Generalized probabilities in statistical theories

Generalized Probabilities in Statis tical Theories Federico Holik 1 ∗ , C ´ esar Mas sri 2 , Angelo Plastino 1 Manuel S ´ aenz 2 August 4, 2021 1 Instituto de F ´ ısica (IFLP-CCT-CONICET), Un ivers idad Nacional de La Plata, C.C. 727, 1900 La Plata, Argen tina 2 Departamento de Matem´ atica, Universidad CAECE (1084), Buenos Aires, Argentina; 3 International Center for Theoretical Physics, UNESCO, (34151) T rieste, Italy; ∗ holik@ﬁsica.unlp.edu.ar Abstract W e discuss diﬀerent formal framew orks for the d escription of general- ized probabilities in statistical theories. W e analyze the particular cases of probabilities app earing in classical and quantum mec hanics and the approac h to generalized probabilities based on conv ex sets. W e argue for considering quantum p robabilities as the natural probabilistic assign- ments for rational agents dealing with contextual probabilistic mo dels. In this w a y , th e formal structure of quantum probabilities as a non-Bo olean probabilistic calculus is endow ed with a natural interpretation. Keywor ds: quantum pr ob ability; lattic e the o ry; information the ory; classic al pr ob ability; Cox’s appr o ach 1 In tro duction In the year 1900, the great ma thematician Da vid Hilb ert presented a famous list of problems at a Conference in Paris. Hilber t s uggested that the eﬀorts of the ma thematicians in the years to come sho uld b e oriented in the solution o f these problems. The complete list w as published later [1]. Remark ably , o ne of these pro blems was dedica ted to the ax io matic treatment of probability theory and physical theor ies. In Hilb ert’s own words ([1], p. 4 5 4): “The in v estigations on the fo undations of geometry sug gest the pr ob- lem: T o tre at in the sa me manner, by mea ns of axioms, those phys- ical sc ie nces in whic h mathematics plays an imp ortant part; in the ﬁrst rank a re the theo r y o f pr obabilities and mechanics. As to the axioms of the theory o f proba bilities, it seems to me desir able that 1 their logical inv estigation should be accompanied b y a rigoro us a nd satisfactory developmen t of the metho d of mean v alues in mathe- matical physics, a nd in par ticula r in the k ine tic theory of g a ses.” After a series of pr eliminary inv estigations by many resear chers (see, for example [2]), an axio matization of pro ba bility theory was ﬁnally presented in the 1930s b y Andrey Ko lmogorov [3]. This contribution, whic h can b e consider ed as the foundation of mo dern probability theory , is based on measure theory . Indeed, in Ko lmogorov’s axioma tic treatment, pr obability is cons ider ed as a measure deﬁned over a suitable collectio n of even ts, or ganized as a sigma-algebr a (whic h is found to b e also a Bo ole an lattic e ). His list of axio ms allows the description of ma ny examples o f interest and was considered as a rea sonable fulﬁllmen t of Hilb ert’s progr a m for probability theory . Hilber t himse lf dedica ted great eﬀorts to s o lve his sixth problem. His con- tributions were inﬂuen tial in the development of Rela tivity Theory , and he also contributed to the development of Quantum Mechanics. Indeed, Qua n tum Me- chanics acquired its rigorous axiomatic formulation after a series of pa per s by Hilber t, J . von Neumann, L. Nordheim, and E. P . Wigner [4]. It can b e said that its deﬁnitive form was accomplished in the b o ok o f von Neumann [5]. This axiomatic a pproach w as extended to the r elativistic setting in subsequent years (see, for exa mple, [6, 7]; see [8] for a more up dated expos ition of the a lgebraic ap- proach; and for a rigorous for mu lation of qua ntum sta tistical mechanics, see [9]). How ev er, the adv ent of Qua nt um Mechanics pres ent ed a mo de l of probabil- ities that had many p e c uliar features . R. P . F eynman stated this clear ly in [1 0], p. 5 33: “I should say , that in spite of the implication of the title of this talk the c oncept o f probability is not a ltered in quantum mec hanics. When I say the probability o f a certain outco me of an exp eriment is p , I mean the conv en tional thing, that is, if the exp eriment is re- pea ted many times one exp ects that the fraction of those which giv e the outcome in q uestion is ro ug hly p . I will not b e at all concerned with analyzing o r deﬁning this co ncept in more detail, for no depar- ture of the concept us ed in classical s tatistics is r equired. What is changed, and changed radic a lly , is the method of calculating pro ba- bilities.” What is the meaning of F eynman’s w ords? F eynman tells us that the wa y of computing freq uencies is not altered in quantum mechanics: the real n um ber s yielded by Bo rn’s rule can b e tested in the lab in the usual way . How ever, the method for c omp uting pr obabilities has changed in a radical wa y . As put in [11], this can be r ephrased as follows: the radical change has to do with the recip e that quantum mechanics gives us for calculating new probabilities fro m old. The radica l change men tioned b y F e y nman lies b ehind all the asto nishing features o f q uantum pheno mena. This w as reco gnized very quickly as a no n- classical feature. These p eculiarities a nd the forma l asp ects of the probabilities inv olv ed in quantum theor y ha ve b een ex tens ively studied in the literature [1 2, 2 13, 14, 15, 16, 17, 18, 19]. W e refer to the pr obabilities rela ted to qua ntu m phenomena as quantum pr ob abilities (QP ). According ly , we r efer to pro babilities ob eying Kolmogorov’s axio ms as classic al pr ob abilitie s (CP ). In this pape r, w e discuss the formal structure of quantum probabilities as measures over a non-Bo o lean algebr a. W e fo cus on a crucial asp ect of quantum probabilities—namely , that there exists a ma jor structura l diﬀerence betw een classical s tates a nd quantum states: • States of classica l pro babilistic systems ca n b e suitably describ e d by Kol- mogorovian measures. This is due to the fact that each classical state deﬁnes a measure in the Bo olean sigma- algebra of mea s urable subsets o f phase s pace. • Co ntrarily to classical states, quantum states cannot b e reduced to a sin- gle K olmogor ovian mea s ure. A density oper ator repres ent ing a quantum state deﬁnes a measure o ver an orthomo dular la ttice of pro jection oper a - tors, which contains (inﬁnitely many) incompatible maximal Bo o lean sub- algebras . These represent diﬀeren t and complementary—in the Bohr ian sense—exp erimental setups. The best w e can do is to consider a quan- tum state as a family o f K olmogor ovian measures , pasted in a har mo nic wa y [20]; ho wev er, there is no joint (classica l) probability distr ibutio n en- compassing a ll pos s ible contexts. W e discuss the ab ov e men tioned diﬀerences in relation to q ua ntu m theory as a non-Kolmo gorovian pr obability calculus. This calculus can b e considered as an extension of classical measure theory to a non-commutativ e s e tting (see, for e x ample [12, 21]; see also [22] fo r a study o f quantum measure theo ry). In this way , the axiomatization o f probabilities arising in QM (and more g eneral probabilistic models ) c an be viewed as a co ntin uation o f the Hilbert’s progr am with re gard to probabilit y theory . W e ar gue that the probabilities in generalized probabilistic mo de ls ca n be interpreted, in a natural wa y , in ter ms of reasonable exp ectations o f a rationa l agent facing even t structures that may deﬁne diﬀeren t and incompatible contexts. This allows us to under s tand other re la ted notions, such as random v ar ia bles a nd information measures, as na tural generalizations of the usual ones. Kolmogo rov’s approa ch to pro bability theory is not the only one. In the attempts to e s tablish foundations for probability , we have to men tion the works of de Finetti [23] and R. T. Cox [24, 25] (in co nnection with R. T. Cox w orks, see also [2 6]). F or a detailed and accessible study of the history of probability theory and its interpretations, we refer the r eader to the Ap endix of [2]. In this pap er, we pay special attention to Co x’s a pproach and make use of its extension to the quantum realm [27]. Cox’s approa ch is based on a study o f the measur e functions compatible with the a lgebraic prop er ties of the logic of a r ational agent trying to make pro bable inferences out of the av aila ble data. Diﬀerent v ariants of this approa ch have be e n used to des crib e proba bilities in QM [2 8, 29, 30, 31, 32, 33 , 34, 27]. 