Theorem Index

Theorem names are useful only when matched to their hypotheses. This index lists the decisive input and conclusion for results used throughout mathematical quantum mechanics. Each link leads to the complete statement, qualifications, and proof or proof architecture. The short entries are navigation aids, not replacements for the full theorem.

Operator and representation theorems

Theorem	Decisive input	Conclusion to use
Spectral theorem	A self-adjoint operator, including its domain	A unique PVM and measurable functional calculus with spectral-integrability domains
Stone’s theorem	A strongly continuous one-parameter unitary group	A unique self-adjoint generator with a specified sign convention
Deficiency-index extension theorem	A densely defined closed symmetric operator	Self-adjoint extensions exist exactly when the two deficiency indices agree; extensions are parametrized by unitaries between deficiency spaces
Friedrichs extension	A densely defined semibounded symmetric operator	A distinguished self-adjoint extension obtained from the closure of its quadratic form
Stone–von Neumann theorem	A regular irreducible Weyl representation for finitely many canonical pairs with fixed nonzero central parameter	Unitary equivalence to the Schrödinger representation
Wigner’s theorem	A bijection of complex-Hilbert-space rays preserving every transition probability, dimension at least two	A unitary or antiunitary implementation, unique up to one scalar phase
GNS construction	A positive normalized functional on a unital C*-algebra	A cyclic representation and vector state, unique up to the appropriate unitary equivalence

The first two rows have different starting data. If a Hamiltonian is already self-adjoint, spectral calculus constructs its unitary group. If a strongly continuous unitary group is given abstractly, Stone’s theorem recovers the generator. Neither statement turns an arbitrary symmetric differential expression into a self-adjoint operator without a domain argument.

Likewise, the Stone–von Neumann hypotheses include regularity, irreducibility, and finitely many canonical pairs. A commutator identity on a test domain alone is not the theorem’s input.

State, measurement, and channel theorems

Theorem	Decisive input	Conclusion to use
Gleason’s theorem	A normalized countably additive measure on all projections of a separable complex Hilbert space of dimension at least three	A unique density-operator trace representation
Naimark dilation	A normalized POVM	Realization as the compression of a PVM on a larger space
Stinespring dilation	A CP map from a unital C*-algebra into $B(\mathcal H)$	Representation followed by compression; a unital map has an isometric embedding
Countable Kraus representation	A normal CP map between type-I algebras on separable Hilbert spaces	A finite or countable operator sum with the specified strong and trace-norm convergence
Schmidt decomposition	A vector in a bipartite Hilbert tensor product	A finite or countable orthogonal Schmidt expansion; not a general mixed-state ensemble formula
Bounded GKSL theorem	A normal unital CP semigroup on $B(\mathcal H)$ continuous in superoperator norm	A bounded Hamiltonian and bounded strongly summed noise-intensity representation
No-broadcasting theorem	Exact broadcasting of every state in a finite-dimensional promised family by one channel	The family is broadcastable exactly when its density operators commute pairwise

Normality and separability enter the Kraus row for specific reasons: normality supplies a trace-class predual, and separability permits a countable environment basis in a minimal spatial dilation. They are not interchangeable with continuity in time. The Kraus Representation Limits page explains why a countable representation does not also guarantee uniform truncation accuracy.

No-go results need their operational assumptions

No cloning excludes one deterministic exact copying device for arbitrary unknown nonorthogonal pure states; it does not exclude preparing a known state repeatedly. Bell’s theorem and CHSH test a specified local hidden-variable factorization and input-independence structure. Kochen–Specker addresses context-independent value assignments respecting the prescribed functional constraints.

The PBR theorem adds assumptions about independently prepared systems in an ontological model. Its conclusion should not be cited after silently dropping that preparation structure. What No-Go Theorems Do Not Prove collects the boundaries between these statements.

Worked theorem selection: a Gibbs operator is an extra claim

Suppose $H$ is self-adjoint and bounded below. Spectral calculus proves that $e^{-\beta H}$ is a bounded positive operator for $\beta>0$. It does not prove that its trace is finite.

For $H=0$ on an infinite-dimensional Hilbert space, $e^{-\beta H}=I$ has infinite trace. For the diagonal Hamiltonian $He_n=n^2e_n$ on $\ell^2(\mathbb N)$, the series $\sum_ne^{-\beta n^2}$ converges and does define a normalized Gibbs state.

The difference is established by a spectral summability estimate, not by repeating “spectral theorem.” A correct proof identifies which conclusion has already been supplied and which new estimate remains.

Exercises

Strong continuity versus the bounded GKSL theorem. A channel semigroup is continuous on every trace-class input. Can the bounded GKSL row be applied immediately?

Solution

No. That row requires continuity in superoperator norm. Strong continuity can coexist with an unbounded generator, as shown by the high-frequency Hamiltonian example in Strong Continuity. One must establish the stronger hypothesis or use an unbounded-generator theory with explicit domain and conservativity arguments.

A qubit probability assignment. A measure on qubit projections is normalized and additive on orthogonal pairs. Which hypothesis prevents direct application of the projection version of Gleason’s theorem listed here?

Solution

The Hilbert dimension is two, below the stated threshold. One cannot infer the conclusion from this version. Enlarged assumptions involving all effects or POVMs can lead to other theorems, but those use different event structures and must be cited separately.

References

• Bratteli, Ola, and Derek W. Robinson. Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed. Springer, 1987. doi:10.1007/978-3-662-02520-8. Algebraic representations and states.
• Hall, Brian C. Quantum Theory for Mathematicians. Graduate Texts in Mathematics 267, Springer, 2013. doi:10.1007/978-1-4614-7116-5. Operator, spectral, symmetry, and representation theorems.
• Holevo, Alexander S. Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. De Gruyter, 2019. doi:10.1515/9783110642490. States, measurements, and channel structure.
• Watrous, John. The Theory of Quantum Information. Cambridge University Press, 2018. Author’s text. Finite-dimensional channel and information-theoretic results.