Big Toy Models: Representing Physical Systems As Chu Spaces

Big T o y M o dels R epr esenting Physic al Systems As Chu Sp ac es Samson Abramsky Oxfor d University Computing L ab or atory Abstract. W e pursue a mod el-oriented rath er than axiomatic approac h t o the foundations of Quantum Mec hanics, with th e idea that new mo dels can often sug- gest new a xioms. This approac h has often been fruitful in Log ic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, w e aim for a ‘big to y mo del’, in whic h b oth quan tum and classi cal systems can b e faithfully represented — as w ell as, possibly , more e xotic k in d s of systems. T o this en d, w e sho w how C hu spaces can b e used to represen t physical systems of v arious k inds. In particular, we sho w ho w quantum systems can b e represented as Ch u spaces ov er the unit in terv al in suc h a w ay that the Chu morphisms corresp ond exactly to the physicall y meaningful symmetries of the systems — the unitaries and antiunitaries. In this w ay w e obtain a full and faithful functor from the groupoid of Hilb ert spaces and their symmetries to Chu spaces. W e also consider whether it is p ossible to use a finite val ue set rather than the unit interv al; w e show that th ree v alues suffice, while the t w o standard p oss ibilistic reductions to tw o v alues b oth fail to p rese rve fullness. 1. In tro duction 1.0.0.1 . Mo dels vs. Axioms The main metho d p u rsued in the found a- tions of qu an tum mec hanics has b een axioma tic ; on e seeks conceptually primitiv e and clea rly motiv ated axioms, sho ws that quan tum systems satisfy these axioms, and then, often, aims for a r epr esentation the or em sho wing th at the axioms essentia lly determine the “sta ndard mo del” of Quant um Mec hanics. Or one ma y admit non-standard in terp reta tions, and seek to lo cate Quan tum Mec hanics in a larger “space” of theories. There is an alternativ e and complemen tary approac h, whic h has b een less explored in the foun d ati ons of Q uan tum Mec hanics, although it has pro v ed very fruitful in mathemat ics, logic and theoretical com- puter science. Namely , one lo oks for conceptually natural c onstructions of mo dels . Often a new mo del constru ction can su gg est new axioms, ar- ticulated in terms of n ew form s of structur e. T here are man y examples of this phenomenon, shea ves and top os theory b eing one case in p oint (MacLane and Mo erdijk, 1992), and domain-theoretic mo dels of the λ -calculus another (Scott, 1970). A successful recen t example of gaining insigh t by mo del constru c- tion is the we ll-kno wn pap er b y Rob Sp ekk ens on a to y mod el for Quant um Mec hanics (Sp ekk ens, 2007), whic h has led to n ov el ideas on c  201 8 Kluwer Ac ademic Publishers. Printe d i n the Netherlands. btm.tex; 24/09/201 8; 17:35; p.1 2 the connections b et w een phase groups and n on-localit y (Co ec ke et al., 2009) . 1.0.0.2 . Big T oy M o d els W e shall also, in a sen s e, b e concerned w ith “to y mo dels” in the presen t p aper; w ith building m odels w hic h exhibit “quan tum-lik e” features with ou t necessarily exactly corresp onding to the standard formalism of Qu an tum Mec hanics. Indeed, the more dif- feren t the mo del constru cti on can b e to the usual formalism, while still repro ducing many quan tum -lik e features, the more inte resting it will b e from this p ersp ectiv e. Ho wev er, th er e will b e an imp ortan t difference b et wee n the kind of mo del we shall study , and the usual idea of a “to y mo del”. Usually , a to y m odel will b e a small, simplified gadget, wh ic h giv es a pictur e of Q uan tum Mec hanics in some “collapsed” form, w ith m uc h detail thrown a w a y . By contrast, w e are aiming f or a bi g to y mo del, in w hic h b oth quantum and classic al systems c an b e faithful ly r e pr esente d — as we ll as, p ossibly , m an y more exotic kin d s of systems. 1.0.0.3 . R e su lts Mo re precisely , we sh al l see ho w the simple, discrete notions of Chu s pace s suffice to determine the app ropriate notions of state equiv alence, an d to pick out the physically s ignifi ca n t symmetries on Hilb ert space in a very striking fashion. This leads to a full and faithful representa tion of the category of quantum sys tems, with the group oid structure of their physical symmetries, in the cat egory of Ch u spaces v alued in the unit interv al. Th e argument s here make use of Wigner’s theorem and the d ualiti es of pr o jectiv e geometry , in the mo dern form develo p ed by F aure and F r¨ olic her (F aure and F r¨ olic her, 2000; Stubb e and v an Steirteghem, 2007). The surpr ising p oin t is that unitarit y/anitunitarit y is essen tially f or c e d by the mere requirement of b eing a Ch u morphism . This eve n extends to su rjectivit y , whic h her e is derive d r ather than assumed. W e also consider the question of whether we can obtain a natural represent ation of this f orm in Chu spaces o ver a finite v alue set. W e sho w that th is can b e done with just three v alues. By con trast, th e t wo stand ard p ossibilistic reductions to t w o v alues b oth fail to pr eserve ful lness . The use of Ch u spaces for r epresen ting physical systems as initiate d in this pap er seems quite pr omising; a num b er of further topics imme- diately suggest themselv es, including mixed states, unive rsal mo dels, the repr esen tation of con v ex theories, linear types, and lo cal logics for quan tum systems. The p la n of the remainder of the pap er is as follo ws . In S ect ion 2, w e s hall provide a brief o v erview of Chu spaces. Section 3 conta ins the main tec hn ical results, leading to a full and f aithfu l r epresen tation of btm.tex; 24/09/201 8; 17:35; p.2 3 quan tum systems and their s ymmetries as Chu spaces and morph ism s of C h u spaces. Section 4 pr ese n ts the results on finite v alue sets. Finally , Section 5 con tains a d iscussion of conceptual and metho dological issues. 