Theories of systems with limited information content

TL;DR

This paper introduces a hierarchical classification of theories describing systems with limited information, highlighting their operational features, mutual complementarity, and computational implications, encompassing classical, quantum, and generalized probabilistic theories.

Contribution

It provides a new hierarchical framework for classifying theories based on information content and complementarity, including operational definitions and implications for quantum and generalized theories.

Findings

Theories can be ordered by the number of mutually complementary measurements.

Multipartite states may contain entanglement in these theories.

Tomography with local measurements is possible in the classified theories.

Abstract

We introduce a hierarchical classification of theories that describe systems with fundamentally limited information content. This property is introduced in an operational way and gives rise to the existence of mutually complementary measurements, i.e. a complete knowledge of future outcome in one measurement is at the expense of complete uncertainty in the others. This is characteristic feature of the theories and they can be ordered according to the number of mutually complementary measurements which is also shown to define their computational abilities. In the theories multipartite states may contain entanglement and tomography with local measurements is possible. The classification includes both classical and quantum theory and also generalized probabilistic theories with higher number of degrees of freedom, for which operational meaning is given. We also discuss thought experiments discriminating standard quantum theory from the generalizations.