Dichotomy between deterministic and probabilistic models in countably additive effectus theory

Kenta Cho National Institute of Informatics, Japan Bas Westerbaan University College London John van de Wetering Radboud University Nijmegen

June 6, 2020 such that § hom-sets tf : A Ñ Bu are convex sets, § and the scalars ts : I Ñ I u are the real unit interval r0, 1s Special operations: § States StpAq :“ tω : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ω is probability that p holds on state ω

Generalized Probabilistic Theories

GPTs are generalisations of quantum theory. They consist of § systems A, B, C,..., § the ‘empty system’ I , § operations f : A Ñ B, Special operations: § States StpAq :“ tω : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ω is probability that p holds on state ω

Generalized Probabilistic Theories

GPTs are generalisations of quantum theory. They consist of § systems A, B, C,..., § the ‘empty system’ I , § operations f : A Ñ B, such that § hom-sets tf : A Ñ Bu are convex sets, § and the scalars ts : I Ñ I u are the real unit interval r0, 1s Generalized Probabilistic Theories

GPTs are generalisations of quantum theory. They consist of § systems A, B, C,..., § the ‘empty system’ I , § operations f : A Ñ B, such that § hom-sets tf : A Ñ Bu are convex sets, § and the scalars ts : I Ñ I u are the real unit interval r0, 1s Special operations: § States StpAq :“ tω : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ω is probability that p holds on state ω Solution: allow more general sets of scalars ts : I Ñ I u. Result: effectus theory.

From GPTs to effectuses

Problem with GPTs: § Can’t describe deterministic models. § Assumptions on classical probability are baked in. From GPTs to effectuses

Problem with GPTs: § Can’t describe deterministic models. § Assumptions on classical probability are baked in.

Solution: allow more general sets of scalars ts : I Ñ I u. Result: effectus theory. § R is a commutative monoid. § Same holds for pRě0, `, 0q § What about pr0, 1s, `, 0q? Definition A partial commutative monoid (PCM) pX , , 0q is a set X with a partial associative commutative operation >with unit 0. > px yq z “ x py zq x y “ y x x 0 “ x > > > > > > > Write x K y when x y is defined. >

Partial commutative monoids

What should we replace R with? § What about pr0, 1s, `, 0q? Definition A partial commutative monoid (PCM) pX , , 0q is a set X with a partial associative commutative operation >with unit 0. > px yq z “ x py zq x y “ y x x 0 “ x > > > > > > > Write x K y when x y is defined. >

Partial commutative monoids

What should we replace R with? § R is a commutative monoid. § Same holds for pRě0, `, 0q Definition A partial commutative monoid (PCM) pX , , 0q is a set X with a partial associative commutative operation >with unit 0. > px yq z “ x py zq x y “ y x x 0 “ x > > > > > > > Write x K y when x y is defined. >

Partial commutative monoids

What should we replace R with? § R is a commutative monoid. § Same holds for pRě0, `, 0q § What about pr0, 1s, `, 0q? Partial commutative monoids

What should we replace R with? § R is a commutative monoid. § Same holds for pRě0, `, 0q § What about pr0, 1s, `, 0q? Definition A partial commutative monoid (PCM) pX , , 0q is a set X with a partial associative commutative operation >with unit 0. > px yq z “ x py zq x y “ y x x 0 “ x > > > > > > > Write x K y when x y is defined. > § A partial function f : X Ñ Y has either f pxq P Y or f pxq undefined. Partial functions form a category Pfn. Homsets PfnpX , Y q are PCMs: Set f K g when domains of definition are disjoint and set f g as the ‘union’ of f and g. > § Let Cstar be category of unital C˚-algebras with positive subunital maps. Homsets CstarpA, Bq are PCMs: f K g when f p1q ` gp1q ď 1, and then f g “ f ` g. > Pfn and Cstar are examples of PCM-enriched categories:

pf gq ˝ h “ pf ˝ hq pg ˝ hq h ˝ pf gq “ ph ˝ f q ph ˝ gq > > > >

PCM examples

§ Real unit interval r0, 1s is PCM. Set f K g when domains of definition are disjoint and set f g as the ‘union’ of f and g. > § Let Cstar be category of unital C˚-algebras with positive subunital maps. Homsets CstarpA, Bq are PCMs: f K g when f p1q ` gp1q ď 1, and then f g “ f ` g. > Pfn and Cstar are examples of PCM-enriched categories:

pf gq ˝ h “ pf ˝ hq pg ˝ hq h ˝ pf gq “ ph ˝ f q ph ˝ gq > > > >

PCM examples

§ Real unit interval r0, 1s is PCM. § A partial function f : X Ñ Y has either f pxq P Y or f pxq undefined. Partial functions form a category Pfn. Homsets PfnpX , Y q are PCMs: § Let Cstar be category of unital C˚-algebras with positive subunital maps. Homsets CstarpA, Bq are PCMs: f K g when f p1q ` gp1q ď 1, and then f g “ f ` g. > Pfn and Cstar are examples of PCM-enriched categories:

pf gq ˝ h “ pf ˝ hq pg ˝ hq h ˝ pf gq “ ph ˝ f q ph ˝ gq > > > >

PCM examples

§ Real unit interval r0, 1s is PCM. § A partial function f : X Ñ Y has either f pxq P Y or f pxq undefined. Partial functions form a category Pfn. Homsets PfnpX , Y q are PCMs: Set f K g when domains of definition are disjoint and set f g as the ‘union’ of f and g. > Pfn and Cstar are examples of PCM-enriched categories:

pf gq ˝ h “ pf ˝ hq pg ˝ hq h ˝ pf gq “ ph ˝ f q ph ˝ gq > > > >

PCM examples

§ Real unit interval r0, 1s is PCM. § A partial function f : X Ñ Y has either f pxq P Y or f pxq undefined. Partial functions form a category Pfn. Homsets PfnpX , Y q are PCMs: Set f K g when domains of definition are disjoint and set f g as the ‘union’ of f and g. > § Let Cstar be category of unital C˚-algebras with positive subunital maps. Homsets CstarpA, Bq are PCMs: f K g when f p1q ` gp1q ď 1, and then f g “ f ` g. > PCM examples

§ Real unit interval r0, 1s is PCM. § A partial function f : X Ñ Y has either f pxq P Y or f pxq undefined. Partial functions form a category Pfn. Homsets PfnpX , Y q are PCMs: Set f K g when domains of definition are disjoint and set f g as the ‘union’ of f and g. > § Let Cstar be category of unital C˚-algebras with positive subunital maps. Homsets CstarpA, Bq are PCMs: f K g when f p1q ` gp1q ď 1, and then f g “ f ` g. > Pfn and Cstar are examples of PCM-enriched categories:

pf gq ˝ h “ pf ˝ hq pg ˝ hq h ˝ pf gq “ ph ˝ f q ph ˝ gq > > > > Definition An effect algebra is a PCM pE, , 0, 1q with special element 1 such that for each a P E: > § there is unique aK P E with a aK “ 1, > § a K 1 implies a “ 0.

Examples § r0, 1s with aK :“ 1 ´ a. § A Boolean algebra: a K b when a ^ b “ 0 and then a b “ a _ b. aK is the regular negation. > K § CstarpC, Aq – r0, 1sA with a :“ 1 ´ a.

Effect algebras

r0, 1s is not just a PCM: the 1 has a special role. Examples § r0, 1s with aK :“ 1 ´ a. § A Boolean algebra: a K b when a ^ b “ 0 and then a b “ a _ b. aK is the regular negation. > K § CstarpC, Aq – r0, 1sA with a :“ 1 ´ a.

Effect algebras

r0, 1s is not just a PCM: the 1 has a special role. Definition An effect algebra is a PCM pE, , 0, 1q with special element 1 such that for each a P E: > § there is unique aK P E with a aK “ 1, > § a K 1 implies a “ 0. Effect algebras

r0, 1s is not just a PCM: the 1 has a special role. Definition An effect algebra is a PCM pE, , 0, 1q with special element 1 such that for each a P E: > § there is unique aK P E with a aK “ 1, > § a K 1 implies a “ 0.

