the Born measure is not optional. It is not a proof that reality obeys the bookkeeping.

The two levels

The principle does its real work in a multiverse setting, where the awkward question is what a single observer inside one branch is entitled to say. Paper 2 separates the levels. Inside a branch, an observer sees information conserved in the ordinary sense, and their statistics obey the Born rule because the bookkeeping is local to their level. The substrate, meanwhile, holds many branches and does its own conservation accounting across all of them. Nothing at the observer’s level needs to know the substrate exists for the observer’s physics to close.

Keeping those two levels apart is the hard part of the whole architecture, and most objections to the framework dissolve once the levels are separated. Critics arrive at the problem of a branch observer knowing about other branches, and the answer is that they do not and need not.

What could kill it

Two things, concretely. A demonstration that information bookkeeping is contextual, meaning the conserved quantity depends on which measurement is performed in a way no redistribution can absorb. Or a competing measure satisfying the same axioms, which would break uniqueness. Neither is known, and both are the kind of result someone would have noticed.

The weaker risk is that the principle is true but empty, a restatement of unitarity with ceremony. The reply is the Born rule: empty principles do not select measures. That argument, and what it does to the oldest argument in quantum foundations, is the next post.