Reconstruction Program Map
Reconstruction programs differ in their starting objects and in the structure they aim to recover. This map helps readers compare their assumptions without treating all uses of the word “derive” as equivalent. Reconstructions of Quantum Theory supplies the conceptual discussion; this page is a lookup route through the structural alternatives.
Match the starting object to the target
Section titled “Match the starting object to the target”| Starting point | Typical target | Assumption to inspect | Entry |
|---|---|---|---|
| Preparations and tests | Quantum state and process representation | Composition and reversible control | Operational axioms |
| Pure extensions of mixed states | Structure of states and transformations | Existence and essential uniqueness of purification | Purification |
| Sharp propositions | Hilbert-space event geometry | Representation hypotheses beyond orthomodularity | Orthomodular lattices |
| Observable algebras | Noncommutativity and compatible subsystem structure | Algebraic framework already assumed | Algebraic perspectives |
| Composable processes | Equational structure of protocols | Scalars, positivity, and physical normalization | Categorical quantum mechanics |
| Rational acts in a quantum model | Born-rule weights | Decision-theoretic axioms and assumed quantum dynamics | Decision theory |
| Families of classical contexts | Contextual propositions and generalized truth | Choice of contexts and internal logical semantics | Topos approaches |
These rows describe broad approaches, not a claim that every program proves the same theorem. In particular, topos and categorical reformulations need not themselves be uniqueness reconstructions.
Four questions for a theorem
Section titled “Four questions for a theorem”What is already assumed? Record finite dimensionality, scalar field, convexity, composition, regularity, and the existence of classical controls. A short advertised axiom list may rest on a substantial framework.
What is recovered? A probability rule on projections is less than a complete dynamical theory. A representation of one system does not automatically fix the composition of several systems.
What alternatives are excluded? A classical simplex, real quantum theory, and a no-signaling box model test different parts of an axiom package.
Where does interpretation enter? A representation theorem can establish equivalence of predictive structures without deciding which mathematical objects constitute reality.
A worked comparison of dimensions
Section titled “A worked comparison of dimensions”For a system with at most perfectly distinguishable states, classical unnormalized states require real coordinates, complex quantum states require , and real quantum states require . Here is the maximum capacity, not the number of labels on an arbitrary detector. A composite rule satisfying is compatible with both the classical and complex cases, while ordinary real-quantum composition fails it.
Thus local tomography excludes one comparison theory but does not choose between all remaining candidates. The rest of the reconstruction must carry that burden.
Using the map
Section titled “Using the map”Follow a row to its detailed explanation, then read the cited original result with its assumption list. When two papers claim different primitive principles, compare the model classes they permit before deciding that their explanations conflict.
References
Section titled “References”- Chiribella, Giulio, Giacomo Mauro D’Ariano, and Paolo Perinotti, “Informational derivation of quantum theory,” Physical Review A 84, 012311 (2011). Paper.
- Clifton, Rob, Jeffrey Bub, and Hans Halvorson, “Characterizing quantum theory in terms of information-theoretic constraints,” Foundations of Physics 33, 1561–1591 (2003). Paper.
- Hardy, Lucien, “Quantum Theory From Five Reasonable Axioms,” arXiv:quant-ph/0101012 (2001). Paper.