Examples § r0, 1s with aK :“ 1 ´ a. § A Boolean algebra: a K b when a ^ b “ 0 and then a b “ a _ b. aK is the regular negation. > K § CstarpC, Aq – r0, 1sA with a :“ 1 ´ a. § CpA, I q is effect algebra @A P C; We write 1A and 0A for the 1 and 0 in CpA, I q.

§ 1B ˝ f “ 0A implies f “ 0 for all f : A Ñ B,

§ 1B ˝ f K 1B ˝ g implies f K g for all f , g : A Ñ B. Examples: § Pfn: I “ t˚u, PfnpA, I q – PpAq. op op § Cstar : I “ C, Cstar pA, I q – r0, 1sA.

Effectus

Definition An effectus is a PCM-enriched category C with designated object I such that: § C has coproducts, § the coproducts are ‘compatible’ with the PCM structure, § 1B ˝ f “ 0A implies f “ 0 for all f : A Ñ B,

§ 1B ˝ f K 1B ˝ g implies f K g for all f , g : A Ñ B. Examples: § Pfn: I “ t˚u, PfnpA, I q – PpAq. op op § Cstar : I “ C, Cstar pA, I q – r0, 1sA.

Effectus

Definition An effectus is a PCM-enriched category C with designated object I such that: § C has coproducts, § the coproducts are ‘compatible’ with the PCM structure, § CpA, I q is effect algebra @A P C; We write 1A and 0A for the 1 and 0 in CpA, I q. Examples: § Pfn: I “ t˚u, PfnpA, I q – PpAq. op op § Cstar : I “ C, Cstar pA, I q – r0, 1sA.

Effectus

Definition An effectus is a PCM-enriched category C with designated object I such that: § C has coproducts, § the coproducts are ‘compatible’ with the PCM structure, § CpA, I q is effect algebra @A P C; We write 1A and 0A for the 1 and 0 in CpA, I q.

§ 1B ˝ f “ 0A implies f “ 0 for all f : A Ñ B,

§ 1B ˝ f K 1B ˝ g implies f K g for all f , g : A Ñ B. op op § Cstar : I “ C, Cstar pA, I q – r0, 1sA.

Effectus

Definition An effectus is a PCM-enriched category C with designated object I such that: § C has coproducts, § the coproducts are ‘compatible’ with the PCM structure, § CpA, I q is effect algebra @A P C; We write 1A and 0A for the 1 and 0 in CpA, I q.

§ 1B ˝ f “ 0A implies f “ 0 for all f : A Ñ B,

§ 1B ˝ f K 1B ˝ g implies f K g for all f , g : A Ñ B. Examples: § Pfn: I “ t˚u, PfnpA, I q – PpAq. Effectus

Definition An effectus is a PCM-enriched category C with designated object I such that: § C has coproducts, § the coproducts are ‘compatible’ with the PCM structure, § CpA, I q is effect algebra @A P C; We write 1A and 0A for the 1 and 0 in CpA, I q.

§ 1B ˝ f “ 0A implies f “ 0 for all f : A Ñ B,

§ 1B ˝ f K 1B ˝ g implies f K g for all f , g : A Ñ B. Examples: § Pfn: I “ t˚u, PfnpA, I q – PpAq. op op § Cstar : I “ C, Cstar pA, I q – r0, 1sA. § Given a state ω : I Ñ A and effect p : A Ñ I , the composition forms a scalar p ˝ ω : I Ñ I . § Recall in GPT, scalars ts : I Ñ I u are r0, 1s. § What are they in an effectus? § PfnpI , I q – t0, 1u & CstaroppI , I q – r0, 1s. § In general: CpI , I q is effect algebra. s t § But also has a ‘multiplication‘ given by composition I Ñ I Ñ I .

Relating effectus to GPTs

Let C be an effectus, and A P C. § Effects EffpAq :“ CpA, I q form effect algebra. § States StpAq :“ CpI , Aq form ‘abstract convex set’. § PfnpI , I q – t0, 1u & CstaroppI , I q – r0, 1s. § In general: CpI , I q is effect algebra. s t § But also has a ‘multiplication‘ given by composition I Ñ I Ñ I .

