Foundational Theorems

Foundational theorems clarify which parts of the quantum formalism follow from declared structural assumptions. Their force comes from stating those assumptions precisely, not from presenting them as premise-free derivations of quantum mechanics.

Enter the chapter

This chapter has two independent prerequisite branches.

Probability-assignment branch

Complete the event-and-probability branch first: State Vectors → Projectors and Probability Amplitudes → the Born Rule, followed by both Born Rule for Continuous Spectra and Expectation Values. In the operator branch, read Self-Adjoint Operators → the Unbounded Spectral Theorem. Rejoin the branches at Projection-Valued Measures, then continue to the theorem.

Gleason’s Theorem starts with a normalized countably additive probability measure $\mu$ on orthogonal projections and, for complex Hilbert-space dimension at least three, proves the unique form

$$\mu(P)=\operatorname{Tr}(\rho P).$$

Information-transformation branch

Complete the current density-operator route before the proof: State Vectors → Projectors and Probability Amplitudes → the Born Rule → Expectation Values → Density Operators. Then continue through Entangled States → Partial Trace → Quantum Operations.

No-Broadcasting Theorem starts with a family of finite-dimensional density operators and asks whether one quantum channel can reproduce every input state in both output marginals. The answer is exact:

$$\text{the family is broadcastable} \quad\Longleftrightarrow\quad [\rho_i,\rho_j]=0 \text{ for every }i,j.$$

The theorem boundary

The treatment makes explicit that:

• one context-independent value is assigned to each projector;
• orthogonal alternatives are additive;
• countable additivity matters in infinite dimension;
• the original projection theorem requires dimension at least three;
• a concrete qubit assignment shows why dimension two is exceptional;
• POVM variants close that exception only by using stronger generalized-effect premises;
• the conclusion constrains probabilities, not state update or dynamics.

No-broadcasting makes a different boundary explicit:

• the main theorem is finite-dimensional, exact, and deterministic;
• one common CPTP map must work for the entire state family;
• the two output states need only agree with the input marginally and may be correlated jointly;
• commuting families can be broadcast in a shared eigenbasis;
• noncommuting families cannot be broadcast;
• pure-state broadcasting reduces to the familiar cloning question.

These pages share theorem-level precision but do not share premises. Gleason constrains probability measures on projectors; no-broadcasting constrains quantum channels on families of density operators.