3 In [2 7], it is shown that the peculia r featur e s of QP arise whenev er the la ttice of prop ositions of Cox’s approach is replaced by a non-distributive one. As is well known, the standard quan tum-logical approach to QM characterizes this theory using a lattice-theo retical fr a mework in whic h the lattices are orthomo dular [35, 36, 21, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 4 7, 48, 49, 50, 51, 52, 53]. In [27], and it is shown that, when Cox’s metho d is a pplied to the orthomo dula r lattice of pro jections o f a Hilb ert spa c e, QP are der ived. W e remark that generalized probabilities can also be studied in what has bee n ca lled the Conv ex Op erationa l Mo dels (COM) a pproach [54, 55, 56, 57, 58, 59, 60, 61 , 62, 63, 64]. These ar e called “ge neralized probabilistic theo ries” (GPTs) or simply , “op erational mo dels” (see also [17]). Diﬀerent mathematical frameworks are used to descr ib e GPTs. Here, we fo cus on the most importa nt ones, paying attention to how they can b e in ter- translated (when poss ible). In the COM appro ach, the prop erties of the sys tems studied and their asso ciated probabilities are e nc o ded in a geometrica l wa y in the form of a co nv ex set and its space of obs erv able magnitudes. The qua ntum formalism and many asp ects of quantum information theor y (such as entangle- men t, discord, and many information proto cols ) ca n be suitably describ ed using this approach [59, 60, 61, 58, 6 4, 63, 5 7, 5 5, 5 6, 62]. Non-linear genera lizations of QM were s tudied us ing the co n vex appro ach in [65, 66, 67]. It is impo rtant to understa nd the r elations b etw een the diﬀer e nt formulations of GPTs. F or example, the measures over complete orthomo dular lattices dis- cussed in Section 4 o f this work deﬁne GPTs (while an arbitr ary GPT might be not be describable in terms o f a measure over an orthomo dula r la ttice). The r ea- son why mo dels deﬁned ov er lattices a re s o imp or tant is that all r elev an t ph ysical theories ca n b e describ ed in such a s etting. Indeed, all relev an t ph ysical mo dels can b e ultimately describ ed using v on Neumann algebr as, that a re g enerated by their lattices of pro jection op erator s (see, for exa mple [9, 8, 21]). This is the case for classical statistical theories, standar d quantum mec hanics, relativistic quantum mechanics, and quantum sta tistical mechanics. As a n example (as we will discuss in Section 3.2), states of mo dels of quan tum mechanics can b e de- scrib ed as measures ov er the orthomo dula r lattice of pro jection op erato rs a c ting on a s eparable Hilbert spa ce. It is interesting, for several reaso ns, to study more general mo dels (that co uld describ e, for exa mple, alter native physical theories ). How ev er, there is always a trade oﬀ b etw een genera lity and particular ities: if our mo dels ar e too general, we can lo ose v aluable information ab out the geo- metric and algebr aic s tructures inv olv ed in relev ant physical theories. On the contrary , if they are to o sp eciﬁc, we might lo os e information ab out the gener al road map for explo ration. It is o ur aim here to shed so me light onto this v ast ﬁeld of r esearch, putting the fo cus on the idea that Kolmogor ov’s framework can cer tainly b e gener a lized in a rea sonable a nd useful wa y . It is a lso imp or tant to mention that quantum-lik e probabilities hav e b een considered outside the quantum domain. Man y probabilistic systems b ehav e in a contextual wa y , a nd then, it is reaso nable to attempt to us e quantum- like probabilities to descr ibed them, since these ar e sp ecially suited to dea l with contextual b ehavior. This is an e xciting ﬁeld of research that has grown 4 int ensively during r ecent years (see, for e x ample [6 8 , 69, 70]). W e start by r e viewing diﬀerent approaches to CP , namely , K olmogor ov’s and Cox’s, in Section 2. Next, we discuss the formalism of QM in Section 3, emphasizing how it can b e co ns idered as a non-Bo olean v ersion of Kolmogo rov’s theory . In Sectio ns 4 and 5, w e discuss generaliza tions using o r thomo dular lattices and COMs, resp ectively . After discussing alternative approa ches in Section 6, w e present the g eneralizatio n of the Cox metho d to genera l non- distributive lattices in Sectio n 7. Finally , our conclus ions are drawn in Section 8. Given that lattice theory is so cen tral to the discussio ns presented here, we hav e included a short review about its e le men tary notions in Appendix A. 2 Classical Probabili ties This Section is devoted to classic al pr ob a bility the ory (CP ). Howev er, what do we mean with this notio n? There exists a v ast literature and tendencies disputing the meaning o f CP . W e will not give a detailed sur vey of the discussion here; how ev er, w e will discuss tw o of the most imp or tant appro aches to CP . These ar e the one given by A. N. Kolmog o rov [3] and the o ne given by R. T. Cox [24, 2 5]. 2.1 Kolmogoro v Kolmogo rov presented his axiomatization of classical probability theory [3] in the 193 0 s. It ca n be formulated as follows. Giv en an outcome set Ω, let us consider a σ -algebra Σ of subsets of Ω. A probability measure will be a function µ such that µ : Σ → [0 , 1] (1a) satisfying µ (Ω) = 1 , (1b) and, if I is a denumerable set of indices, for any pa irwise disjoint family { A i } i ∈ I , µ ( [ i ∈ I A i ) = X i µ ( A i ) (1c) Conditions (1a)–(1c) a re known as axioms of Kolmo gor ov [3]. The triad (Ω , Σ , µ ) is calle d a pr ob ability sp ac e . Pro bability spaces ob eying E quations (1a)– (1c) are usua lly referred as K olmogo r ovian, class ic a l, co mmutative, or Bo o lean probabilities [1 6], due to the Bo olean character of the σ -a lgebra in which they are deﬁned. It is p os sible to show that, if (Ω , Σ , µ ) is a Kolmogo rovian probability space, the inclus ion–exclusion principle holds µ ( A ∪ B ) = µ ( A ) + µ ( B ) − µ ( A ∩ B ) (2) 5 or, as ex pr essed in logica l terms, by replacing “ ∨ ” instea d of “ ∪ ” a nd “ ∩ ” ins tead of “ ∧ ”: µ ( A ∨ B ) = µ ( A ) + µ ( B ) − µ ( A ∧ B ) (3) As r emarked in [71], E q uation (2) w as considered a s crucial b y von Neumann for the interpretation o f µ ( A ) and µ ( B ) as relative frequencie s. If N ( A ∪ B ) , N ( A ) , N ( B ) , N ( A ∩ B ) are the num ber of times for eac h e ven t to o ccur in a ser ies of N rep etitions, then (2) trivially holds. Notice that (3) implies that: µ ( A ) + µ ( B ) ≤ µ ( A ∨ B ) (4) The inequality (4) no lo ng er holds in QM, a fact linked to its non-Bo olean character (see, for example [16]), Sectio n 2.2. Indeed, for a suitably chosen state and events A a nd B (i.e., for a non-co mmu tative pair ), (4) c an be vi- olated. If N ( A ∨ B ), N ( A ), N ( B ) and N ( A ∧ B ) ar e the num ber o f times for each even t to o ccur in a series of N rep etitions, then the sum rule should trivially hold (but it do es not). This p oses problems to a rela tive-frequencies’ int erpretation of quantum pro babilities (see, for example, the disc ussion po sed in [71]). The QM example s hows that non-distributive pr op ositional s tr uctures give rise to pro bability theories that a ppe ar to b e very diﬀer ent fr om those of Kolmogo rov. Not withstanding, it is impo rtant to mention tha t so me author s hav e managed to develop a relative frequencies in terpretations (see, for e x am- ple [72 ]). If all p oss ible measures satisfying (1a)–(1c) are