2. Ch u Spaces W e sh all assum e that the reader is familiar with a few basic n ot ions of category theory . 1 The bare defi n itio ns of category and functor will suffice for the most part. Ch u spaces are a sp ecial case of a construction whic h originally app eared in (Ch u , 1979) , written b y Po-Hsia ng Chu as an app endix to Mic hael Barr’s monograph on ∗ -autonomous catego ries (Barr, 1979). In terest in ∗ -autonomous categories increased with the adv ent of Linear Logic (Girard, 1987), since ∗ -autonomous categ ories provide mo dels for Classical Multiplicativ e Linear Logic (and w ith add itio nal assumptions, for th e whole of Classical Linear Logic ) (Seely , 198 9). The Ch u constru cti on applied to the categ ory Set of sets and functions w as indep endentl y introdu ced (und er the n ame of ‘games’) b y Yv es Lafon t and T homas Streic her (Lafon t and Streic her, 1991), and subsequently (under the name of Chu sp ac es ) formed the sub ject of a ser ies of pa- p ers by V aughan Pratt and h is collab orat ors, e.g. (Dev ara jan et al., 1999; Pratt, 1995; Pr at t, 1999). Recent pap ers on Chu spaces in clude (Droste and Zh ang, 2007; Palmigi ano and V enema, 2007) . Ch u spaces hav e sev eral int eresting asp ects: − They ha v e a ric h t yp e structure, and in particular form mo dels of Linear Logic. − They ha v e a rich represent ation theory; m any concrete categories of inte rest can b e fully embedd ed into Chu sp ac es. − There is a natur al notion of ‘lo cal logic’ on Chu spaces (Bar- wise and Seligman, 1997), and an interesting c h arac terization of information transf er across Chu m orphisms (v an Benthem, 2000). Applications of Chu s pace s ha ve b een prop osed in a num b er of ar- eas, including concurren cy (Pratt, 2003), hardw are v erification (Iv ano v, 2008) , game theory (V annucci, 2004) and fuzzy sys tems (P apadop ou- los and Syrop oulos, 2000; Nguye n et al., 2001). Mathematical stu d ies concerning th e general Ch u construction include (Barr, 1998; Giuli and Tholen, 2007). W e briefly review the basic d efinitions. btm.tex; 24/09/201 8; 17:35; p.3 4 Fix a set K . A C h u space ov er K is a structur e ( X, A, e ), where X is a set of ‘p oin ts’ or ‘ob jects’, A is a set of ‘attributes’, and e : X × A → K is an ev aluation f unction. A morphism of Chu spaces f : ( X, A, e ) → ( X ′ , A ′ , e ′ ) is a pair of functions f = ( f ∗ : X → X ′ , f ∗ : A ′ → A ) suc h that, for all x ∈ X an d a ′ ∈ A ′ : e ( x, f ∗ ( a ′ )) = e ′ ( f ∗ ( x ) , a ′ ) . Ch u morphisms comp ose comp onen twise: if f : ( X 1 , A 1 , e 1 ) → ( X 2 , A 2 , e 2 ) and g : ( X 2 , A 2 , e 2 ) → ( X 3 , A 3 , e 3 ), then ( g ◦ f ) ∗ = g ∗ ◦ f ∗ , ( g ◦ f ) ∗ = f ∗ ◦ g ∗ . Ch u spaces o v er K and th eir morphisms form a category Chu K . Giv en a C h u space C = ( X, A, e ), we sa y that C is: − extensional if for all a 1 , a 2 ∈ A : [ ∀ x ∈ X . e ( x, a 1 ) = e ( x, a 2 )] ⇒ a 1 = a 2 − sep ar ate d if for all x 1 , x 2 ∈ X : [ ∀ a ∈ A. e ( x 1 , a ) = e ( x 2 , a )] ⇒ x 1 = x 2 − biextensional if it is extensional and separated. W e define a relation on X by: x 1 ∼ x 2 ⇐ ⇒ ∀ a ∈ A. e ( x 1 , a ) = e ( x 2 , a ) . This is evidently an equiv alence relation: C is separated exactly when this relation is the ident it y . There is a Ch u morphism ( q , id A ) : ( X , A, e ) → ( X/ ∼ , A, e ′ ) where e ′ ([ x ] , a ) = e ( x, a ) and q : X → X/ ∼ is the qu otient map. The sp ac e ( X/ ∼ , A, e ′ ) is separated; if ( X , A, e ) is extensional, it is biextensional. Prop osition 2.1 If f : ( X , A, e ) → ( X ′ , A ′ , e ′ ) is a Chu morphism, then f ∗ pr eserves ∼ . That is, for al l x 1 , x 2 ∈ X , x 1 ∼ x 2 ⇒ f ∗ ( x 1 ) ∼ f ∗ ( x 2 ) . Pro of F or an y a ′ ∈ A ′ : e ′ ( f ∗ ( x 1 ) , a ′ ) = e ( x 1 , f ∗ ( a ′ )) = e ( x 2 , f ∗ ( a ′ )) = e ′ ( f ∗ ( x 2 ) , a ′ ) .  btm.tex; 24/09/201 8; 17:35; p.4 5 W e shall write eCh u K , sChu K and bChu K for the full sub cate- gories of Ch u K determined b y the extensional, separated and biexten- sional Chu sp ace s. W e s h all mainly w ork with extensional and b iextensional Ch u spaces. Ob viously bChu K is a fu ll sub-category of eChu K . Prop osition 2.2 The inclusion bChu K ⊂ ✲ eCh u K has a left ad- joint. Pro of The unit of the adjun ctio n is the Chu morphism ( q , id A ) : ( X , A, e ) → ( X/ ∼ , A, e ′ ) w e ha v e already describ ed, while Prop osition 2.1 guaran tees that giv en a Ch u morph ism f : ( X, A, e ) → ( Y , B , r ) to a biextensional Chu space, we can factor it through th e quotien t space ( X/ ∼ , A, e ′ ). The fun cto r Q : eChu K → bChu K pro vided by this adju nction sends morphism s ( f ∗ , f ∗ ) : ( X , A, e 1 ) → ( X ′ , A ′ , e 2 ) to ( f ∗ / ∼ , f ∗ ) : ( X/ ∼ , A, e ′ 1 ) → ( X ′ / ∼ , A ′ , e ′ 2 ) where f ∗ / ∼ ([ x ]) = [ f ∗ ( x )].  W e refer to the functor Q as the biextensional c ol lapse . W e can define an equ iv alence relation on the Ch u morphism s in eac h hom-set in eC h u K b y: f ∼ g ⇐ ⇒ ∀ x. f ∗ ( x ) ∼ g ∗ ( x ) . Then Qf = Qg ⇔ f ∼ g . 