Ontological models as functors Andru Gheorghiu Chris Heunen - PowerPoint PPT Presentation

Ontological models as functors Andru Gheorghiu Chris Heunen arXiv:1905.09055 1 / 20

(Finite-dimensional) Quantum theory unit vector in | ψ � ∈ H, �| ψ �� 2 = 1 state complex Hilbert space uu † = u † u = 1 transformation unitary operator composition tensor product H AB = H A ⊗ H B observation orthonormal basis {| i �} , � i | j � = δ ij 2 / 20

Ontological interpretation Are quantum states real? 3 / 20

Ontological interpretation Are quantum states real? Hilbert space − → ontic (measurable) space �− → H (Λ , Σ Λ ) 3 / 20

Ontological interpretation Are quantum states real? Hilbert space − → ontic (measurable) space �− → H (Λ , Σ Λ ) µ ψ | ψ � �− → Λ 3 / 20

Ontological interpretation probability measure state | ψ � �− → µ ψ : Σ Λ → [0 , 1] 4 / 20

Ontological interpretation probability measure state | ψ � �− → µ ψ : Σ Λ → [0 , 1] response function measurement {| i �} 1 ≤ i ≤ dim( H ) �− → ξ i : Λ → [0 , 1] 4 / 20

Ontological interpretation probability measure state | ψ � �− → µ ψ : Σ Λ → [0 , 1] response function measurement {| i �} 1 ≤ i ≤ dim( H ) �− → ξ i : Λ → [0 , 1] Λ ξ i ( λ )d µ ψ ( λ )) = |� i | ψ �| 2 � 4 / 20

Ontological interpretation probability measure state | ψ � �− → µ ψ : Σ Λ → [0 , 1] response function measurement {| i �} 1 ≤ i ≤ dim( H ) �− → ξ i : Λ → [0 , 1] Λ ξ i ( λ )d µ ψ ( λ )) = |� i | ψ �| 2 � ∀ λ ∈ Λ: � dim( H ) ξ i = 1 i =1 4 / 20

Ontological interpretation µ ψ µ φ 0 < |� ψ | φ �| < 1 �− → Λ 5 / 20

Ontological interpretation µ ψ µ φ 0 < |� ψ | φ �| < 1 �− → Λ Epistemic model 5 / 20

Ontological interpretation µ ψ µ φ 0 < |� ψ | φ �| < 1 �− → Λ Epistemic model “Quantum state is state of knowledge about underlying ontic reality” 5 / 20

Ontological interpretation µ ψ µ φ 0 < |� ψ | φ �| < 1 �− → Λ Epistemic model (otherwise ontic model) “Quantum state is state of knowledge about underlying ontic reality” [Leifer arXiv:1409.1570] 5 / 20

No-go results for epistemic models [Pusey-Barrett-Rudolph arXiv:1111.3328] ◮ Preparation independence: {| ψ � ⊗ | φ �} ψ ∈ H A ,φ ∈ H B �→ (Λ A × Λ B , Σ Λ A ⊗ Σ Λ B ) µ ψ ⊗ φ = µ ψ ⊗ µ φ 6 / 20

No-go results for epistemic models [Pusey-Barrett-Rudolph arXiv:1111.3328] ◮ Preparation independence: {| ψ � ⊗ | φ �} ψ ∈ H A ,φ ∈ H B �→ (Λ A × Λ B , Σ Λ A ⊗ Σ Λ B ) µ ψ ⊗ φ = µ ψ ⊗ µ φ [Leifer-Maroney arXiv:1208.5132] ◮ Maximally epistemic: ∀| ψ � , | φ � : |� ψ | φ �| 2 = � supp( µ φ ) d µ ψ ( λ ) 6 / 20

