Principle of Interval Knowledge in Science Methodology

The article discusses the Principle of Interval Knowledge designed to fill the gap in the methodology of science, associated with the problem of the knowledge boundaries. Over the course of half a century, several versions of the Intervality Principle have been created, different interpretations of its notions have arisen, in­cluding the key concept of “abstraction interval”, but they encountered a number of contradictions. Therefore, the task arises of creating a unified concept and in­troducing precise definitions. The author shows that overcoming contradictions is possible on the way of studying the meaning of introduced notions. This meaning should be correlated with the goal of the intervality concept: to reflect the birth of partial knowledge under the conditions of total restrictions of all components in­volved in the creation of knowledge. The article provides definitions of interval notions, formulation of the Principle and interpretation of the key concept “ab­straction interval”. The author believes that the expansion of the notion of “inter­val” allows accepting it as a generic concept for all types of restrictions. The crite­rion for the meaning of a key concept is the requirement of informational completeness of knowledge representation. Then the concept of “abstraction in­terval” can be understood as a complete informational characteristic of knowledge about a fragment of reality and the conditions for its creation. Two comple­mentary concepts are introduced: “conditions interval” and “content interval”. The author shows that the essence of the Principle consists of asserting condition­ality of the content interval by the conditions interval. The introduced refinements remove the contradictions between different directions and lead to the creation of a unified interval concept.

Probing quantum state space: does one have to learn everything to learn something?

Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement set-up that reveals this property with as little effort as possible. Here, we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem, we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.