2.0.0.4 . R e pr esentations Recall that a fun ct or F : C → D is faithful if for eac h pair of ob jects A , B of C , the indu ced map F AB : C ( A, B ) → D ( F A, F B ) is injectiv e; it is ful l if eac h F AB is surjectiv e; and it is an emb e dding if F is faithful and inj ec tiv e on ob jects. W e refer to a fu ll and faithful functor as a r epr esentation , and to a fu ll em b edd ing as a strict r e pr esentation . Note that if F is a r epresen tation, it can only identify isomorphic ob jects. If F is a r epresen tation, then C is equiv alen t to a full sub-category of D , wh ile if F is a strict represent ation, then C is isomorphic to a full sub -category of D . btm.tex; 24/09/201 8; 17:35; p.5 6 As a first examp le of the representat ional capacit y of Chu spaces, supp ose that { 0 , 1 } ⊆ K . F or any set X , define the follo win g Chu space on K : ( X , P X, e X ), where: e X ( x, S ) =    1 , x ∈ S 0 otherw ise Giv en a f unction f : X → Y , we send it to the Chu space morphism ( f , f − 1 ) : ( X , P X, e X ) → ( Y , P Y , e Y ) . It is easy to see that this defin es a f u ll emb edding of Set int o Ch u K . 3. Represen tation of Quan tum Syste ms Our p oin t of view in mo delling ph ysical sy s te ms as Chu spaces will b e as follo ws. W e tak e a system to b e sp ecified b y its set of states S , and the set of questions Q whic h can b e ‘ask ed’ of the system. W e shall consid er only ‘y es/no’ questions; ho w ev er, the resu lt of asking a question in a giv en state will in general b e pr ob abilistic . Th is will b e represent ed by an ev aluation fu nctio n e : S × Q → [0 , 1] where e ( s, q ) is the pr obabilit y that the qu estio n q will receiv e the answ er ‘yes’ when the sy s te m is in state s . Th is is essentially the p oin t of view tak en by Mac ke y in his classic p ionee ring work on the foun - dations of Qu an tum Mec hanics (Mac k ey , 1963). Note that, follo wing Mac ke y , w e p r efer the term ‘question’ to ‘prop ert y’, since in the case of Qu an tum Mec hanics we cannot think in terms of static prop erties whic h are determinately p ossessed b y a giv en state; questions imply a dynamic act of asking. It is stand ard in the foun datio nal literature on quantum mec h an ics to fo cus on y es/no qu estio ns. How ev er, the usu al app roac h es to quan- tum logic a vo id the direct introdu cti on of probabilities. W e shall retur n to the issue of whether it is necessary to take probabilities as our v alue set in S ec tion 4. W e can tak e th e category Set itself as a crude v ersion of discrete deterministic classical systems, with arbitrary irreversible transforma- tions allo w ed. W e now consider the quantum case, in the pur e state form ulation. Mixed states will b e considered in a s equ el to the pr esen t pap er. btm.tex; 24/09/201 8; 17:35; p.6 7 Let H b e a complex Hilb ert space. 2 W e define the follo wing Chu space o ver [0 , 1]: ( H ◦ , L ( H ) , e H ) where: − H ◦ = H \ { 0 } , the s et of non-zero v ectors. W e shall regard all such v ectors, not necessarily normalized, as represent ations of states of the system. Note th at the zero v ector is not a legitimat e state; its rˆ ole in Quantum Mechanics prop er (as opp osed to linear-algebraic calculatio ns) is largely as an ‘error element ’ wh en op erations can- not legitimately b e p erform ed . − L ( H ) is the lattice of closed su bspaces of H . This is the standard notion of y es/no qu estio ns in Quantum Mec hanics. The observ able corresp onding to the subsp ace S is the self-adjoin t op er ator w hose sp ectral decomp osition is S ⊕ S ⊥ ∼ = H . T o eac h sub s pace S there corresp onds the pr o jector P S . − The ev aluation e H is the fundamenta l formula or ‘statistical al- gorithm’ (Redhead, 1987) giving the b asic predictiv e con tent of Quant um Mec hanics: e H ( ψ , S ) = h ψ | P S ψ i h ψ | ψ i = h P S ψ | P S ψ i h ψ | ψ i = k P S ψ k 2 k ψ k 2 . Note that e H ( ψ , S ) = e H ( ψ k ψ k , S ), so this is equiv alen t to working with normalized vecto rs only . W e h a v e thus directly trans cribed the basic ingredients of the Dirac/v on Neumann-st yle formulation of Quant um Mec hanics (Dirac, 1947; v on Neumann, 1955) in to th e d efinition of the Ch u space corresp onding to a giv en Hilb ert space. 3.1. Characterizing Chu Morphisms on Qu antum Chu Sp aces Recall firstly th e follo wing explicit expression for the p ro jection of a v ector ψ on a subs p ace S . Let { e i } b e an orth onormal basis for S . Then P S ψ = X i h ψ | e i i e i . It follo ws th at ψ ⊥ S if and only if P S ψ = 0 . W e b egin with a basic fact which w e record explicitly . btm.tex; 24/09/201 8; 17:35; p.7 8 Lemma 3.1 F or ψ ∈ H ◦ and S ∈ L ( H ) : ψ ∈ S ⇐ ⇒ e H ( ψ , S ) = 1 . Pro of Firstly , if ψ ∈ S , then P S ( ψ ) = ψ , so e H ( ψ , S ) = 1. Next, w e recall that P S ⊥ = I − P S . Hence e H ( ψ , S ⊥ ) = 1 h ψ | ψ i h ψ − P S ψ | ψ − P S ψ i = 1 h ψ | ψ i ( h ψ | ψ i − h ψ | P S ψ i − h P S ψ | ψ i + h P S ψ | P S ψ i ) = 1 h ψ | ψ i ( h ψ | ψ i − h P S ψ | P S ψ i ) . Hence e H ( ψ , S ) + e H ( ψ , S ⊥ ) = 1 h ψ | ψ i ( h P S ψ | P S ψ i + h ψ | ψ i − h P S ψ | P S ψ i ) = 1 h ψ | ψ i h ψ | ψ i = 1 . So if ψ 6∈ S , it su ffices to show that e H ( ψ , S ⊥ ) > 0. I n this case, ψ = θ + χ , wh ere θ ∈ S and χ ∈ S ⊥ \ { 0 } ; so P S ⊥ ( θ ) = 0 and P S ⊥ ( χ ) = χ . T hen e H ( ψ , S ⊥ ) = 1 h ψ | ψ i h P S ⊥ ( θ ) + P S ⊥ ( χ ) | P S ⊥ ( θ ) + P S ⊥ ( χ ) i = 1 h ψ | ψ i h χ | χ i > 0 .  Prop osition 3.2 The Chu sp ac e ( H ◦ , L ( H ) , e H ) is extensional but not sep ar ate d. The e q u ivalenc e classes of the r elation ∼ on states ar e exactly the rays of H . That is: φ ∼ ψ ⇐ ⇒ ∃ λ ∈ C . φ = λψ . Pro of E xte nsionalit y follo w s d irectl y fr om Lemma 3.1, sin ce if t w o subspaces ha v e th e same ev aluations on all states, th ey ha ve the s ame elemen ts. W e ha v e e H ( λψ , S ) = | λ | 2 | λ | 2 h P S ψ | P S ψ i h ψ | ψ i = e H ( ψ , S ) so φ = λψ ⇒ φ ∼ ψ . F or th e conv erse, let S b e th e one-dimensional subspace (ray) sp anned b y ψ , and s upp ose that φ 6∈ S . By Lemma 3.1, e H ( ψ , S ) = 1, while e H ( φ, S ) 6 = 1. Hence φ 6∼ ψ .  