considered as forming a s et ∆(Σ) (with Σ ﬁxe d), then it is straightforward to show that it is co n vex. As we shall see b elow, it is a simplex, and its form will b e re lated to the Bo olean character of the lattice of classica l even ts. 2.2 Random V ariables and Classical Stat es It is imp ortant to recall here how random v a riables are deﬁned in Kolmog orov’s setting (according to the measure theoretic approa ch). See [1 6] for a detailed exp osition. A random v a riable f can be deﬁned as a measura ble function f : Ω − → R . In this context, by a measurable function f , we mean a function satisfying that, for every Borel subset B of the real line (the b or el sets ( B ( R )) are deﬁned a s the family of subsets of R s uch that (a) it is closed under set theoretical co mplements, (b) it is closed under denumerable unions, a nd (c) it includes all op en in terv als [73]), w e have that f − 1 ( B ) ∈ Σ (i.e., the pre- image of every Borel set B under f b elo ng s to Σ, and thus it has a deﬁnite probability measure given by µ ( f − 1 ( B ))). Notice that a random v aria ble f deﬁnes an in verse map f − 1 f − 1 : B ( R ) − → Σ (5a) satisfying f − 1 ( ∅ ) = ∅ (5b) f − 1 ( R ) = Ω (5c) 6 f − 1 ( [ j B j ) = [ j f − 1 ( B j ) (5d) for any disjoint den umerable family of B o rel sets B j . Denoting the co mplement of a set X by X c , w e hav e that, for every Bor el set B : f − 1 ( B c ) = ( f − 1 ( B )) c . (5e) T o illustrate ideas , let us consider a classical pr obabilistic system. A classica l observ able H (such as the energy) will be a function fro m the s tate space Γ to the real n um bers. The state of the s y stem, given by a proba bilit y dens it y  (i.e.,  ≥ 0 and with Leb esg ue integral R Γ  ( x ) dx = 1)), w ill deﬁne a mea sure µ ov er the mea surable subs e ts of Γ a s follows. F or each subse t S ∈ Γ, deﬁne µ ( S ) := Z S  ( x ) dx. (6) Measurable subsets o f Γ will be those for which the ab ove integral conv erges. The function µ will ob ey K o lmogor ov’s axio ms, provided that w e ta ke Γ = Ω and Σ as the set of measurable s ubsets o f Γ. The above formula is suﬃcien t to compute the mean v alues and the probabilities of a ny even t of interest. Given a n elementary testable prop o sition such as: “the v a lues of the observ able H lie in the int erv al ( a, b )”, the rea l n um ber µ ( H − 1 (( a, b ))) gives us the pr obability that this prop osition is true. In this sense, each observ able of a classic a l probabilistic system can b e considered a s a r andom v ariable. This has to b e a neces sary condition for any admiss ible classical state: a state must s pe c ify deﬁnite prob- abilities for every e le men tary test that we may p erform on the s ystem. In this sense, each classica l (probabilistic) state can be de s crib ed by a K o lmogor ovian measure, with the observ ables repr esented a s r andom v ariables. A t the same time, b y asso ciating “ ∨ ” with “ ∪ ”, “ ∧ ” with “ ∩ ”, “ ¬ ” with “( . . . ) c ” (set theor etical co mplemen t), a nd “ ≤ ” with “ ⊆ ” (set theoretical in- clusion), w e see that the B o olean structure a s so ciated to measura ble subsets is coincident with the distributive character of class ical logic. The fact that the logic asso ciated to a classical s y stem is B o olean (in the ab ov e op erational sense), was one of the main observ ations in [35]. As we will see in the following sections, the qua nt um formalis m can b e considered as a n extension o f the classica l one, provided that we replace the measurable subsets of phase space with P ( H ) (the lattice of pro jection op er- ators on a Hilb ert space H ), the mea sure µ by a quantum state represented by a dens it y op erator , and the classic al random v ariables b y pro jection v a lued measures as so ciated to s e lf-adjoint op er ators. As a co nsequence, the op era tional logic asso ciated to a quan tum system will fail to be Bo olean [35], due to the non- distributive c haracter of P ( H ). The set of states of a quantum system will be conv ex to o. How ev er, the geometrical shape will b e very diﬀerent fro m that o f a classical one, due to the non-Bo o lean character of the lattice o f e ven ts in v olved. 7 2.3 Co x’s A pproac h Since the b eginning of pro bability theory , ther e has b een a schoo l of though t known as Bay esianism, which treats pro babilities in a diﬀere nt manner fro m the one discus sed in the previous section. F or the Bay esian approa ch, proba bilities are not to be reg arded as a pr o p erty of a system but as a prop erty of our knowl- edge ab out it. This p ositio n is pre s ent as ea rly as in the XIX cent ury in o ne of the milestones in the developmen t of pr obability theory [74]. In his w ork, Laplace prop osed a wa y to assig n proba bilities in situations of ignora nce that would even tually b e kno wn as the“Laplace principle”. Later w orks would attempt to formalize and give coherence to the Bay esian a pproach, as for example, [75, 23]. In this section, we center our attention on one o f these attempts [24, 25], of R. T. Cox. While a ttaining equiv alent r esults to those o f Kolmo gorov, Cox’s a ppr oach is conceptually very diﬀerent. In the Ko lmo gorovian a pproach, proba bilities can b e naturally interpreted (though not neces s arily) as relativ e freq ue nc ie s in a sample s pa ce. O n the other ha nd, in the appr oach developed b y Cox, probabilities are considered as a meas ure o f the degree of b elief o f a ra tional agent—whic h may be a ma chine—on the truth of a pro p o sition x , if it is known that propo sition y is true. In this wa y , Cox intended to ﬁnd a set of rule s for inferential reasoning that would b e coherent with cla ssical lo gic and that would reduce to it whenev er a ll the premises ha ve deﬁnite truth v alues. T o do this, he started with t w o very g eneral ax io ms and presupp osed the calculus of classical pr op ositions, which, as is well known, for ms a Bo o le a n lattice [76]. By doing so, he derived cla ssical probability theory as an inferential calculus on Bo olea n lattices. W e sketc h her e the arg umen ts present ed in his bo ok [24]. F or a more detaile d expo sition on the deductions, the reader is referred to [25, 24, 77, 78, 30, 31, 3 3]. See [79] for dis cussions on a r igoriza tio n of Cox’s metho d. The t w o axio ms used b y Cox [24] are • C1 —T he probabilit y of an infer ence on given evidence determines the probability of its contradiction from the same evidence. • C2 —T he probability on a given evidence that b oth of t w o inferences ar e true is deter mined by their separate probabilities, one from the given ev - idence and the other from this evidence with the additio na l ass umption that the ﬁrst inference is true. A real v alued function ϕ represe nting the degree to which a prop osition h (usually called hyp othesis ) implies ano ther prop os itio n a is p ostula ted. Thus, ϕ ( a | h ) will repres ent the degree of b elief of an in telligen t a gent re garding how likely it is that a is true giv en that the agent knows that the h ypothesis h is true. Then, requiring the function ϕ to b e coherent with the pro pe r ties of the calculus of class ical prop os itio ns, the ag ent derives the rules for manipula ting probabilities. Using axio m C2 , the asso cia tivity of the conjunction ( a ∧ ( b ∧ c ) = 8 ( a ∧ b ) ∧ c )), and deﬁning the function F [ ϕ ( a | h ) , ϕ ( b | h )] ≡ ϕ ( a ∧ b | h ) : R 2 → R , the a gent arr ives at a functional equation for F ( x, y ): F [ x, F ( y , z )] = F [ F ( x, y ) , z ] (7) Whic h, after a resca ling a nd a prop er deﬁnition of the pr obability P ( a | h ) in terms o f ϕ ( a | h ), leads to the well known pr o duct rule of pro bability theo ry: P ( a ∧ b | h ) = C P ( a | h ∧ b ) P ( b | h ) (8) The deﬁnitio n of P ( a | h ) in terms of ϕ ( a | h ) is omitted, as o ne ultimately ends up using only the function P ( a | h ) a nd never ϕ ( a | h ). In an analog ous manner, using a xiom C1, the law o f double negation ( ¬¬ a = a ), Morgan’s law for disjunction ( ¬ ( a ∨ b ) = ¬ a ∧ ¬ b ), and deﬁning the function f [ P ( a | h )] ≡ P ( ¬ a | h ) : R → R , we ar rive at the following functional equation for P ( a | h ) P ( a | h ) r + P ( ¬ a | h ) r = 1 (9) With r as a n ar bitrary constant. Although, in principle, diﬀerent v alues of r w ould g ive r ise to diﬀeren t r ules for the co mputation of the proba bilit y of the negation of a prop osition, as taking diﬀeren t v a lues o f r ac count for a rescaling of P ( a | h ), one could a s well call P ′ ( a | h ) ≡ P ( a | h ) r pr ob abili ty and work with this function instead o f P ( a | h ). F or simplicit y , Cox decided to take r = 1 and to contin ue using P ( a | h ). Using E q uations (7) and (9), the law of double negation and Morg an’s law for co njunction ( ¬ ( a ∧ b ) = ¬ a ∨ ¬ b ), w e ar r ive a t the sum rule o f proba bility theory: P ( a ∨ b | h ) = P ( a | h ) + P ( b | h ) − P ( a ∧ b | h ) (10) As it turns out, P ( a | h )—if suitably no rmalized—satisﬁes the prop erties of an additive Ko lmogorovian pro bability (E q uations (1a)– (1c)). Due to the imp or tance of Cox’s theor em to the foundations of pr obability , it has b een the targ e t of thorough s crutiny by many authors . Some hav e po inted out inco nsistencies b ehind the implicit a s sumptions made during its deriv ations, most notably the as sumptions behind the v a lidity of Equatio n (7). Since then, there hav e b een diﬀere nt prop osals to sav e Co x’s approach by proving it using less restrictive axioms. In [80], a discussion of the status of Cox prop os al is presented as well as a co unterexample to it. F or a r eview o n the sub ject, it is recommended to consult [78]. Once the genera l pr o p erties of the function P ( a | h ) ar e es tablished, the next problem is to ﬁnd a way to determine prio r proba bilities (i.e., proba bilities con- ditional only to the hypothesis h ). Although, fo r mally , one could assign prior probabilities in any w a y coherent with the normaliza tion used, in practica l situ- ations, o ne is comp elled to as sign them in a wa y that they reﬂect the information contained in the h ypothesis h . A pos sible w a y to do this is by using the MaxEnt principle [7 7, 2 6], which we will review sho rtly in the next section. O ther ways of assigning prio r proba bilities include the Laplace pr inciple [75] a nd coherence with symmetry transformations [81]. Nevertheless, the existence of a general algorithm for assigning prior probabilities is still an op en question. 9 2.4 MaxEn t P rinciple This principle ass e rts that the assignment of the prio r pr o babilities from a hy- po thesis h should be do ne by maximizing the uncertaint y a s so ciated with its dis- tribution while resp ecting the constrains impos ed ov er them by h . Although this may so und pa radoxical, by maximizing the uncertaint y of the prior probabilities one avoids as suming more information than that strictly con tained in h . T aking Shannon’s information mea sure S [ P ] = − P i P ( a i | h ) l og [ P ( a i | h )] as the measure o f the uncertaint y asso ciated with the dis tr ibution P , the Max- Ent principle can b e resta ted as: the pr ior proba bilities cor resp onding to the hypothesis h are given by the distribution that maximizes S [ P ] sub ject to the constraints imp osed by h on P . The simplest exa mple is given by the hy- po thesis h that imp oses no constr a ints on P , in which case P r esults as the uniform distr ibutio n, and the MaxEnt pr inciple reduces to Laplace’s. Diﬀeren t kinds of constraints result in diﬀerent prio r probability dis tr ibutions (PP D) [26]. In [82], a table of some of the distributions obtained in this wa y is pre s ented. Although, given a set of constr aints, the cor r esp onding P PD ca n b e r eadily computed, there is no g eneral metho d of translating a hypothesis h into equiv- alent co nstraints. By means of the MaxEnt principle, class ical and quan tum equilibrium sta- tistical mechanics can b e fo r mulated on the basis o f information theo ry [77]. Assuming that the prio r knowledge ab out the system is given b y n exp ecta - tion v a lues of a collectio n of ph ysical quantities R j , i.e., h R 1 i , . . . , h R n i , then, the most un biased probability distribution ρ ( x ) is uniquely ﬁxed by maximizing Shannon’s lo garithmic entropy S sub ject to the n constra ints h R i i = r i ; for all i . (11) In order to solve this problem, n Lag range multipliers λ i m ust b e int ro duced. In the pro cess of employing the Ma x Ent pro cedure, one discov ers that the information quantiﬁer S can b e identiﬁed with the equilibrium e ntropy of ther- mo dynamics if o ur prior knowledge h R 1 i , . . . , h R n i refers to extensiv e quan- tities [77]. S ( maximal ), once determined, yields complete thermo dyna mica l information with resp ect to the system of interest [77]. The MaxEnt proba- bilit y distribution function (PDF), asso ciated to Bo ltzmann–Gibbs–Shannon’s logarithmic entropy S , is giv en by [77] ρ max = exp ( − λ 0 1 − λ 1 R 1 − · · · − λ n R n ) , (12) where the λ ’s are Lag range multipliers gua ranteeing that r i = − ∂ ∂ λ i ln Z , (13) while the partition function reads Z ( λ 1 · · · λ n ) = X i exp − λ 1 R 1 ( x i ) −···− λ n R n ( x i ) , (14) 10 and the normaliza tion condition λ 0 = ln Z . (15) In a quantum setting, the R ’s are o p e r ators on a Hilb ert spa ce H , while ρ is a density matrix (o pe r ator). The sum in the pa rtition function must b e replaced by a trace, and Sha nno n’s entropy must b e replaced by von Neumann’s. 3 The F ormalism Of QM In this Section, we discuss some sp eciﬁc features of the qua nt um forma lis m [21, 38, 39 , 5] that are relev an t for the problem of QP . 3.1 Elemen tary Measu remen ts And Pro jection Oper ators In QM, obser v able physical magnitudes are represented b y compact self-adjoint op erator s in a Hilber t space H (we deno te this set b y A ). Due to the spectra l decomp osition theor em [73, 5], a key role is play ed by the no tion of pr oje ction value d me asu re (PVM): the set of PVMs ca n b e put in a bijective corre s po ndence with the s et A of self adjoint op erator s of H . Intuitiv ely s pe aking, a P VM is a map that as signs a pr o jection oper ator to each interv al of the real line. I n this sense, pr o jection op er a tors are the building blo cks out of which any observ able can b e built. It is imp orta nt to recall that pro jection op era tors ha ve a very clear op er ational meaning : they represent elementary empirica l tests with only t wo outputs (zero and one, o r YES a nd NO). In a forma l way , a PVM is a map M deﬁned ov er the Borel sets (see Sectio n 2 .2) a s follows M : B ( R ) → P ( H ) , (16a) satisfying M ( ∅ ) = 0 ( 0 := n ull subspace) (16b) M ( R ) = 1 (16c) M ( ∪ j ( B j )) = X j M ( B j ) , (16d) for a ny disjoint denumerable family B j . M ( B c ) = 1 − M ( B ) = ( M ( B )) ⊥ (16e) As we will see in the following Section, a PVM is the natural gener alization of the notio n of r a ndom v ariable to the non-Bo o lean setting. In o rder to rea lize why this is so , it is imp or ta nt to compar e Eq uations (5a)–(5e) and (16a)–(16 e). It is also imp orta nt to remar k that the s et of pro jectio ns in the image of a PVM ar e alwa ys orthogo nal: this implies that this set can always b e endow ed with a Bo olean lattice structure. This allows us to asso ciate, to each complete observ able, a particular empirical co n text represented by a Bo olea n algebra 11 of even ts. Thus, in this sense, complete obser v a ble s ar e always referr e d to a particular c o ntext. Fixing an element A ∈ A , the in tended interpretation of the asso ciated PVM ( M A ( . . . )), ev aluated in an interv al ( a, b ) (i.e., M A (( a, b )) = P ( a,b ) ) is: “the v alue of A lies b etw een the in terv a l ( a, b )”. In this s ense, pro jection op era tors represent elementary tests or pro p o sitions in QM. In o ther words, they can b e considered as the