No-go results for epistemic models [Pusey-Barrett-Rudolph arXiv:1111.3328] ◮ Preparation independence: {| ψ � ⊗ | φ �} ψ ∈ H A ,φ ∈ H B �→ (Λ A × Λ B , Σ Λ A ⊗ Σ Λ B ) µ ψ ⊗ φ = µ ψ ⊗ µ φ [Leifer-Maroney arXiv:1208.5132] ◮ Maximally epistemic: ∀| ψ � , | φ � : |� ψ | φ �| 2 = � supp( µ φ ) d µ ψ ( λ ) [Aaronson-Bouland-Chua-Lowther arXiv:1303.2834] ◮ Symmetric and maximally nontrivial: Λ = H u | ψ � = ψ = ⇒ µ uψ ( uλ ) = µ ψ ( λ ) ∀| ψ � , | φ � : |� ψ | φ �| 2 > 0 ⇐ ⇒ � supp( µ ψ ) d µ φ ( λ ) > 0 6 / 20

No-go results for epistemic models [Pusey-Barrett-Rudolph arXiv:1111.3328] ◮ Preparation independence: {| ψ � ⊗ | φ �} ψ ∈ H A ,φ ∈ H B �→ (Λ A × Λ B , Σ Λ A ⊗ Σ Λ B ) µ ψ ⊗ φ = µ ψ ⊗ µ φ [Leifer-Maroney arXiv:1208.5132] ◮ Maximally epistemic: ∀| ψ � , | φ � : |� ψ | φ �| 2 = � supp( µ φ ) d µ ψ ( λ ) [Aaronson-Bouland-Chua-Lowther arXiv:1303.2834] ◮ Symmetric and maximally nontrivial: Λ = H u | ψ � = ψ = ⇒ µ uψ ( uλ ) = µ ψ ( λ ) ∀| ψ � , | φ � : |� ψ | φ �| 2 > 0 ⇐ ⇒ � supp( µ ψ ) d µ φ ( λ ) > 0 ◮ [Gheorghiu-Heunen arXiv:1905.09055]: one approach to rule them all 6 / 20

Category theory Explicitly invented to translate structure between different areas: ◮ Algebraic topology: topology �→ groups ◮ Algebraic geometry: varieties �→ schemes ◮ Logic: theories �→ models ◮ Computer compilers: high-level language �→ assembly ◮ Complexity theory: algorithm �→ function ◮ Semantics: computer programs �→ mathematical model ◮ Physics: physical systems �→ mathematical abstractions Here: quantum physics �→ statistical physics 7 / 20

Categorical approach H C N N ◦ M H B M H A FHilb 8 / 20

Categorical approach H C bounded linear maps N N ◦ M H B M Hilbert space H A FHilb 8 / 20

Categorical approach ( X C , Σ C ) f ( X B , Σ B ) g ◦ f g ( X A , Σ A ) BoRel 8 / 20

Categorical approach ( X C , Σ C ) Markov kernels f ( X B , Σ B ) g ◦ f g Borel spaces ( X A , Σ A ) BoRel 8 / 20

Categorical approach ( X C , Σ C ) Borel space: topological measurable space f Markov kernels: ( X B , Σ B ) g ◦ f f : X A × Σ B → [0 , 1] f ( − , W ): X A → [0 , 1] bounded measurable f ( x, − ): Σ B → [0 , 1] probability measure g ( X A , Σ A ) BoRel 8 / 20

Categorical approach H C ( X C , Σ C ) f N ( X B , Σ B ) g ◦ f N ◦ M H B g M ( X A , Σ A ) H A functor F FHilb BoRel 8 / 20

States H | ψ � C 9 / 20

States (Λ , Σ Λ ) H F µ ψ | ψ � � {•} , {∅ , {•}} � C 9 / 20

States (Λ , Σ Λ ) H F µ ψ | ψ � � {•} , {∅ , {•}} � C F ( | ψ � )( • , − ): Σ Λ → [0 , 1] probability measure 9 / 20

Effects H � ψ | C 10 / 20

Effects (Λ , Σ Λ ) H F � ψ | ξ i � {•} , {∅ , {•}} � C 10 / 20

Effects (Λ , Σ Λ ) H F � ψ | ξ i � {•} , {∅ , {•}} � C F ( � ψ | )( − , {•} ): Λ → [0 , 1] response function 10 / 20