btm.tex; 24/09/201 8; 17:35; p.8 9 Th us w e ha ve r ec o vered the standard notion of pure states as the ra y s of the Hilb ert sp ac e from the general notion of state equiv alence in Chu sp ac es. W e sh all n ow use some notions and results from pro jectiv e geom- etry . W e shall use the very nice Handb o ok article (Stubb e and v an Steirteghem, 2007) as a con v enien t r eferen ce . Giv en a vecto r ψ ∈ H ◦ , we write ¯ ψ = { λψ | λ ∈ C } for the ray whic h it generates. The r a ys are the atoms in the lattice L ( H ). W e wr ite P ( H ) for the set of ra ys of H . By vir tu e of Prop osi- tion 3.2, w e can write the biextensional collapse of ( H ◦ , L ( H ) , e H ) give n b y Prop osition 2.2 as ( P ( H ) , L ( H ) , ¯ e H ) where ¯ e H ( ¯ ψ , S ) = e H ( ψ , S ). W e restate Lemma 3.1 for the biextensional case. Lemma 3.3 F or ψ ∈ H ◦ and S ∈ L ( H ) : ¯ e H ( ¯ ψ , S ) = 1 ⇐ ⇒ ¯ ψ ⊆ S. Pro of Since S is a subs p ace , ¯ ψ ⊆ S iff ψ ∈ S , and the result follo ws from Lemma 3.1.  W e n o w turn to the issue of charac terizing the Chu morp hisms b e- t ween these b ie xtensional Chu representa tions of Hilb ert spaces. This will lead to our first r ep resen tation theorem. T o fix n ota tion, sup p ose we h av e Hilb ert sp ac es H and K , and a Ch u morphism ( f ∗ , f ∗ ) : ( P ( H ) , L ( H ) , ¯ e H ) → ( P ( K ) , L ( K ) , ¯ e K ) . Prop osition 3.4 F or ψ ∈ H ◦ and S ∈ L ( K ) : ¯ ψ ⊆ f ∗ ( S ) ⇐ ⇒ f ∗ ( ¯ ψ ) ⊆ S. Pro of By Lemma 3.3: ¯ ψ ⊆ f ∗ ( S ) ⇔ ¯ e H ( ¯ ψ , f ∗ ( S )) = 1 ⇔ ¯ e K ( f ∗ ( ¯ ψ ) , S ) = 1 ⇔ f ∗ ( ¯ ψ ) ⊆ S.  Note that P ( H ) ⊆ L ( H ). Prop osition 3.5 The fol lowing ar e e quivalent: − f ∗ is inje ctive − The fol lowing diagr am c ommutes: btm.tex; 24/09/201 8; 17:35; p.9 10 P ( H ) f ∗ ✲ P ( K ) L ( H ) ❄ ∩ ✛ f ∗ L ( K ) ❄ ∩ (1) That is, for al l ψ ∈ H ◦ : ¯ ψ = f ∗ ( f ∗ ( ¯ ψ )) . Pro of Clearly , (1) imp lie s th at f ∗ is injectiv e. F or the conv erse, Prop osition 3.4 implies that ¯ ψ ⊆ f ∗ ( f ∗ ( ¯ ψ )). No w s u pp ose that ¯ φ ⊆ f ∗ ( f ∗ ( ¯ ψ )). Applying Prop osition 3.4 agai n, this implies that f ∗ ( ¯ φ ) ⊆ f ∗ ( ¯ ψ ). Since f ∗ ( ¯ φ ) and f ∗ ( ¯ ψ ) are atoms, this imp lies that f ∗ ( ¯ φ ) = f ∗ ( ¯ ψ ), whic h sin ce f ∗ is injectiv e implies that ¯ φ = ¯ ψ . Thus the only atom b elo w f ∗ ( f ∗ ( ¯ ψ )) is ¯ ψ . Since L ( H ) is atomistic (Stubb e and v an Steirteghem, 2007) , this imp lies th at f ∗ ( f ∗ ( ¯ ψ )) ⊆ ¯ ψ .  W e state another imp ortan t basic prop erty of th e ev aluation. Lemma 3.6 F or any φ, ψ ∈ H ◦ : ¯ e H ( ¯ φ, ¯ ψ ) = 0 ⇐ ⇒ φ ⊥ ψ . Pro of ¯ e H ( ¯ φ, ¯ ψ ) = 0 ⇔ h P ¯ ψ ( φ ) | P ¯ ψ ( φ ) i = 0 ⇔ P ¯ ψ ( φ ) = 0 ⇔ φ ⊥ ψ .  Prop osition 3.7 If f ∗ is inje ctive, it pr eserves and r eflect s orthogo- nalit y . That is, for al l φ, ψ ∈ H ◦ : φ ⊥ ψ ⇐ ⇒ f ∗ ( ¯ φ ) ⊥ f ∗ ( ¯ ψ ) . Pro of φ ⊥ ψ ⇐ ⇒ ¯ e H ( ¯ φ, ¯ ψ ) = 0 Lemma 3.6 ⇐ ⇒ ¯ e H ( ¯ φ, f ∗ ( f ∗ ( ¯ ψ ))) = 0 Prop osition 3.5 ⇐ ⇒ ¯ e K ( f ∗ ( ¯ φ ) , f ∗ ( ¯ ψ )) = 0 ⇐ ⇒ f ∗ ( ¯ φ ) ⊥ f ∗ ( ¯ ψ ) Lemma 3.6  btm.tex; 24/09/201 8; 17:35; p.10 11 W e define a map f → : L ( H ) → L ( K ): f → ( S ) = _ { f ∗ ( ¯ ψ ) | ψ ∈ S ◦ } where S ◦ = S \ { 0 } . Lemma 3.8 The map f → is left adjoint to f ∗ . Pro of W e must sho w th at , for all S ∈ L ( H ) and T ∈ L ( K ): f → ( S ) ⊆ T ⇐ ⇒ S ⊆ f ∗ ( T ) . Using Prop osition 3.4, we ha ve: f → ( S ) ⊆ T ⇐ ⇒ ∀ ψ ∈ S ◦ . f ∗ ( ¯ ψ ) ⊆ T ⇐ ⇒ ∀ ψ ∈ S ◦ . ¯ ψ ⊆ f ∗ ( T ) ⇐ ⇒ S ⊆ f ∗ ( T ) .  W e can now extend the diagram (1) : P ( H ) f ∗ ✲ P ( K ) L ( H ) ❄ ∩ f → ✲ ⊥ ✛ f ∗ L ( K ) ❄ ∩ (2) By construction, f → extends f ∗ : this sa ys that f → preserve s atoms. Since f → is a left adj oi n t, it preserves sup s. Hence f → and f ∗ are paired un der the dualit y of pro jectiv e lattices and pro jectiv e geome- tries, for whic h see T h eorem 16 of (Stubb e and v an Steirteghem, 2007). In particular, we ha ve the follo wing. Prop osition 3.9 f ∗ is a total map of p ro jectiv e geometries ( Stub b e and v an S teirteghem, 2007 ). It follo w s that we can apply Wig ne r’s The or em , in the form give n as Theorem 4.1 in (F aure, 2002). In ord er to state this, we n eed some additional notions. Let V 1 b e a ve ctor space o v er the field F and V 2 a v ector sp ace o v er the field G . A semiline ar map from V 1 to V 2 is a pair ( f , α ) where btm.tex; 24/09/201 8; 17:35; p.11 12 α : F → G is a field homomorphism , and f : V 1 → V 2 is an additiv e map such that, for all λ ∈ F and v ∈ V 1 : f ( λv ) = α ( λ ) f ( v ) . Note that semilinear maps comp ose: if ( f , α ) : V 1 → V 2 and ( g , β ) : V 2 → V 3 , then ( g ◦ f , β ◦ α ) : V 1 → V 2 is a semilinear map. This notion is usually defi ned in greater generalit y , for division rings, but w e are only concerned with Hilb ert spaces o ve r the complex n um b ers. Giv en a s emilinear map g : V 1 → V 2 , w e defin e P g : P V 1 → P V 2 b y P ( g )( ¯ ψ ) = g ( ψ ) . W e can now state Wigner’s Theorem in the form w e shall us e it. Theorem 3.10 L et f : P ( H ) → P ( K ) b e a total map of pr oje ctive ge ometries, wher e dim H > 2 . If f pr eserves ortho g on ality, me aning that ¯ φ ⊥ ¯ ψ ⇒ f ( ¯ φ ) ⊥ f ( ¯ ψ ) then ther e is a semiline ar map g : H → K such that P ( g ) = f , and h g ( φ ) | g ( ψ ) i = σ ( h φ | ψ i ) , wher e σ is the homomorp hism asso ciate d with g . M or e over, this homo- morphism is either the identity or c omplex c onjugation, so g is either line