simplest quantum mec hanical observ ables. As we reviewed in App endix A, pro jection oper a tors can be endowed with a lattice structure and, th us, also elementary tests. This lattice was called “Qua nt um L o gic” b y Birkhoﬀ and von Neumann [3 5]. W e r efer to it a s the von N eu mann-lattic e ( P ( H )) [21]. As shown in [35], a n analo gous treatment can b e done for classical systems. As w e hav e seen in Section 2.2, prop ositions a sso ciated to a classica l system a re endowed with a natural Bo olea n structure . During the thir ties, von Neumann and co llab orator s con tin ued studying for- mal developmen ts related to the quantum formalism. One of the results of this inv estigation was the developmen t of the theory of rings of op er ators (b etter known as von Neumann algebr as [21, 83, 84, 8 5, 8 6]), as a n attempt of gener al- izing certain a lgebraic pr op erties of Jordan a lg ebras [4]. The subsequent study of von Neumann a lg ebras show ed that they are closely related to lattice theory . Murray and von Neumann provided a classiﬁcation of factors (v on Neumann algebras whose center is formed b y the m ultiples of the identit y) using ortho- mo dular lattices in [83, 8 4, 8 5, 8 6]. On the o ther ha nd, la ttice theory is deeply connected to pro jectiv e geometry [87], and one of the ma jor discov eries of von Neumann was that of c ontinuous ge ometries , whic h do no t p osses s “ po ints” (or “atoms”) and ar e r e lated to type II factors. F ar from b eing a mere mathemati- cal curiosity , type II factor s found applica tions in sta tistical mec hanics a nd t ype II I factors play a key r ole in the axiomatic appro a ch to Quantum Field Theory (QFT) [12, 21]. The quantum logica l approa ch o f Birkhoﬀ and von Neumann was contin ued by other resea r chers [16 , 50, 42, 4 3, 1 7, 3 8, 3 9] (se e [36, 41, 4 4] for complete ex- po sitions). One of the key res ults of this appr o ach is the r epr esentation the or em of C. P iron [43]. He show ed that any pro p o sitional system ca n be co o rdinatized in a generalized Hilb ert space. A later result by Sol` er sho wed that, by adding extra a ssumptions, it can only b e a Hilb ert space over the ﬁelds of the real nu m be rs, c o mplex num ber s , or quaternions [88]. 3.2 Quan tum Stat es And Quan tum Probabilities In this Section we discuss QP . W e do this by r eviewing the usual appro ach, in which K olmogor ov’s axioms are ex tended to non-Bo olean lattices (or a lge- bras) [12]. As we ha ve see n in Section 3.1, elemen tary tests in QM are repr esented by closed subs paces of a Hilb ert space. These subspa ces form a n o rthomo dular atomic la ttice P ( H ). In o rder to assign pro babilities to these elementary tests or pro ces s es, man y texts pro ceed by p o stulating axioms that a re similar to those of Kolmogorov [21, 5 , 37]. The B o olean Σ-algebra a pp e a ring in Kolmogorov’s 12 axioms (Equa tions (1 a)–(1c)) is replaced by P ( H ), and a measur e s is deﬁned as fo llows: s : P ( H ) → [0; 1] (17a) such that: s ( 1 ) = 1 ( 1 := H ) (17b) and, for a denumerable a nd pairwise ortho g onal family of pr o jections P j , s ( X j P j ) = X j s ( P j ) . (17c) In this way , a r eal num ber b etw een 0 and 1 is a ssigned to any elemen tary test. Despite the similar ity with Kolmo gorov’s axioms, the pro babilities deﬁned ab ov e a re very diﬀerent, due to the non- B o olean character of the lattice in volv ed. Gleason’s theo rem [8 9, 9 0] a sserts tha t if di m ( H ) ≥ 3, any measur e s satisfying (17a)–(17c) can b e put in co rresp ondence with a trace class o p e r ator (of tr ace one) ρ s : s ( P ) := tr( ρ s P ) (18) for ev ery orthogonal pro jection P . On the other hand, using Equation (18), any trace class op er ator of trace o ne deﬁnes a measure as in (1 7 a)–(17c), a nd th us the corres po ndence is o ne to one for dim ( H ) ≥ 3 (so mething that is not true for the t wo dimensio nal case). In this wa y , Equations (17a)–(1 7 c) deﬁne the usual probabilities of QM and cons titute a natural g eneraliza tio n of Kolmogo rov’s axioms to the quantum cas e. The set C ( H ) of all p oss ible measures s a tisfying Equations (1 7a)–(17c) is indeed conv ex, as in the class ical case. How ever, these sets a re very diﬀerent. As an ex ample, let us compa r e a cla s sical bit (to ﬁx ideas, think ab out the p os- sible pro babilistic s tates o f a coin) and a qubit (a quantum sy stem repr esented by a tw o-dimensional mo del). While the state space of the ﬁrst o ne is a line segment, it is well kno wn tha t the sta te space of the second is homeomorphic to a sphere [9 1]. F or more discussion abo ut the conv ex set of quan tum states, see [9 2, 9 3]. A state sa tisfying Equations (17 a)–(17c) will yie ld a Kolmogor ovian proba - bilit y when res tricted to a maximal Bo o lean suba lgebra o f P ( H ). In this wa y , a quantum state can be considered a s a co herent pas ting of diﬀer ent Kolmo goro - vian mea sures. T his has a na tural physical interpretation a s follows. Each em- pirical setup will deﬁne a maximal Bo olea n algebra . The fact that q uantum states yield the corr ect observed frequencies (via the Bor n rule), a llows deﬁning consistent K olmogor ovian probabilities on each Bo olean setting. Ho w ever, do- ing sta tistics on rep ea ted mea surements in identical prepara tions using a single empirical setup, is not suﬃcien t to determine a quantum state completely . F or a general state, it will be mandatory to p er form measur e men ts in diﬀer- ent and complemen tary (in Bohr’s s e nse) empirical setups. Notice that there are many wa ys in which o ne could deﬁne a family of Kolmo g orovian probabilities 13 in P ( H ). Howev er, the probabilities deﬁned by a quantum state (or equiv a- lent ly , by Equations (17a)–(17c)), hav e a very particular mathematical form. The existence of uncertaint y relatio ns b etw een non-co mpa tible observ ables [94] is nothing but an e x pression of this fact. The fact that a q uantum state µ can b e cons idered as a coher ent collection of Ko lmogorovian measur es can be summarized in a diagra m a s follows. If Σ is an a rbitrar y Bo o lean subalgebr a of P ( H ), let µ be a quantum state and µ Σ the restriction of µ to Σ. Then, for every Bo olean subalgebr a Σ, we have the following commutativ e diagram: Σ   / / µ Σ ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ P ( H ) µ   [0 , 1 ] The fact that there exists a global quantum state µ that mak es the above dia- gram commute for every Bo olean suba lgebra, is a quite remark able fact abo ut the qua nt um for malism. Notice that, given tha t the intersection o f Bo ole an subalgebra s may be non-trivial (see examples in the ne x t Sec tion), the proba- bilit y assignments m ust s atisfy cer tain co mpa tibilit y conditions. Th us, ev en if an ev ent x belo ng s to t w o diﬀerent measurement contexts, the quant um s tate assigns to it the same probability . Tha t is, the pr o bability is ass igned inde- pendently o f the context to which it b elongs . This is known as the no-signal condition, whic h will not necessarily hold outside ph ysics (for example, in cog- nition exper imen ts). The ab ov e g eneralizatio n a lso includes quantum observ ables in a natural way . Indeed, by app ealing to the sp ectra l decompos ition theorem, there is a one-to- one corresp o ndence betw een q uantum obser v ables represented by self-adjoint op erator s and PVM’s. Then, these notions are interc hangeable. Ho wev er, a quick lo o k to Equa tions (16a)–(16e) reveals tha t PMS are very similar to class i- cal random v ariables: while class ic a l random v ar iables map Borel sets into the Bo olean lattice of measura ble sets, PVMs map Bor el sets into the non- B o olean lattice P ( H ). Thus, quantum observ ables can b e reasona bly in terpreted as non- Kolmogo rovian random v ariables. W e mentioned ab ov e that (1a)–(1c ) and (17a)–(1 7 c) are no t eq uiv alent prob- ability theor ies. F or example, E quation (2) is no longer