Quantum theory from questions

TL;DR

This paper reconstructs the formal structure of qubit quantum theory from simple, operational rules based on an observer's information, revealing new insights into the informational foundations of quantum mechanics.

Contribution

It provides a purely operational reconstruction of quantum theory from elementary rules on information acquisition, deriving state spaces, evolution, and measurement outcomes.

Findings

States correspond to density matrices over N-qubit Hilbert space

States evolve unitarily under the group PSU(2^N)

Binary questions align with projective Pauli measurements

Abstract

We reconstruct the explicit formalism of qubit quantum theory from elementary rules on an observer's information acquisition. Our approach is purely operational: we consider an observer O interrogating a system S with binary questions and define S's state as O's `catalogue of knowledge' about S. From the rules we derive the state spaces for N elementary systems and show that (a) they coincide with the set of density matrices over an N-qubit Hilbert space; (b) states evolve unitarily under the group $\rm{PSU}(2^N)$ according to the von Neumann evolution equation; and (c) O's binary questions correspond to projective Pauli operator measurements with outcome probabilities given by the Born rule. As a by-product, this results in a propositional formulation of quantum theory. Aside from offering an informational explanation for the theory's architecture, the reconstruction also unravels new structural insights. We show that, in a derived quadratic information measure, (d) qubits satisfy inequalities which bound the information content in any set of mutually complementary questions to 1 bit; and (e) maximal sets of mutually complementary questions for one and two qubits must carry precisely 1 bit of information in pure states. The latter relations constitute conserved informational charges which define the unitary groups and, together with their conservation conditions, the sets of pure quantum states. These results highlight information as a `charge of quantum theory' and the benefits of this informational approach. This work emphasizes the sufficiency of restricting to an observer's information to reconstruct the theory and completes the quantum reconstruction initiated in arXiv:1412.8323.