ar or antiline ar. The map g is unique up to a ph ase , i.e. a sc alar of mo dulus 1. The final statemen t follo w s from the Second F un damen tal Theorem of Pro jectiv e Geometry , Theorem 3.1 in (F aure, 2002) or Theorem 46 in (Stubb e and v an Steirteghem, 2007) . Note that in our case, taking f ∗ = f , P g is j u st the action of the biextensional collapse f unctor on Ch u morph isms. Note that a tota l map of pr o jectiv e geo metries m u st necessarily come f rom an inje ctive map g on the un derlying v ector spaces, since P ( g ) maps r ays to r a ys, and hence g m ust hav e trivial ke rnel. F or this reason, partial maps of pro jectiv e geometries are considered in the F aure-F r ¨ olic h er app roac h (F aure and F r¨ olic h er, 200 0; Stubb e and v an Steirteghem, 2007 ). Ho wev er, we are simply f ol lo win g the ‘log ic’ of Ch u space morph isms h ere. Prop osition 3.11 L et g : H → K b e a semiline ar morphism such that P ( g ) = f ∗ wher e f is a Chu sp ac e morphism, and dim( H ) > 0 . Supp ose that the endomo rphism σ : C → C asso ciate d with g i s surje ctive, and henc e an automorphism. Then g is surje ctive. btm.tex; 24/09/201 8; 17:35; p.12 13 Pro of W e wr ite Im g for the set-theoretic d irect image of g , wh ich is a linear subsp ac e of K , since σ is an automorphism. In particular, g carries ra ys to r a ys, since λg ( φ ) = g ( σ − 1 ( λ ) φ ). W e claim that for any ve ctor ψ ∈ K ◦ whic h is not in the image of g , ψ ⊥ Im g . Given suc h a ψ , for an y φ ∈ H ◦ it is not the case that f ∗ ( ¯ φ ) ⊆ ¯ ψ ; for otherw ise, for some λ , g ( φ ) = λψ , and hence g ( σ − 1 ( λ − 1 ) φ ) = ψ . Then b y Prop osition 3.4, f ∗ ( ¯ ψ ) = { 0 } . It follo ws that for all φ ∈ H ◦ , ¯ e K ( f ∗ ( ¯ φ ) , ¯ ψ ) = ¯ e H ( ¯ φ, { 0 } ) = 0 , and hence by Lemma 3.6 that ψ ⊥ Im g . No w sup pose for a con tradiction that su c h a ψ exists. Cons id er the v ector ψ + χ where χ is a non-zero ve ctor in Im g , which must exist since g is injectiv e and H has p ositiv e d imension. Th is ve ctor is not in Im g , nor is it orthogonal to Im g , sin ce e.g. h ψ + χ | χ i = h χ | χ i 6 = 0. This yields the r equired con tradiction.  W e can no w put the p iec es toget her to obtain the main result of th is section. W e sa y that a m ap U : H → K is semiunitary if it is either unitary or antiunitary; that is, if it is a bijectiv e map satisfying U ( φ + ψ ) = U φ + U ψ , U ( λφ ) = σ ( λ ) U φ, h U φ | U ψ i = σ ( h φ | ψ i ) where σ is the identit y if U is un itary , and complex conjugation if U is an tiunitary . Note that semiunitaries pr eserv e norm, so if U and V are semiunitaries and U = λV , then | λ | = 1. Theorem 3.12 L et H , K b e Hilb ert sp ac e s of dimension gr e ater than 2. Consider a Chu morphism ( f ∗ , f ∗ ) : ( P ( H ) , L ( H ) , ¯ e H ) → ( P ( K ) , L ( K ) , ¯ e K ) . wher e f ∗ is inje ctive. Then ther e is a semiunitary U : H → K such that f ∗ = P ( U ) . U is unique up to a phase. Pro of By th e proviso on injectivit y , w e can apply Prop osition 3.7. By th is and Prop osition 3.9, together with the pro viso on dimension, w e can apply Wigner’s Theorem 3.10. Since the semilinear map in Wigner’s Theorem has an asso ciat ed automorphism , w e can apply Prop osition 3.11.  3.2. The Repres ent a t ion Theorem W e n o w tur n to the big picture. W e define a category SymmH as follo ws: btm.tex; 24/09/201 8; 17:35; p.13 14 − The ob jects are Hilb ert sp ac es of dimension > 2. − Morphisms U : H → K are semiunitary ( i.e. u nitary or an tiuni- tary) maps. − Semiunitaries comp ose as explained more generally for semilinear maps in the previous subsection. S ince complex conju ga tion is an in v olution, semiunitaries comp ose according to the rule of signs: t wo antiunitaries or tw o u nitaries comp ose to form a u nitary , while a unitary an d an anti unitary comp ose to form an an tiu n ita ry . This catego ry is a group oid, i.e. ev ery arr o w is an isomorph ism. The semiunitaries are the physic al ly signific ant symmetries of Hilb e rt sp ac e from the p oin t of view of Quantum Mec hanics. Th e usual d y- namics acco rding to the Sc h r ¨ odinger equation is give n b y a contin uous one-parameter group { U ( t ) } of these symm et ries; the r equiremen t of con tinuit y forces the U ( t ) to b e un ita ries. 3 Ho wev er, some imp ortant physic al symmetries are represente d by antiunitaries, e.g. time r ev ersa l and char ge c onjugation . By the results of the pr evio us sub s ec tion, Chu morph isms essentia lly force us to consider the symmetries on Hilb ert space. As p oin ted out there, linear maps w hic h can b e represent ed as Chu morphisms in the biextensional category m u st b e injectiv e; and if g : H → K is an injectiv e linear or antili near map, then P ( g ) is inj ectiv e. Our results then show that if g can b e rep r esen ted as a Chu morphism, it must in fact b e semiunitary . This delineation of the physical ly significant symmetries by the logic of Chu morphisms should b e seen as a strong p oin t in fa v our of this repr esentati on by Chu sp ace s. W e define a functor R : SymmH → eChu [0 , 1] : R : U : H → K 7− → ( U ◦ , U − 1 ) : ( H ◦ , L ( H ) , e H ) → ( K ◦ , L ( K ) , e K ) where U ◦ is the r estriction of U to H ◦ . As n oted in Prop osition 2.2, the inclusion bC h u [0 , 1] ⊂ ✲ eCh u [0 , 1] has a left adjoint , w hic h we name Q . By Prop osition 3.2, this can b e defined on the image of R as follo w s: Q : ( H ◦ , L ( H ) , e H ) 7→ ( P H , L ( H ) , ¯ e H ) and for ( U ◦ , U − 1 ) : ( H ◦ , L ( H ) , e H ) → ( K ◦ , L ( K ) , e K ), Q : ( U ◦ , U − 1 ) 7− → ( P U, U − 1 ) . W