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 :“ t! : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ! is probability that p holds on state ! 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 :“ t! : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ! is probability that p holds on state ! 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 :“ t! : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ! is probability that p holds on state ! 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 pR¥0; `; 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 pR¥0; `; 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 pR¥0; `; 0q § What about pr0; 1s; `; 0q? Partial commutative monoids What should we replace R with? § R is a commutative monoid. § Same holds for pR¥0; `; 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 ¤ 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 ¤ 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 ¤ 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 ¤ 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 ¤ 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.