Relating effectus to GPTs

Let C be an effectus, and A P C. § Effects EffpAq :“ CpA, I q form effect algebra. § States StpAq :“ CpI , Aq form ‘abstract convex set’. § Given a state ω : I Ñ A and effect p : A Ñ I , the composition forms a scalar p ˝ ω : I Ñ I . § Recall in GPT, scalars ts : I Ñ I u are r0, 1s. § What are they in an effectus? § In general: CpI , I q is effect algebra. s t § But also has a ‘multiplication‘ given by composition I Ñ I Ñ I .

Relating effectus to GPTs

Let C be an effectus, and A P C. § Effects EffpAq :“ CpA, I q form effect algebra. § States StpAq :“ CpI , Aq form ‘abstract convex set’. § Given a state ω : I Ñ A and effect p : A Ñ I , the composition forms a scalar p ˝ ω : I Ñ I . § Recall in GPT, scalars ts : I Ñ I u are r0, 1s. § What are they in an effectus? § PfnpI , I q – t0, 1u & CstaroppI , I q – r0, 1s. Relating effectus to GPTs

Let C be an effectus, and A P C. § Effects EffpAq :“ CpA, I q form effect algebra. § States StpAq :“ CpI , Aq form ‘abstract convex set’. § Given a state ω : I Ñ A and effect p : A Ñ I , the composition forms a scalar p ˝ ω : I Ñ I . § Recall in GPT, scalars ts : I Ñ I u are r0, 1s. § What are they in an effectus? § PfnpI , I q – t0, 1u & CstaroppI , I q – r0, 1s. § In general: CpI , I q is effect algebra. s t § But also has a ‘multiplication‘ given by composition I Ñ I Ñ I . Examples: § r0, 1s. § Any Boolean algebra: a b :“ a _ b, a ¨ b :“ a ^ b. > § tf : X Ñ r0, 1s continuousu for a compact Hausdorff space X (i.e. unit interval of commutative unital C˚-algebra).

Scalars in effectus

Definition An effect monoid pM, , 0, 1, ¨q is an effect algebra with associative distributive multiplication:>

pa bq ¨ c “ pa ¨ cq pb ¨ cq c ¨ pa bq “ pc ¨ aq pc ¨ bq > > > > Scalars in effectus

Definition An effect monoid pM, , 0, 1, ¨q is an effect algebra with associative distributive multiplication:>

pa bq ¨ c “ pa ¨ cq pb ¨ cq c ¨ pa bq “ pc ¨ aq pc ¨ bq > > > > Examples: § r0, 1s. § Any Boolean algebra: a b :“ a _ b, a ¨ b :“ a ^ b. > § tf : X Ñ r0, 1s continuousu for a compact Hausdorff space X (i.e. unit interval of commutative unital C˚-algebra). § An M-effect module E is an effect algebra with suitable left M-action ¨ : M ˆ E Ñ E. op op § EModM is an effectus. Eff: C Ñ EModM is a functor.

§ StpAq has ‘weight function’ |ω| :“ 1A ˝ ω. § A weight M-module X is a PCM with M-action ¨ : M ˆ X Ñ X and suitable weight function |¨| : X Ñ M.

§ WModM is an effectus. St: C Ñ WModM is a functor.

Any effect monoid is the set of scalars of some effectus Corollary: There are some weird effectuses out there

Effectus over an effect monoid

Let C be effectus, M :“ CpI , I q an effect monoid and A P C. § EffpAq has action of scalars via s ¨ p :“ s ˝ p. § StpAq has ‘weight function’ |ω| :“ 1A ˝ ω. § A weight M-module X is a PCM with M-action ¨ : M ˆ X Ñ X and suitable weight function |¨| : X Ñ M.

§ WModM is an effectus. St: C Ñ WModM is a functor.

Any effect monoid is the set of scalars of some effectus Corollary: There are some weird effectuses out there

Effectus over an effect monoid

Let C be effectus, M :“ CpI , I q an effect monoid and A P C. § EffpAq has action of scalars via s ¨ p :“ s ˝ p. § An M-effect module E is an effect algebra with suitable left M-action ¨ : M ˆ E Ñ E. op op § EModM is an effectus. Eff: C Ñ EModM is a functor. § A weight M-module X is a PCM with M-action ¨ : M ˆ X Ñ X and suitable weight function |¨| : X Ñ M.