e wr ite emChu , bmChu for the sub categories of eCh u [0 , 1] and bCh u [0 , 1] obtained b y restricting to Chu morp h isms f for w hic h f ∗ is injectiv e. Th e f unctors R and Q factor through th ese su bcategories. btm.tex; 24/09/201 8; 17:35; p.14 15 Prop osition 3.13 R : SymmH → emChu and Q : emCh u → bmCh u ar e functors. R is faithful but not ful l; Q is fu l l but not faithful. Pro of W e verify th at if U : H → K is semiunitary , R U is a w ell- defined morp hism in e mCh u . Firstly , we v erify the C h u morphism condition. Since U is semiunitary , f or ψ ∈ H ◦ and S ∈ L ( K ): P S ( U ψ ) = U ( P U − 1 ( S ) ψ ) . Indeed, if U is unitary , let { e i } b e an orthonorm al b asis for S . Then { U − 1 e i } is an orthon orm al basis for U − 1 S . Now U ( P U − 1 ( S ) ψ ) = U ( P i h ψ | U − 1 e i i U − 1 e i ) = P i h ψ | U − 1 e i i e i = P i h U ψ | e i i e i = P S U ψ where th e third equation holds because U − 1 = U † . A similar cal- culation h olds if U is anti unitary . In this case, th e inner pro duct is comm u te d when we apply conjugate linearit y in the s econd equation, and comm u te d bac k in the third, since for an an tiunitary we ha ve h U − 1 e i | ψ i = h U − 1 e i | U − 1 U ψ i = h U ψ | e i i , leading to the same result. Moreo ve r, U preserv es norms, so k U ψ k = k ψ k . Now h P S U ψ | P S U ψ i = h U ( P U − 1 ( S ) ψ ) | U ( P U − 1 ( S ) ψ ) i = h P U − 1 ( S ) ψ | P U − 1 ( S ) ψ i . Hence e H ( ψ , U − 1 ( S )) = e K ( U ψ , S ), so ( U ◦ , U − 1 ) is a Chu morphism. Finally , U is b iject iv e, so U ◦ is injectiv e.  W e can analyze the non-fullness of R more precisely as follo ws. Prop osition 3.14 L et ( U ◦ , U − 1 ) : ( H ◦ , L ( H ) , e H ) → ( K ◦ , L ( K ) , e K ) b e a Chu morphism in the image of R . Given an arbitr ary function f : H → C \ { 0 } , define f U : H ◦ → K ◦ by: f U ( ψ ) = f ( ψ ) U ( ψ ) . Then ( f U, U − 1 ) ∼ ( U ◦ , U − 1 ) . Mor e over, the ∼ -e q uivalenc e class of U is exactly the set of functions of this form. btm.tex; 24/09/201 8; 17:35; p.15 16 Th us b efore biextensional collapse, C h u morp hisms can in tro duce ar- bitrary scalar f ac tors p oin t wise. Once we mo ve to the biextensional catego ry , we kn ow by T heorem 3.12 that our representa tion will b e full, and essen tially faithful — u p to a global p hase. This p oin ts to the need for a pro j ective v er s io n of the symmetry group oid. The mathematical ob ject underlyin g phases is the c i r cle gr oup U (1): U (1) = { λ ∈ C | | λ | = 1 } = { e iθ | θ ∈ R } Since λ has mo dulus 1 if and only if λ ¯ λ = 1, U (1) is the unitary group on the one-dimensional Hilb ert space. The circle group acts on the symmetry group oid SymmH b y scalar m ultiplication. F or Hilb ert sp ac es H , K we can define U (1) × SymmH ( H , K ) → SymmH ( H , K ) :: ( λ, U ) 7→ λU. Moreo ve r, this is a category action, meaning th at ( λU ) ◦ V = U ◦ ( λV ) = λ ( U ◦ V ) . It follo ws that w e can form a quotien t category (in fact again a group oid) with the same ob jects as SymmH , an d in which the morp hisms will b e th e orbits of this group action: U ∼ V ⇔ ∃ λ ∈ U (1) . U = λV . W e call the resulting cate gory P SymmH , the pr oje ctive quantum sym- metry gr oup oid . It is a natural generalization of th e standard n oti on of the pr oje ctive unitary gr oup on Hilb ert space. T here is a qu oti en t functor P : SymmH → P SymmH , and by virtue of Th eo rem 3.12, w e can factor Q ◦ R through this quotient to obtain a f unctor P R : P SymmH → bmChu . The situation can b e summarized by th e follo w ing diagram: SymmH > R > emCh u P SymmH P ∨ ∨ > P R > > bmCh u Q ∨ ∨ Theorem 3.15 The functor P R : P SymmH → bmCh u i s a r epr e- sentation. btm.tex; 24/09/201 8; 17:35; p.16 17 Pro of This follo ws fr om Theorem 3.12 . T o see that P R is essen tially injectiv e on ob jects, we can u s e the r epresen tation theorems of Piron and Sol ` er (Stub b e and v an Steirteghem, 2007), which tell us that w e can reconstru ct H as a Hilb ert s pace f rom L ( H ). Th is reconstruction will giv e us a Hilb ert space H ′ suc h that L ( H ) ∼ = L ( H ′ ), and P ( H ) ∼ = P ( H ′ ). W e can app ly Wigner’s theorem to this isomorphism to obtain a semiu n ita ry U : H ∼ = H ′ from wh ic h w e can reco v er the Hilb ert space structure on H . Th is means that w e ha v e reco v ered H un iquely to within the coset of id H in P SymmH .  4. Reducing T he V alue Set W e no w return to the issue of wh et her it is n ece ssary to use the full unit in terv al as the v alue set for our C h u spaces. W e b egin with some generalities. Given a fun cti on v : K → L , we define a fun cto r F v : Ch u K → C h u L : F v : ( X , A, e ) 7→ ( X , A, v ◦ e ) and F v f = f for Chu morphisms f . Prop osition 4.1 F v is a f aithful functor. If v is inje ctive, it is fu l l. Pro of T his is easily v erifi ed. Th e Chu morphism condition is pre- serv ed by comp osing with any function on v alues, while F v is eviden tly faithful. F or fullness, note that the only v alues in L relev ant to whether a pair of f u nctions ( f , g ) : ( X , A, v ◦ e ) → ( X ′ , A ′ , v ◦ e ′ ) satisfies the Ch u morph ism condition are those in th e ranges of v ◦ e and v ◦ e ′ , w h ic h if v is injectiv e are in bijection with those in the r an ges of e and e ′ .  