Information Invariance and Quantum Probabilities

TL;DR

This paper explores probabilistic theories with two-dimensional systems containing one bit of information, deriving quantum-like properties through invariance principles and information measures quadratic in probabilities.

Contribution

It introduces a unique information measure invariant under measurement changes and derives quantum state conditions from information and post-selection assumptions.

Findings

Quadratic information measure is uniquely determined by invariance.

Quantum state positivity emerges from information constraints.

Higher-dimensional states relate to two-dimensional post-selected systems.

Abstract

We consider probabilistic theories in which the most elementary system, a two-dimensional system, contains one bit of information. The bit is assumed to be contained in any complete set of mutually complementary measurements. The requirement of invariance of the information under a continuous change of the set of mutually complementary measurements uniquely singles out a measure of information, which is quadratic in probabilities. The assumption which gives the same scaling of the number of degrees of freedom with the dimension as in quantum theory follows essentially from the assumption that all physical states of a higher dimensional system are those and only those from which one can post-select physical states of two-dimensional systems. The requirement that no more than one bit of information (as quantified by the quadratic measure) is contained in all possible post-selected two-dimensional systems is equivalent to the positivity of density operator in quantum theory.