§ WModM is an effectus. St: C Ñ WModM is a functor.

Any effect monoid is the set of scalars of some effectus Corollary: There are some weird effectuses out there

Effectus over an effect monoid

Let C be effectus, M :“ CpI , I q an effect monoid and A P C. § EffpAq has action of scalars via s ¨ p :“ s ˝ p. § An M-effect module E is an effect algebra with suitable left M-action ¨ : M ˆ E Ñ E. op op § EModM is an effectus. Eff: C Ñ EModM is a functor.

§ StpAq has ‘weight function’ |ω| :“ 1A ˝ ω. Any effect monoid is the set of scalars of some effectus Corollary: There are some weird effectuses out there

Effectus over an effect monoid

Let C be effectus, M :“ CpI , I q an effect monoid and A P C. § EffpAq has action of scalars via s ¨ p :“ s ˝ p. § An M-effect module E is an effect algebra with suitable left M-action ¨ : M ˆ E Ñ E. op op § EModM is an effectus. Eff: C Ñ EModM is a functor.

§ StpAq has ‘weight function’ |ω| :“ 1A ˝ ω. § A weight M-module X is a PCM with M-action ¨ : M ˆ X Ñ X and suitable weight function |¨| : X Ñ M.

§ WModM is an effectus. St: C Ñ WModM is a functor. Effectus over an effect monoid

Let C be effectus, M :“ CpI , I q an effect monoid and A P C. § EffpAq has action of scalars via s ¨ p :“ s ˝ p. § An M-effect module E is an effect algebra with suitable left M-action ¨ : M ˆ E Ñ E. op op § EModM is an effectus. Eff: C Ñ EModM is a functor.

§ StpAq has ‘weight function’ |ω| :“ 1A ˝ ω. § A weight M-module X is a PCM with M-action ¨ : M ˆ X Ñ X and suitable weight function |¨| : X Ñ M.

§ WModM is an effectus. St: C Ñ WModM is a functor.

Any effect monoid is the set of scalars of some effectus Corollary: There are some weird effectuses out there n k § In r0, 1s a sum i“0 xi exists when i“0 xi ď 1 for all k P N. Definition (informal)ř ř A σ-PCM is a PCM where some sums of countable sets are defined. Examples: § PfnpA, Bq. § Let Wstar be category of von Neumann algebras with normal positive subunital maps. Then WstarpA, Bq is σ-PCM. In fact: Pfn and Wstar are σ-PCM enriched.

σ-PCMs

§ pR, `, 0q does not just have finite sums. Some countable sums exist too! Definition (informal) A σ-PCM is a PCM where some sums of countable sets are defined. Examples: § PfnpA, Bq. § Let Wstar be category of von Neumann algebras with normal positive subunital maps. Then WstarpA, Bq is σ-PCM. In fact: Pfn and Wstar are σ-PCM enriched.

σ-PCMs

§ pR, `, 0q does not just have finite sums. Some countable sums exist too! n k § In r0, 1s a sum i“0 xi exists when i“0 xi ď 1 for all k P N. ř ř Examples: § PfnpA, Bq. § Let Wstar be category of von Neumann algebras with normal positive subunital maps. Then WstarpA, Bq is σ-PCM. In fact: Pfn and Wstar are σ-PCM enriched.

σ-PCMs

§ pR, `, 0q does not just have finite sums. Some countable sums exist too! n k § In r0, 1s a sum i“0 xi exists when i“0 xi ď 1 for all k P N. Definition (informal)ř ř A σ-PCM is a PCM where some sums of countable sets are defined. σ-PCMs

§ pR, `, 0q does not just have finite sums. Some countable sums exist too! n k § In r0, 1s a sum i“0 xi exists when i“0 xi ď 1 for all k P N. Definition (informal)ř ř A σ-PCM is a PCM where some sums of countable sets are defined. Examples: § PfnpA, Bq. § Let Wstar be category of von Neumann algebras with normal positive subunital maps. Then WstarpA, Bq is σ-PCM. In fact: Pfn and Wstar are σ-PCM enriched. It turns out that σ-effectuses are way more well-behaved.

