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PhyslibAlpha.ProbabilisticTheory.Classical.BauerSimplex

Bauer simplices

i. Overview

A Bauer simplex is a classical state space whose pure states form a closed set. The pure states are then a compact phase space, and observables become continuous functions on it.

Closed pure states already give every state a pure decomposition. A state is a positive functional on the observables. The M. Riesz extension theorem extends it to a positive functional on all continuous functions on the pure states, and the Riesz–Markov–Kakutani theorem turns that into a probability measure. Together with the uniqueness of pure decompositions, the state space is a Bauer simplex exactly when ensembles refine and the pure states are closed.

ii. Key results

- `PureState.hasPureDecomposition` proves that, with compact pure states, every state has a pure decomposition. - `isSimplexStateSpace_iff_ensemblesRefine_of_isClosed` proves that, with closed pure states, the system is classical exactly when ensembles refine. - `isBauerSimplexStateSpace_iff` proves that the state space is a Bauer simplex exactly when ensembles refine and the pure states are closed.

iii. Table of contents

  • A. Pure decompositions
  • B. Bauer simplices

iv. References

  • E. M. Alfsen, *Compact Convex Sets and Boundary Integrals*, Springer, 1971, ch. II.4.

A. Pure decompositions

B. Bauer simplices

3 declarations

theorem

Compactness of Pure States   ⟹  \implies Existence of Pure Decomposition

Let EE be an Archimedean order-unit space. If the space of pure states PureState E\text{PureState } E (the set of extreme points of the state space S(E)\mathcal{S}(E) equipped with the weak-star topology) is compact, then every state ω∈S(E)\omega \in \mathcal{S}(E) has a pure decomposition. That is, there exists a regular probability measure μ\mu on PureState E\text{PureState } E such that for every f∈Ef \in E, the value of the state is given by the integral: ω(f)=∫PureState Eϕ(f) dμ(ϕ)\omega(f) = \int_{\text{PureState } E} \phi(f) \, d\mu(\phi)

theorem

With closed pure states, Simplex State Space   ⟺  \iff Ensemble Refinement

Let EE be an Archimedean order-unit space with state space S(E)\mathcal{S}(E) (the set of normalized positive continuous linear functionals). Suppose that the set of pure states of EE (the extreme points of S(E)\mathcal{S}(E)) is closed in the weak* topology. Then the state space S(E)\mathcal{S}(E) is a Choquet simplex if and only if EE satisfies the ensemble refinement property. The ensemble refinement property states that for any two finite families of positive linear functionals {αi}i∈I\{\alpha_i\}_{i \in \mathcal{I}} and {βj}j∈J\{\beta_j\}_{j \in \mathcal{J}} on EE such that ∑i∈Iαi=∑j∈Jβj\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j, there exists a family of positive linear functionals {γij}i∈I,j∈J\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}} such that ∑j∈Jγij=αi\sum_{j \in \mathcal{J}} \gamma_{ij} = \alpha_i for all i∈Ii \in \mathcal{I} and ∑i∈Iγij=βj\sum_{i \in \mathcal{I}} \gamma_{ij} = \beta_j for all j∈Jj \in \mathcal{J}.

theorem

Bauer Simplex   ⟺  \iff Ensemble Refinement and Closed Pure States

Let EE be an Archimedean order-unit space. The state space S(E)\mathcal{S}(E), consisting of normalized positive continuous linear functionals on EE, is a Bauer simplex if and only if the following two conditions hold: 1. EE satisfies the ensemble refinement property: for any two finite families of positive linear functionals {αi}i∈I\{\alpha_i\}_{i \in \mathcal{I}} and {βj}j∈J\{\beta_j\}_{j \in \mathcal{J}} on EE such that ∑i∈Iαi=∑j∈Jβj\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j, there exists a family of positive linear functionals {γij}i∈I,j∈J\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}} such that ∑j∈Jγij=αi\sum_{j \in \mathcal{J}} \gamma_{ij} = \alpha_i for all i∈Ii \in \mathcal{I} and ∑i∈Iγij=βj\sum_{i \in \mathcal{I}} \gamma_{ij} = \beta_j for all j∈Jj \in \mathcal{J}. 2. The set of pure states of EE (the extreme points of S(E)\mathcal{S}(E)) is closed in the weak-* topology.