W e can now state the question we wish to p ose more p recisel y: Is there a mapp ing v : [0 , 1] → K fr om the unit inte rv al to s ome finite set K suc h that the restriction of the fu ncto r F v to the image of P R is full, and thus the comp ositio n F v ◦ P R : P SymmH → bmCh u K is a represent ation? There is no gener al reason to supp ose that this is p ossible; in fact, we shall sho w that it is, although not quite in the obvious fashion. btm.tex; 24/09/201 8; 17:35; p.17 18 W e shall w r ite n = { 0 , . . . , n − 1 } for the finite ord inals. T h e m ost p opular c hoice of v alue set for Chu spaces, by f ar, has b een 2 , and indeed m an y in teresting categories can b e strictly (and ev en concretely) represent ed in Ch u 2 (Pratt, 1995). Th is mak es the follo wing question natural: Question 4.2 Is ther e a fu nction v : [0 , 1] → 2 such that F v ◦ P R is ful l and faithful? What w e can sho w is that the m ost plausible candidates for su c h functions, yielding the t wo canonical forms of p ossibilistic semantics as a coarse-graining of p robabilistic seman tics, b oth in f act fail . Note that any fu nctio n v : [0 , 1] → { 0 , 1 } m ust lose information either on 0 or on 1 – or b oth. I n this sense, the t wo ‘sharp est’ mappings 4 will b e: v 0 : 0 7→ 0 , (0 , 1] 7→ 1 v 1 : [0 , 1) 7→ 0 , 1 7→ 1 . These are the tw o canonical reductions of probabilistic to p ossibilistic information: the firs t maps ‘defin itel y not’ to ‘no’, and an ything else to ‘y es’, wh ic h is to b e r ead as ‘p ossibly y es’; the second maps ‘definitely y es’ to ‘y es’, and anything else to ‘no’, to b e r ea d as ‘p ossibly no’. Note that, un d er the first of these, Lemma 3.1 will n o longer h old, while under the second, Lemma 3.6 w ill fail. Prop osition 4.3 F or ne i th er v = v 0 nor v = v 1 is F v ◦ P R fu l l. Pro of Le t H b e a Hilb ert s pace with 2 < dim H < ∞ , an d let ( g , σ ) b e an y semilinear automorphism of H , where σ can b e any au- tomorphism of th e complex field. 5 F or eac h of the ab o ve t wo mapp ings of the u n it in terv al to 2 , we shall construct a Ch u 2 endomorphism f : F v ◦ P R ( H ) → F v ◦ P R ( H ) with f ∗ = P ( g ). This will s h o w the non-fullness of F v . Case 1 Here we consider the mapp ing v 1 whic h sends [0 , 1) to 0 and fixes 1. In this case: ¯ e H ( ¯ ψ , S ) = 1 ⇐ ⇒ ψ ∈ S and h ence the Chu morphism condition on ( f ∗ , f ∗ ), wh ere f ∗ = P ( g ), is: ψ ∈ f ∗ ( S ) ⇐ ⇒ g ( ψ ) ∈ S. T aking f ∗ = g − 1 ob viously fulfills this condition. Note that, since g is a semilinear automorphism, and H is finite-dimensional, g − 1 : L ( H ) → L ( H ) is w ell-defined. Case 2 No w consider the mapp ing v 0 k eeping 0 fi xed and sen ding (0 , 1] to 1. In this case: ¯ e H ( ¯ ψ , S ) = 0 ⇐ ⇒ ψ ⊥ S btm.tex; 24/09/201 8; 17:35; p.18 19 and h ence the Chu morphism condition on ( f ∗ , f ∗ ), wh ere f ∗ = P ( g ), is: ψ ⊥ f ∗ ( S ) ⇐ ⇒ g ( ψ ) ⊥ S. W e defin e f ∗ ( S ) = g − 1 ( S ⊥ ) ⊥ . Note that f ∗ : L ( H ) → L ( H ) is w ell defined, and also that g − 1 ( S ⊥ ) is a subspace of H ; hence g − 1 ( S ⊥ ) ⊥⊥ = g − 1 ( S ⊥ ). No w: ¯ e H ( ¯ ψ , f ∗ S ) = 0 ⇐ ⇒ ψ ⊥ f ∗ S ⇐ ⇒ ψ ∈ g − 1 ( S ⊥ ) ⊥⊥ = g − 1 ( S ⊥ ) ⇐ ⇒ g ( ψ ) ∈ S ⊥ ⇐ ⇒ g ( ψ ) ⊥ S ⇐ ⇒ ¯ e H ( f ∗ ( ¯ ψ ) , S ) = 0 and hence ( f ∗ , f ∗ ) is a Chu morphism as required.  Ho wev er, this negativ e result immediately su gg ests a remedy: to ke ep the interpr etations of b o th 0 and 1 sharp . W e can do this w ith th r ee v alues! Namely , we define v : [0 , 1] → 3 b y 0 7→ 0 , (0 , 1) 7→ 2 , 1 7→ 1 Th us w e lose inform at ion only on the probabilities strictly b et we en 0 and 1, w hic h are lump ed together as ‘maybe’ — repr ese n ted here, b y arbitrary con v en tion, by 2. Wh y is th is adequate? Giv en a v ector ψ and a subspace S , we can write ψ uniquely as θ + χ , w here θ ∈ S and χ ∈ S ⊥ . F or non-zero ψ , there are only three p ossibilities: θ = 0 and χ 6 = 0 , which yields e H ( φ, S ) = 0 b y Lemm a 3.6 ; θ 6 = 0 and χ = 0 whic h yields e H ( φ, S ) = 1 b y L emm a 3.1; and θ 6 = 0 and χ 6 = 0 , whic h yields e H ( ψ , S ) ∈ (0 , 1) b y these Lemmas again, and h ence v ◦ e H ( ψ , S ) = 2. These ar e the only c ase discriminatio ns which ar e use d in our r esults le ading to the R e pr esentation The or em 3.15 . Hence we ha v e: Theorem 4.4 The functor F v ◦ P R : P SymmH → bmCh u 3 is a r e pr esentation. W e note that Ch u 3 has found some u ses in concurren cy and v eri- fication (Pratt, 200 3; Iv ano v, 2008), under a temp oral interpretati on: the three v alues are read as ‘b efore’, ‘durin g’ and ‘after’, whereas in our setting the th ree v alues represent ‘definitely yes’, ‘definitely n o’ and ‘ma yb e’. Theorem 4.4 ma y suggest some interesting uses for 3-v alued ‘lo cal logics’ in the sen s e of Jon Barwise (Barwise and Seligman, 1997). btm.tex; 24/09/201 8; 17:35; p.19 20 5. Discussion W e sh ou ld understand Ch u spaces as pro vidin g a v ery general (and, w e migh t reasonably sa y , rather simple) ‘logic of s ystems or stru ctur es’. Indeed, they hav e b een prop osed by Barwise and S elig man as the v e- hicle for a general logic of ‘distributed systems’ and inform at ion fl o w (Barwise and Seligman, 1997 ). This logic of Chu spaces w as in no w a y biassed in its conception to w ards the description of quan tum mec hanics or an y other kind of physical sys te m. Just for this reason, it is inte rest- ing to see h o w m u ch of quantum-mec hanical structur e and concepts can b e absorb ed and essenti ally determine d by this more general systems logic. It m igh t b e argued that our r epresen tation of quantum systems as Ch u spaces has already sp ecified the essentia l ingredien ts of the quan- tum structure ‘by h and’. T he conceptual s ignifi ca nce of our tec hnical results is precisely to sh o w that there is a n on-trivia l ‘capturing’ of quan tum structure by the general notions of Chu spaces: − Firstly , Prop osition 3.2 sho ws that the