σ-effectus

Definition σ-effectus is σ-PCM-enriched effectus with countable coproducts. Examples: § Pfn § Wstarop σ-effectus

Definition σ-effectus is σ-PCM-enriched effectus with countable coproducts. Examples: § Pfn § Wstarop

It turns out that σ-effectuses are way more well-behaved. Theorem (Westerbaan, Westerbaan & vdW, LICS’20)

An ω-directed-complete effect monoid M embeds into M1 ‘ M2 where M1 is a ω-complete Boolean algebra and M2 :“ tf : X Ñ r0, 1s continuousu for a basically disconnected X . Corollary Scalars in a σ-effectus are commutative.

σ-effect monoids

Proposition EffpAq in a σ-effectus is ω-directed-complete, i.e. increasing sequences a1 ď a2 ď ... have suprema. Corollary Scalars in a σ-effectus are commutative.

σ-effect monoids

Proposition EffpAq in a σ-effectus is ω-directed-complete, i.e. increasing sequences a1 ď a2 ď ... have suprema. Theorem (Westerbaan, Westerbaan & vdW, LICS’20)

An ω-directed-complete effect monoid M embeds into M1 ‘ M2 where M1 is a ω-complete Boolean algebra and M2 :“ tf : X Ñ r0, 1s continuousu for a basically disconnected X . σ-effect monoids

Proposition EffpAq in a σ-effectus is ω-directed-complete, i.e. increasing sequences a1 ď a2 ď ... have suprema. Theorem (Westerbaan, Westerbaan & vdW, LICS’20)

An ω-directed-complete effect monoid M embeds into M1 ‘ M2 where M1 is a ω-complete Boolean algebra and M2 :“ tf : X Ñ r0, 1s continuousu for a basically disconnected X . Corollary Scalars in a σ-effectus are commutative. Furthermore, if any and thus all these conditions hold then M – t0u, M – t0, 1u or M – r0, 1s.

Normalisation in σ-effectuses

Theorem Let C be a σ-effectus with M “ CpI , I q. The following are equivalent. § States in C can be normalized. § Non-zero scalars are epi. § M has a ‘division’ operation. § M has no zero divisors (a ¨ b “ 0 ùñ a “ 0 or b “ 0).

§ M is irreducible (M1 ‘ M2 “ M ùñ M1 “ 0 or M2 “ 0). Normalisation in σ-effectuses

Theorem Let C be a σ-effectus with M “ CpI , I q. The following are equivalent. § States in C can be normalized. § Non-zero scalars are epi. § M has a ‘division’ operation. § M has no zero divisors (a ¨ b “ 0 ùñ a “ 0 or b “ 0).

§ M is irreducible (M1 ‘ M2 “ M ùñ M1 “ 0 or M2 “ 0). Furthermore, if any and thus all these conditions hold then M – t0u, M – t0, 1u or M – r0, 1s. § CpI , I q – t0, 1u: C is deterministic, i.e. the probability p ˝ ω that an effect p holds on a state ω is either 0 or 1. § CpI , I q – r0, 1s: C is probabilistic, i.e. the probability p ˝ ω is an actual real probability. When appropriate operational separation properties are satisfied, we can say even more about these latter two cases.

Dichotomy between deterministic and probabilistic models

Hence: σ-effectuses with normalization come in three types: § CpI , I q – t0u: only holds when C is equivalent to the trivial single-object category with a single morphism. § CpI , I q – r0, 1s: C is probabilistic, i.e. the probability p ˝ ω is an actual real probability. When appropriate operational separation properties are satisfied, we can say even more about these latter two cases.

Dichotomy between deterministic and probabilistic models

Hence: σ-effectuses with normalization come in three types: § CpI , I q – t0u: only holds when C is equivalent to the trivial single-object category with a single morphism. § CpI , I q – t0, 1u: C is deterministic, i.e. the probability p ˝ ω that an effect p holds on a state ω is either 0 or 1. When appropriate operational separation properties are satisfied, we can say even more about these latter two cases.