Operational category ◮ is monoidal ( ⊗ ,I) ◮ has distinguishing object 2 ◮ has set Ω of elements called probabilities ◮ has evaluation �−� : C ( I, 2) → Ω 11 / 20

Operational category ◮ is monoidal ( ⊗ ,I) ◮ has distinguishing object 2 ◮ has set Ω of elements called probabilities ◮ has evaluation �−� : C ( I, 2) → Ω FHilb is operational: ◮ 2 = C 2 , Ω = [0 , 1] ◮ η : C → C 2 � η � = | a | 2 if η (1) = ( a, b ), | a | 2 + | b | 2 = 1 11 / 20

Operational category ◮ is monoidal ( ⊗ ,I) ◮ has distinguishing object 2 ◮ has set Ω of elements called probabilities ◮ has evaluation �−� : C ( I, 2) → Ω FHilb is operational: ◮ 2 = C 2 , Ω = [0 , 1] ◮ η : C → C 2 � η � = | a | 2 if η (1) = ( a, b ), | a | 2 + | b | 2 = 1 BoRel is operational: ◮ 2 = � { 0 , 1 } , � ∅ , { 0 } , { 1 } , { 0 , 1 } �� , Ω = [0 , 1] ◮ f : I → 2, � f � = f ( • , { 0 } ) if f ( • , { 0 } ) = 1 − f ( • , { 1 } ) 11 / 20

Operational model is functor F : C → D between operational categories satisfying: F ( I ) = I F (2) = 2 � F ( η ) � = � η � 12 / 20

Operational model is functor F : C → D between operational categories satisfying: F ( I ) = I F (2) = 2 � F ( η ) � = � η � For C = FHilb and D = BoRel : � ξ i ( λ )d µ ψ ( λ ) = |� i | ψ �| 2 Λ F ( | ψ � ) = µ ψ F ( � i | ) = ξ i 12 / 20

Distinguishability If C operational category with Ω = [0 , 1], Ψ ⊆ C ( I, A ) collection of states χ : A → 2 measurement, χ distinguishes ψ from Ψ when � χ ◦ ψ � = 1 � � χ ◦ φ = 0 φ ∈ Ψ ,φ � = ψ 13 / 20

Epistemic operational models Operational model is epistemic when there are distinct states ψ � = φ : I → A such that F ( ψ ) and F ( φ ) are not distinguishable 14 / 20

Epistemic operational models Operational model is epistemic when there are distinct states ψ � = φ : I → A such that F ( ψ ) and F ( φ ) are not distinguishable i.e. “distributions overlap”: F ( ψ ) F ( φ ) Λ 14 / 20

Operational vs ontological ◮ operational model is more restrictive ◮ composition needs to be preserved ◮ trivial ontic models can be constructed ◮ not clear whether ontic operational models exist at all 15 / 20

No-go results: Pusey-Barrett-Rudolph No epistemic ontological model when: preparation independence {| ψ � ⊗ | φ �} ψ ∈ H A ,φ ∈ H B �→ (Λ A × Λ B , Σ Λ A ⊗ Σ Λ B ) µ ψ ⊗ φ = µ ψ ⊗ µ φ 16 / 20

No-go results: Pusey-Barrett-Rudolph No epistemic ontological model when: preparation independence {| ψ � ⊗ | φ �} ψ ∈ H A ,φ ∈ H B �→ (Λ A × Λ B , Σ Λ A ⊗ Σ Λ B ) µ ψ ⊗ φ = µ ψ ⊗ µ φ Monoidal operational model implies this So cannot have monoidal epistemic operational model! 16 / 20

No-go results: Leifer-Maroney No maximally epistemic ontological model � ∀| ψ � , | φ � : |� ψ | φ �| 2 = d µ ψ ( λ ) supp( µ φ ) 17 / 20