general Ch u space n ot ion of biextensionalit y sub sumes the standard identificat ion of quantum states with rays in Hilb ert space. This is scarcely su r prising, but it is a first sign of the p rop er alignmen t of concepts. − The main tec h nical r esult of the present pap er is the Represen- tation Theorem 3.15. It is w orth sp elling out the con ten t of this in more elementa ry terms. Once we hav e r ep resen ted our qu an- tum systems as b iextensional Chu spaces ( P ( H ) , L ( H ) , e H ), all we ha v e, fr om the viewp oin t ‘inside’ the catego ry Ch u [0 , 1] , is a pair of sets and an ev aluation fu nction, with all information ab out their pro v enance lost. A Chu m orphism ( f ∗ , f ∗ ) : ( P ( H ) , L ( H ) , e H ) → ( P ( K ) , L ( H ) , e K ) is given by any pair of set-theoretic functions ( f ∗ , f ∗ ) satisfying the Chu morphism condition: ¯ e H ( ¯ ψ , f ∗ ( S )) = ¯ e K ( f ∗ ( ¯ ψ ) , S ) . The Representa tion Th eo rem sa ys that the lo gic of this Chu mor- phism c ondition is str ong enough to guar ante e that any such p air of functions must arise fr om a unitary or antiunitary map U : H → K on the underlying H ilb ert sp ac es , with the sole p r o viso of injectivit y of f ∗ . 6 Moreo ve r, U is uniquely determined by f ∗ up to a phase factor. Of cours e, w e are u sing one of the ‘big guns’ of the su b ject, Wigner’s Theorem, to establish this result. It is w orth noting, btm.tex; 24/09/201 8; 17:35; p.20 21 though, that there is some d ista nce to trav el b et w een th e Chu morphism condition and the h yp otheses of Wigner’s Theorem; and there are sur p rises along the wa y , m ost notably Prop osition 3.11, whic h derives surjectivit y fr om the C h u morphism condition — whereas it m ust inv ariably b e added as a hyp othesis to the many v ersions of Wigner’s Th eorem. 7 − The r esults on reduction to finite v alue sets are also in triguing. Not only is the bare Ch u condition on morphisms sufficient to whittle them do wn to the semiu n itarie s, this is ev en th e case when the discriminations on wh ich th e cond iti on is b ased are r educed to three v alues. T he general case for t wo v alues remains op en, b ut we ha v e shown that the t wo standard p ossibilistic redu ctions b oth fail to pr eserve ful lness . A n egativ e answ er for tw o-v alued semantics in general would suggest an un expected rˆ ole for three-v alued logic in the f oundations of Quan tum Mec h anics. 5.0.0.5 . Wher e N ext? Of course, the devel opmen ts describ ed in the present p ap er are only a first step. W e shall briefly discuss some of the n at ural con tinuatio ns of these ideas, sev eral of wh ic h are already in progress. − There are some inte resting and surp rising connections b et w een Ch u spaces and another imp ortan t paradigm for categorical sys- tems mo delling, namely c o algebr a (Rutten, 2000) . These connec- tions, wh ich s eem not to ha v e b een explored previously , arise b oth at the general lev el, and also with sp ecific reference to the rep- resen tation of physica l sy s te ms. Th ey are describ ed in a sequel to the p resen t p aper (Abr amsk y , 2010), which lifts the results of the p resen t p ap er to a coalgebraic setting. T he b iv ariant nature of Ch u spaces is r eflect ed in a no vel fibred form of coalgebra, in whic h con tra v ariance is repr esen ted as i ndexing . − A natur al next s te p as regards physica l mo delling is to consider mixe d states . There is a general construction on Ch u spaces whic h allo ws mixed states to b e studied in a uniform fash ion, applicable to b oth classical and quantum systems. Th is will b e describ ed in a forthcoming sequel to th e p resen t pap er. − There are int riguing connections b et w een our approac h , and the w ork of the ‘Genev a Sc ho ol’ of Jauc h and Piron (Jauc h, 1968; Piron, 1976), particularly (F aure et al., 1995). W e plan to explore these in a j oint p ap er with Bob C oec k e, Isar Stubb e and F rank V alc k enb orgh. btm.tex; 24/09/201 8; 17:35; p.21 22 − It is also of inte rest to consid er universal Ch u s pace s; single sys- tems in whic h all Chu spaces of a giv en class can b e embed ded, and whic h therefore provi de a single mo del for a large class of systems. W e ma y additionally ask f or suc h systems to b e homo gene ous , whic h means that they exhib it a maxim um degree of s y m metry; suc h universal, h omog eneous spaces are un ique up to isomorphism . Univ ersal homogeneous Chu sp ac es ha v e b een constructed for bifi- nite Chu sp ac es in r ece n t w ork by Manfred Droste and Guo-Qiang Zhang (Droste and Zhan g, 2007). That con text is to o limited for our purp oses here. It remains to b e seen if univ ersal homogeneous mo dels can b e constructed for larger su b categories of Chu spaces, encompassing those in v olv ed in our representat ion r esu lts. − The relation of the rich logica l and t yp e-theoretic asp ects of Chu spaces to quan tum and other physic al systems should also b e in v estigated. Notes 1 The c harming in trodu ctory text (Pierce, 1991) should b e more t h an sufficient. 2 A useful reference for the mathematical bac kground is (Jordan, 1969). 3 Indeed, t h e Sc hr¨ odinger equation can actually be reco ver ed from this group via Stone’s Theorem (S imon, 1976). 4 W e consider only funct ions whic h fix 0 an d 1, to exclude irrelev ant p ermutatio ns and the trivial case of constant maps. 5 W e can ext end the argument to infinite-dimensional Hilbert space by requiring g to b e continuous. 6 The injectivity assumption on f ∗ is annoying. 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