Dichotomy between deterministic and probabilistic models

Hence: σ-effectuses with normalization come in three types: § CpI , I q – t0u: only holds when C is equivalent to the trivial single-object category with a single morphism. § CpI , I q – t0, 1u: C is deterministic, i.e. the probability p ˝ ω that an effect p holds on a state ω is either 0 or 1. § CpI , I q – r0, 1s: C is probabilistic, i.e. the probability p ˝ ω is an actual real probability. Dichotomy between deterministic and probabilistic models

Hence: σ-effectuses with normalization come in three types: § CpI , I q – t0u: only holds when C is equivalent to the trivial single-object category with a single morphism. § CpI , I q – t0, 1u: C is deterministic, i.e. the probability p ˝ ω that an effect p holds on a state ω is either 0 or 1. § CpI , I q – r0, 1s: C is probabilistic, i.e. the probability p ˝ ω is an actual real probability. When appropriate operational separation properties are satisfied, we can say even more about these latter two cases. Similarly we can define effect-separation for an effectus. Examples § Pfn has both state- and effect-separation. § Cstar and Wstar both have state- and effect-separation.

Non-Example op op Category of effect algebras EA – EModt0,1u. Let PpH q P EA be the projections on a Hilbert space H with dimpH q ą 2. Then Kochen-Specker theorem says StpPpH qq “ t0u.

Separation properties

Definition An effectus C has state-separation when f ˝ ω “ g ˝ ω for all states ω P EffpAq implies f “ g for any f , g : A Ñ B. (i.e. iff St: C Ñ WModM faithful) Examples § Pfn has both state- and effect-separation. § Cstar and Wstar both have state- and effect-separation.

Non-Example op op Category of effect algebras EA – EModt0,1u. Let PpH q P EA be the projections on a Hilbert space H with dimpH q ą 2. Then Kochen-Specker theorem says StpPpH qq “ t0u.

Separation properties

Definition An effectus C has state-separation when f ˝ ω “ g ˝ ω for all states ω P EffpAq implies f “ g for any f , g : A Ñ B. (i.e. iff St: C Ñ WModM faithful) Similarly we can define effect-separation for an effectus. Non-Example op op Category of effect algebras EA – EModt0,1u. Let PpH q P EA be the projections on a Hilbert space H with dimpH q ą 2. Then Kochen-Specker theorem says StpPpH qq “ t0u.

Separation properties

Definition An effectus C has state-separation when f ˝ ω “ g ˝ ω for all states ω P EffpAq implies f “ g for any f , g : A Ñ B. (i.e. iff St: C Ñ WModM faithful) Similarly we can define effect-separation for an effectus. Examples § Pfn has both state- and effect-separation. § Cstar and Wstar both have state- and effect-separation. Separation properties

Definition An effectus C has state-separation when f ˝ ω “ g ˝ ω for all states ω P EffpAq implies f “ g for any f , g : A Ñ B. (i.e. iff St: C Ñ WModM faithful) Similarly we can define effect-separation for an effectus. Examples § Pfn has both state- and effect-separation. § Cstar and Wstar both have state- and effect-separation.

Non-Example op op Category of effect algebras EA – EModt0,1u. Let PpH q P EA be the projections on a Hilbert space H with dimpH q ą 2. Then Kochen-Specker theorem says StpPpH qq “ t0u. Then there is a faithful morphism of σ-effectuses F : C Ñ Pfn, and StpAq – F pAq for all A P C. Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ ωBAop, the category of ω-complete Boolean algebras.

Hence, such effectuses are entirely classical. Corollary Nonclassical σ-effectuses w/ normalisation must have scalars r0, 1s.

Classical deterministic effectuses

Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ ωBAop, the category of ω-complete Boolean algebras.

Hence, such effectuses are entirely classical. Corollary Nonclassical σ-effectuses w/ normalisation must have scalars r0, 1s.

Classical deterministic effectuses

Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ Pfn, and StpAq – F pAq for all A P C. Hence, such effectuses are entirely classical. Corollary Nonclassical σ-effectuses w/ normalisation must have scalars r0, 1s.

Classical deterministic effectuses

Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ Pfn, and StpAq – F pAq for all A P C. Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ ωBAop, the category of ω-complete Boolean algebras. Classical deterministic effectuses

Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ Pfn, and StpAq – F pAq for all A P C. Theorem Let C be a σ-effectus with CpI , I q – t0, 1u and state-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ ωBAop, the category of ω-complete Boolean algebras.

