A theory of Inaccessible Information

Out of the many lessons quantum mechanics seems to teach us, one is that it seems there are things we cannot experimentally have access to. There is, indeed, a fundamental limit to our ability to experimentally explore the world. In this work we accept this lesson as a fact and we build a general theory based on this principle. We start by assuming the existence of statements whose truth value is not experimentally accessible. That is, there is no way, not even in theory, to directly test if these statements are true or false. We further develop a theory in which experimentally accessible statements are a union of a fixed minimum number of inaccessible statements. For example, the value of truth of the statements a and b is not accessible, but the value of truth of the statement “a or b" is accessible. We do not directly assume probability theory, we exclusively define experimentally accessible and inaccessible statements and build on these notions using the rules of classical logic. We find that an interesting structure emerges. Developing this theory, we relax the logical structure, naturally obtaining a derivation of a constrained quasi-probabilistic theory rich in structure that we name theory of inaccessible information. Surprisingly, the simplest model of theory of inaccessible information is the qubit in quantum mechanics. Along the path for the construction of this theory, we characterise and study a family of multiplicative information measures that we call inaccessibility measures. arXiv:https://arxiv.org/abs/2305.05734

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