Hence, such effectuses are entirely classical. Corollary Nonclassical σ-effectuses w/ normalisation must have scalars r0, 1s. A Banach OUS is furthermore complete in its canonical norm. A σ-OUS furthermore has countable directed suprema. Theorem Let C be a σ-effectus with CpI , I q – r0, 1s and effect-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ σBOUSop, the category of Banach σ-OUSes. Hence: Probabilistic effectuses embed into the well-studied land of real ordered vector spaces.

Convex embeddings of effectuses

A similar sort of embedding holds in the probabilistic setting. Definition (informal) An order unit space (OUS) pV , ď, 1q is a real ordered vector space such that V has no infinitesimals and r0, 1sV generates V . Theorem Let C be a σ-effectus with CpI , I q – r0, 1s and effect-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ σBOUSop, the category of Banach σ-OUSes. Hence: Probabilistic effectuses embed into the well-studied land of real ordered vector spaces.

Convex embeddings of effectuses

A similar sort of embedding holds in the probabilistic setting. Definition (informal) An order unit space (OUS) pV , ď, 1q is a real ordered vector space such that V has no infinitesimals and r0, 1sV generates V . A Banach OUS is furthermore complete in its canonical norm. A σ-OUS furthermore has countable directed suprema. Convex embeddings of effectuses

A similar sort of embedding holds in the probabilistic setting. Definition (informal) An order unit space (OUS) pV , ď, 1q is a real ordered vector space such that V has no infinitesimals and r0, 1sV generates V . A Banach OUS is furthermore complete in its canonical norm. A σ-OUS furthermore has countable directed suprema. Theorem Let C be a σ-effectus with CpI , I q – r0, 1s and effect-separation. Then there is a faithful morphism of σ-effectuses F : C Ñ σBOUSop, the category of Banach σ-OUSes. Hence: Probabilistic effectuses embed into the well-studied land of real ordered vector spaces. § When it is deterministic and has state-separation then it embeds into Pfn and BAop, and hence is classical. § When it is probabilistic and has effect-separation it embeds into BOUSop, and hence reduces to a standard GPT-like object. § So we managed to go from an abstract categorical framework to concrete well-studied standard settings.

Summary

§ A non-trivial σ-effectus with normalisation is either deterministic or probabilistic. § When it is probabilistic and has effect-separation it embeds into BOUSop, and hence reduces to a standard GPT-like object. § So we managed to go from an abstract categorical framework to concrete well-studied standard settings.

Summary

§ A non-trivial σ-effectus with normalisation is either deterministic or probabilistic. § When it is deterministic and has state-separation then it embeds into Pfn and BAop, and hence is classical. § So we managed to go from an abstract categorical framework to concrete well-studied standard settings.

Summary

§ A non-trivial σ-effectus with normalisation is either deterministic or probabilistic. § When it is deterministic and has state-separation then it embeds into Pfn and BAop, and hence is classical. § When it is probabilistic and has effect-separation it embeds into BOUSop, and hence reduces to a standard GPT-like object. Summary

§ A non-trivial σ-effectus with normalisation is either deterministic or probabilistic. § When it is deterministic and has state-separation then it embeds into Pfn and BAop, and hence is classical. § When it is probabilistic and has effect-separation it embeds into BOUSop, and hence reduces to a standard GPT-like object. § So we managed to go from an abstract categorical framework to concrete well-studied standard settings. Future work

§ Study general σ-effectuses (without normalization). § Find inherent categorical characterisation of σ-effectuses. Thank you for your attention References

Kenta Cho, Bas Westerbaan & vdW 2020, arXiv:2003.10245 Dichotomy between deterministic and probabilistic models in countably additive effectus theory

Abraham Westerbaan, Bas Westerbaan & vdW 2019, arXiv:1912.10040 A characterisation of ordered abstract probabilities

Kenta Cho PhD Thesis 2019, arXiv:1910.12198 Effectuses in Categorical Quantum Foundations

Bas Westerbaan PhD Thesis 2018, arXiv:1803.01911 Dagger and Dilation in the category of Von Neumann Algebras