v alid in QM. Indeed, for s uita bly chosen s and quan tum even ts A and B , we have s ( A ) + s ( B ) ≤ s ( A ∨ B ) (19) The ab ov e ineq uality should b e compar ed with the classica l one, g iven by (4 ). As an example, co nsider a tw o dimensional quantum system (in current jargon: a qubit), the even ts A = | ↑ z ih↑ z | (spin up in direction ˆ z ) and B = | ↓ x ih↓ x | (spin down in direction ˆ x ), and the state | ψ i = 1 √ 2 ( | ↑i z + | ↓i z ) (“cat state” in the ba sis ˆ z , which is the sa me as “spin up in direc tio n ˆ x ”). Th us, using some simple math, we o btain that p | ψ i ( A ) = tr( | ψ ih ψ || ↑ z ih↑ z | ) = 1 2 , and p | ψ i ( B ) = tr( | ψ ih ψ || ↓ x ih↓ x | ) = 0. On the o ther ha nd, we ha ve that A ∨ B is the linear 14 span of | ↑ z i and | ↓ x i , a nd then, A ∨ B = I . Since p | ψ i ( I ) = tr( | ψ ih ψ | I ) = 1, we have that p | ψ i ( A ) + p | ψ i ( B ) = 1 2 + 0 < 1 = p | ψ i ( A ∨ B ). The proba bility theor y deﬁned by (17 a)–(1 7 c) can also b e considered as a non-commutativ e g eneralizatio n of clas sical probability theor y in the following sense: while when in an a rbitrary statistica l theory , a s tate will b e a normal- ized measure o ver a suitable C ∗ -algebra , the clas s ical case is recov ered when the algebra is c ommutative [16, 12]. W e end this Sectio n by noting that some tech- nical complications app ea r when o ne attempts to deﬁne a quantum co nditio na l probability in the non-co mm utative setting. F or a complete discussion about these matters and a comparis on b etw een classica l and qua nt um pr obabilities, see [1 6, 1 2]. 3.3 Some E xamples In order to understand b etter the mathematical structure (and the physical in- terpretation) underlying qua nt um probabilities, we discuss here some ex amples in detail. W e relate the ev ent structures ass o ciated to physical systems with diﬀerent notions of lattice theory . W e do this b y en umerating diﬀer ent exam- ples that are relev an t for the dis cussion presented in this work. The r eader unfamiliar w ith lattice theory can consult Appendix A. 1. Fi ni te Probability m o del : a dice. Consider the throw o f a dice. The po ssible outcomes are given by Ω = { 1 , 2 , 3 , 4 , 5 , 6 } . A probabilis tic state of the dice is determined b y a ssigning real num ber s p i , i = 1 , ..., 6, to e ach elemen t of Ω. If the dice is not loaded, then p i = 1 6 for all i ; how ev er, a rea listic dice will not s atisfy this. An e ven t will be repr esented by a subset of Ω. As examples, consider the even t “the outcome is ev en” or “ the o utcome is grea ter than 2”. These are represented b y { 2 , 4 , 6 } and { 3 , 4 , 5 , 6 } , resp ectively . All p ossible subs e ts of Ω form a Bo olea n lattice (see Apendix A), with regar d to the set-theoretical operatio ns: “ ∪ ” (in terpreted a s “ ∨ ”), “ ∩ ” (interpreted as “ ∧ ”), and the set theoretical complement (interpreted as “ ¬ ”). The ex a mple of a σ − algebr a asso ciated to a measur able space (Ω , Σ , µ ) works in a similar way . 2. Hi lb ert lattice: As discus sed above, the event s as so ciated to quant um systems can b e put in one to one corres po ndence with an orthomo du- lar lattice: the one formed by the set o f closed subspaces of a Hilb e rt space H . They can be endow ed with a lattice structure as follows [2 1]. The opera tion“ ∨ ” is taken as the closure o f the direct sum “ ⊕ ” of sub- spaces, “ ∧ ” as the intersection “ ∩ ”, and “ ¬ ” as the or thogonal complement “ ⊥ ”, 0 = ~ 0, 1 = H , a nd we denote by P ( H ) the set of c lo sed subspaces . The order “ ≤ ” is deﬁned b y subspace inclusion: we sa y that S ≤ T , when- ever S ⊆ T . The subspaces 0 and 1 pla y the role of the b ottom and top elements of the lattice, since, for any subspace S , we have 0 ≤ S ≤ 1 . Then, the a lgebraic structure ( P ( H ) , ∩ , ⊕ , ¬ , 0 , 1 ) will be a complete bo unded o rthomo dular la ttice (which we denote simply b y P ( H )). It is 15 complete beca use the intersections and (the closure of ) sums of arbitra ry families of closed subspa ces yields clo sed subspa ces. It is b ounded due to the existence of the top a nd b ottom elements. It is orthomo dula r, bec ause for any pair of subspaces S , and T , whenever w e hav e S ≤ T , then S ∨ (( S ) ⊥ ∧ T ) = T (see Appendix A fo r more details). As close d subspaces ar e in o ne to one corresp ondence with pr o jection op erator s, we take P ( H ) to b e the lattice of closed subspa ces or the la ttice of pro jections interc hangeably . One of the most imp ortant features o f P ( H ) is that the distributive law (49) do es no t ho ld (se e App endix A). P ( H ) is mo dular if H is ﬁnite dimensional. If H is inﬁnite dimensiona l, then P ( H ) is always or thomo dular. Gleaso n’s theorem (men tioned in the previous section) gra n ts that, for dim ( H ) ≥ 3, quantum states can be considered as mea s ures over the lattice of closed subspa ces. This is a remark able fact, since it implies that quan tum pr obabilities ar e desc r ib ed by a very sp eciﬁc mathematical framework. An y measur e men t context can b e r epresented by an or thogonal basis of H . It is easy to chec k that, by applying the lattice op er ations deﬁned ab ove to a ﬁxed bas is , we o btain a Boo lean alg ebra. The cas e s of Hilb ert s paces of dimension 2 and 3 are easy to c heck; ho wev er, this is true in general. It turns o ut that the whole lattice o f subspac e s c an b e des crib ed as a family of int ertwined Bo o lean algebras (more ab out this b elow). F o r more discussion regarding the notion of “intert wined contexts”, see [95]. The following ta ble summarizes the main diﬀerences b etw een Q uantum and Kolmogor ovian pr obabilities: Kolmogo ro v Probability Quan tum Probabilit y Lattice: Σ P ( H ) (Bo olean-a lgebra) (orthomo dular, non-Bo olean) States: Measures over Σ Measures over P ( H ) Ev en ts: Subsets of Ω Closed s ubspaces of H There is a geo metrical underlying quan tum probability: the one dimen- sional subspaces of a Hilb ert spac e form a pr oje ctive ge ometry . The higher dimensional subspaces ar e elements of the pr oj e c tive lattic e a sso ciated with this geometry (see [38, 39]). 3. Fi reﬂy Mo de l: The ﬁreﬂy mo del [40] is used in quantum lo gic to show an example of a system that is not a full quan tum mo del but has certain features that serve to illustrate wha t happ ens with quantum systems. It consists of a ﬁreﬂy that is fr eed inside a b ox. W e are asked to p erfor m an exp eriment to detect the lo ca tion of the ﬁreﬂy but with constrains . W e are only allow ed to loo k at tw o diﬀerent faces of the b ox (and w e can only choose one o n each run of the exper iment). The ﬁr st one is to measure face C 1 with three p oss ible outcomes: the “ﬁreﬂy is detec ted on the left” 16 (l), on the r ig ht (r), and “no-signal” (n) (which means that the light of the ﬁr eﬂy was o ﬀ ). The second p oss ibilit y is to measur e on face C 2 , with the p ossibilities “ﬁreﬂy is on front” (f ), “ﬁreﬂy is in the b ottom” (b), and “ no-signa l” . Notice that the “no -signal” o utco me (n) is present in b oth exp er iments— this will b e impo rtant so on. These co nstraints ar e, of course, silly , given that we can alwa ys lo o k at ev ery place in the box and detect the exact lo cation of the ﬁreﬂy . Howev er, they a re tho ught oﬀ as an artiﬁcial mea- surement pro cedure that resembles what happ ens with quantum sy stems. If we cho ose context C 1 , we can check whether the ﬁreﬂy is on the left, the right, o r no signal. If we c ho ose con text C 2 , we ca n check whether it is front, b otto m, or no-signal. How ever, we canno t check b oth things in the same exper iment, as happ ens with the p o sition and momentu m o f a quantum system. If we c hoo se to measur e in the face C 1 , the thr ee outcomes for m an out- come set Ω 1 = { ( l ) , ( r ) , ( n ) } . This gives rise to a Bo o lean algebra Σ 1 , formed by all p os s ible subsets of Ω 1 . Ea ch of these s ubsets represents an even t, such as “the ﬁreﬂy is not detected on the rig ht” (which is repre- sented by the set { ( l ) , ( n ) } ), and so o n. A probabilis tic state o f the ﬁreﬂy —a throw in which we do not k now the outcome a prio ri—will give a classical probability s pace (Ω 1 , Σ 1 , µ 1 ). Similarly , w e hav e a probability space (Ω 2 , Σ 2 , µ 2 ) for the second option C 2 , where Ω 2 = { ( t ) , ( b ) , ( n ) } . Notice that Ω 1 ∩ Ω 2 = { 0 , ( n ) , ( n ) ′ , 1 } . Since the even t ( n ) b elongs to bo th co ntexts o f measurement, for the sake of consistency , w e must im- po se µ 1 (( n )) = µ 2 ( n ). Now, let ( n ′ ) = { ( l ) , ( r ) } , ( l ′ ) = { ( n ) , ( r ) } and ( r ′ ) = { ( n ) , ( l ) } (i.e., the set of theore tical complements of ( n ), ( l ) and ( r ), resp ectively). The Hasse diagram o f Σ 1 is then given by 1 ( l ) ′ ( r ) ′ ( n ) ′ ( l ) ( r ) ( n ) 0 17 In the above diagram, a line joining tw o elements x and y , means that x ≤ y (i.e., the partial order is repres ent ed by the lines connecting the diﬀerent elements). Th us, for ex ample, ( l ) ≤ ( r ) ′ (whic h is equiv alent to { ( l ) } ⊆ { ( n ) , ( l ) } ). The join of t w o elements is the least element that lie s ab ov e b oth of them (with regar ds to the pa rtial order ). The conjunction is the greatest element that lies b elow b oth. Thus, for example, ( l ) ∨ ( r ) = ( n ) ′ (whic h means { ( l ) } ∪ { ( r ) } = ( { ( n ) } ) c ) and ( l ) ∧ ( r ) = 0 (which means { ( l ) } ∩ { ( r ) } = ∅ ). A similar conv en tion holds for the rest of the diagrams below. Similarly , the Hasse diagram of Σ 2 is then given by 1 ( n ) ′ ( f ) ′ ( b ) ′ ( n ) ( f ) ( b ) 0 A direct chec k shows tha t Σ 1 and Σ 2 are Bo o lean alge br as (also, Bo olean lattices—see Appendix A). Now, we can join all po ssible even ts together, taking into account that Σ 1 ∩ Σ 2 = { 0 , ( n ) , ( n ) ′ , 1 } . W e obtain the follow- ing Hasse diag ram: 18 1 ( l ) ′ ( r ) ′ ( n ) ′ ( f ) ′ ( b ) ′ ( l ) ( r ) ( n ) ( f ) ( b ) 0 The above diag ram deﬁnes a lattice L , whic h—lik e Σ 1 and Σ 2 —is non- distributive (and thus, non-Bo olea n). The lattice join of tw o g iven ele- men ts is the lea st elemen t that lies ab ov e them, and the co njunction is the gre atest element from b elow. The rea der can check non-distributivity by insp ection. The Bo olean algebra s Σ 1 and Σ 2 are sublattices of L . It is v ery important to r e ma rk that they contain an element in common: L can be seen a pasting of Σ 1 and Σ 2 . In other words, L is formed from tw o Bo olean subalgebra s tha t are intertwine d . The ass o ciated lattices of fully quantum mo dels are just like tha t: they a r e for med b y a collectio n of int ertwined Bo olean subalgebras—o ne for each context. The diﬀerence betw een the lattice of the ﬁreﬂy and the lattice of a three - dimensional q uantum system is that there are inﬁnitely man y contexts for the latter, and thus the int ertwining—for dim ( H ) ≥ 3—is muc h mor e co mplicated. This intricate algebraic structur e asso cia ted to quantum sy s tems lies at the core of the celebrated Ko chen–Speck er theor em [96] (which we discus s below). 4. The lattice of Q-bit: Given the incredible adv a nces of q ua ntum infor - mation theory in recent decades, the r eader may wonder what the la ttice of a q-bit lo ok s lik e. It is the simplest quantum mo del conce iv able. Sup- po se then that we are given a spin 1 2 system. As is well known, the set of all pos sible states of a qubit is isomorphic to a spher e, namely , the Blo ch spher e [91]. Each pur e state of a qu bit c orr esp onds to a one dimensional subsp ac e of a two dimensional c omplex Hilb ert sp ac e, and c an b e r epr e- sente d as a p oi nt in the surfac e of the Blo ch spher e . A one dimensional subspace is called a r ay . The diﬀerent sets of o b jective prop erties, whic h are of the form “ the particle has s pin ↑ (o r ↓ ) in dir ection ˆ n ”, a re r epresented by those rays (or, equiv a lent ly , by p oints on the s urface of the sphere). Notice that each direction in space ˆ n deﬁnes tw o rays in 19 the Hilber t space (represented by the pr o jection o p erators P ↑ ˆ n = | ↑ih↑ | ˆ n and P ↓ ˆ n = | ↓ih↓ | ˆ n ). The subspa c e s a s so ciated to P ↑ ˆ n and P ↓ ˆ n are orthogo nal: this mea ns, liter- ally , that we must imagine them as or thogonal lines in the Hilb ert spa c e . As there are inﬁnitely man y directions in space, there are inﬁnitely many such pair s of o rthogona l ev en ts. All these event s will b e included in the lattice of a qubit. In addition to all po ssible rays (asso ciated to one dimensional subspaces), we also have tw o distinguished subspaces, represented by the even ts 0 (the nu ll subspace of the Hilb ert spa ce), and 1 (the maximal subspace, which equals H ). Each one dimensional s ubspace con tains 0 as a subspace, and is contained in 1 . If we chose a dir ection in space ˆ n , co nsider the set B ˆ n = { 0 , P ↑ ˆ n , P ↓ ˆ n , 1 } , and co nsider the ab ove deﬁned lattice op era tions for subspaces, w e o bta in a tw o-elemen t Bo olea n alg e bra. All contexts of a qubit are of this for m: eac h measurement dire c tion ˆ n in spa c e deﬁnes a tw o-elemen ts Bo o lean algebr a B ˆ n . The Hasse diagr am o f a con text represented by B ˆ n is then given by 1 P ↓ ˆ n P ↑ ˆ n 0 Thu s, the Hass e diag ram of a q-bit will hav e the form: 1 · · · P ↓ ˆ n P ↑ ˆ n · · · P ↓ ˆ n ′ P ↑ ˆ n ′ · · · 0 where ˆ n , ˆ n ′ , etc., deﬁne diﬀeren t dire ctions in spa ce. The dots represent the inﬁnitely many other Bo ola n algebra s asso ciated with all p o s sible di- rections in s pace. Again, we obtain a lattice, which is non-distributive. In this example, B ˆ n ∩ B ˆ n ′ = { 0 , 1 } whenever ˆ n and ˆ n ′ deﬁne diﬀerent dire c - tions. Thus, only the top and b otto m elements are shared by the Bo o lean subalgebra s. Th us, this ex ample is degenera ted, s ince there is no (non- trivial) intert wining b etw een the diﬀeren t Bo o lean a lgebras asso cia ted to the measurement co ntexts. I n the following e x ample, we will consider a higher dimensional example, for which the intert wining is highly non- trivial. 5. Ko c hen–Sp ec k er theorem (in a four dimensional m o del ): A nice 20 example of ho w the diﬀeren t contexts of a quantum system ar e intert wined was pr esented in [97] (of cour se, for the orig inal version of the Ko chen– Specker theorem, the rea der is r eferred to [9 6]). Given a four-dimensiona l quantum system, each measurement context has four p ossible outcomes. Each o ne of them is mathematically represe nt ed by a o ne dimensio nal subspace of a four-dimensiona l Hilbert space. Each one dimensional subspace is g enerated by a vector ˆ v . Let us then represent the outco me given b y the vector ˆ v by the pr o jection oper ator P ˆ v (whic h is the pro jection op erato r that pro jects into the subspace gener ated by ˆ v ). Then, ea ch measur ement co ntext is represented by four pro jection op erator s, say P ˆ v 1 P ˆ v 2 , P ˆ v 3 and P ˆ v 4 . The s e are a ll orthogo nal, b eca use they represent mutually exclusive o utcomes (you cannot hav e, in the same mea- surement co nt ext, t w o diﬀerent outputs at the same time). As in the qubit case, b y using the Hilbert la ttice op erations, these pro jections gener ate a Bo olean algebra with four atoms (that has 2 4 elements). Thu s, each measurement context has an asso ciated Bo olean algebra with sixteen elements. When a quantum state is prepared, the probabilities assigned to each cont ext can b e thought o ﬀ as K olmogor ovian. It also happ ens that P ˆ v 1 + P ˆ v 2 + P ˆ v 3 + P ˆ v 4 = 1 (whic h, using the Hilber t lattice op erations, r eads W i P ˆ v i = 1 ). Now, consider a family o f nine diﬀerent measurement contexts, giving place to the equa tions: P 0 , 0 , 0 , 1 + P 0 , 0 , 1 , 0 + P 1 , 1 , 0 , 0 + P 1 , − 1 , 0 , 0 = ˆ 1 P 0 , 0 , 0 , 1 + P 0 , 1 , 0 , 0 + P 1 , 0 , 1 , 0 + P 1 , 0 , − 1 , 0 = ˆ 1 P 1 , − 1 , 1 , − 1 + P 1 , − 1 , − 1 , 1 + P 1 , 1 , 0 , 0 + P 0 , 0 , 1 , 1 = ˆ 1 P 1 , − 1 , 1 , − 1 + P 1 , 1 , 1 , 1 + P 1 , 0 , − 1 , 0 + P 0 , 1 , 0 , − 1 = ˆ 1 P 0 , 0 , 1 , 0 + P 0 , 1 , 0 , 0 + P 1 , 0 , 0 , 1 + P 1 , 0 , 0 , − 1 = ˆ 1 (20) P 1 , − 1 , − 1 , 1 + P 1 , 1 , 1 , 1 + P 1 , 0 , 0 , − 1 + P 0 , 1 , − 1 , 0 = ˆ 1 P 1 , 1 , − 1 , 1 + P 1 , 1 , 1 , − 1 + P 1 , − 1 , 0 , 0 + P 0 , 0 , 1 , 1 = ˆ 1 P 1 , 1 , − 1 , 1 + P − 1 , 1 , 1 , 1 + P 1 , 0 , 1 , 0 + P 0 , 1 , 0 , − 1 = ˆ 1 P 1 , 1 , 1 , − 1 + P − 1 , 1 , 1 , 1 + P 1 , 0 , 0 , 1 + P 0 , 1 , − 1 , 0 = ˆ 1 , Each line of Equa tion (20 ) below r epresents a diﬀerent mea surement con- text. The subindexes r epresent non-nor malized vectors in the four-dimensional Hilber t space (each vector deﬁnes a ray , and repr e s ents a measurement outcome in that cont ext). Each meas ur ement context has four or tho go- nal pro jections, that r epresent the outcomes o f a pro jective measurement (and, as such, add up to the identit y oper ator). The co nt exts are chosen in s uch a wa y that each pair of them shares one e le men t in common. This r e presents the intert wining of the Bo o le an algebr as asso cia ted to the contexts since, for example, the even t represented by P 0 , 0 , 0 , 1 belo ngs to the ﬁrst and seco nd contexts. Similar ly , the even t P 0 , 0 , 1 , 0 belo ngs to the ﬁrst 21 and the ﬁfth co ntexts. The family o f contexts of this e x ample is chosen in such a w ay that each ev ent be lo ngs to exa c tly t wo diﬀerent contexts. Thu s, sinc e there are nine equations, there a re eighteen diﬀerent ev ent s in total. The Bo olean algebra s asso ciated to the nine contexts a r e related in a non- trivial wa y (since the intersection o f any t w o of them is strictly greater than { 0 , 1 } ). In o rder to illustr a te the Kochen–Sp eck er theorem, let try to assig n truth v a lues to ea ch of these event s, which can be repr esented as 0 vs . 1 assignments to the diﬀerent outcomes of the exper iment s. Thus, for example, we can assig n 1 (true) to P 0 , 0 , 0 , 1 , or 0 false, and pro ceed similarly to the other events. W e represent this by a function ν : ν ( P 0 , 0 , 0 , 1 ) = 1, ν ( P 0 , 0 , 1 , 0 ) = 0, etc. A truth v alue assignment would mean tha t e a ch po s sible exp eriment out- come has a deﬁnite v alue previo us to measure ment. This is related to asking ab out the e x istence of a disp ersio n fr e e state , that is, a state that only assigns the probabilities zero and one to all p ossible outcomes. Thus, the function ν must satisfy one condition: given that all the outcomes in a co nt ext ar e mut ually exclusive, the function m ust b e deﬁned in such a wa y that there are no tw o truth v a lue as signments in the same context. Thu s, for example, if we assig n the truth v alue 1 to P 0 , 0 , 0 , 1 (i.e., ν ( P 0 , 0 , 0 , 1 ) = 1), then, all o ther members of that context must hav e the tr uth v alue 0 as- signed ( ν ( P 0 , 0 , 1 , 0 ) = ν ( P 1 , 1 , 0 , 0 ) = ν ( P 1 , − 1 , 0 , 0 ) = 0). Equation (20) implies that the v aluations must satisfy P i ν ( P i ) = 1 on each line (this is known as the FUNC condition in the literature; see the discussion and refer e nc e s in [98]). W e m ust a s sume that the v aluations prese rve their v alues from context to context (if we a ssign a ce r tain truth v alue to a pro jection in a given context, we must use that same v a lue when it app ea rs in a diﬀer ent context). How ev er, it is easy to chec k that s uch a compatible truth v alue as sign- men t is not pos sible. The rea son is as follows. Ther e ar e nine equations and eighteen ev en ts. If we sum all e quations (of the form P i ν ( P i ) = 1 ), on the rig ht, we o btain an o dd nu mber (nine), and on the left, we obta in an even n um ber , sinc e there is a n even num ber of ones. How ever, this is imp ossible . The non-exis tence of such a truth v a lue as signment shows one of the most imp o rtant implications of the intert wining betw een the Bo olean algebras ass o ciated to the con texts. This is known a s the Ko c hen–Sp eck er theorem (see [97] for details). This example illustrates clearly how the Bo o lean a lgebras of even ts asso cia ted to quantum systems are intertwined a nd how this complex str ucture gives place to interpretational issues. As is w ell known, the Ko chen–Spe ck er the- orem is a corners tone in the dis cussions ab o ut the foundations of quan tum mechanics (see, for exa mple [98] and the references therein). 22 3.4 Quan tal E ﬀects Pro jective measur es are not the only wa y in whic h observ able quantities can be describ ed in QM. Ther e exists a more general notio n, namely , that of the quantal eﬀe ct . This notion can be ge ner alized to arbitra ry statistical theories . The generalization of the notion of PVM (whic h is based on pro jections) to an observ able ba sed o n eﬀects is c alled a p ositive op er ator value d me asur e (POVM)) [99, 100, 1 01, 102, 103, 104, 105] and, in QM, will be repr esented b y a mapping ( B ( H ) stands for the set o f b ounded op erator s in H ). E : B ( R ) → B ( H ) . (2 1 a) such that E ( R ) = 1 (21b) E ( B ) ≥ 0 , for any B ∈ B ( R ) (21c) E ( ∪ j ( B j )) = X j E ( B j ) , for any disjoint familly B j (21d) The reade r should co mpare Equations (21a)–(21d) with (5a)–(5e) a nd (16 a)– (16e). A P OVM is, thus, a meas ur e whose v alues a re non-negative self-adjoint op erator s o n a Hilber t spa ce, and the ab ove deﬁnition reduces to the PVM case when these o p e rators are a lso orthog onal pro jections. It is the mo s t general formulation of the description of a meas urement in the framework of quantum ph ysics. P ositiv e op er ators E satisfying 0 ≤ E ≤ 1 are ca lled eﬀe cts a nd generate a n eﬀe ct algebr a [102, 99]). W e denote this a lgebra b y E ( H ). It is also impo rtant to remark that POVMs can b e ass o ciated to fuzzy measur ements (and thus with fuzzy sets; see [106, 104]). In QM, a POVM deﬁnes a family of aﬃne functionals on the quantum state space C of all p ositive hermitian trace- class op erator s o f trace one. Thus, g iven a Bo rel set B , we hav e: E ( B ) : C → [0 , 1] (22a) ρ 7→ tr( E ( B ) ρ ) (22b) for every Bor el set B . This will b e r elev an t in certain gener alizations of quantum probabilities, whic h we will discuss b elow. 4 Generalization to Orthomo d ular Lattices In the algebraic form ulation of relativistic quan tum theor y , there app ear a lge- bras that are diﬀere nt from the ones used in non- r elativistic QM [8]. In the non-relativis tic case, the algebra B ( H ) of all b ounded op erator s acting on a separable Hilbert space generates—via the sp ectral theorem—all po ssible ob- serv ables. How ever, the study o f quantum systems with inﬁnitely many degrees of freedo m revealed that other algebr as ar e needed. Murray a nd von Neumann provided a classiﬁca tion of these algebras, whic h are called T yp e I, T yp e II, a nd 23 Type I I I. F o r the non-relativistic case with ﬁnitely man y degr ees o f freedom, it suﬃces to use Type I factors. How ever, in the general case, Type I I a nd Typ e II I factor s appea r. The existence of diﬀerent a lgebraic mo dels of quantum theories sugg ests that, in principle, one could co nc e ive more gener al probabilistic models than those o f standar d Q M. W e describ e here a p ossible genera lization, based in orthomo dular lattices. Let L be an arbitra ry or thomo dular lattice (standing for the la ttice of all poss ible empirica l even ts of a given mo del). Then, w e deﬁne s : L → [0; 1] , (23 a) such that: s ( 1 ) = 1 . (23b) and, for a denumerable a nd pairwise ortho g onal family of e ven ts E j , s ( X j E j ) = X j s ( E j ) . (23c) If we put L = Σ and L = P ( H ) in Equations (23 a)–(23c), w e recov er the Kolmogo rovian and quantum ca s es, resp ectively . F or a discus s ion on the c o n- ditions under which measures those deﬁned in Equations (23a)–(23c) a re w ell deﬁned, se e [37], Cha pter 11. The fact that pro jection op erators of arbitr ary von Neumann algebra s deﬁne o rthomo dular lattices [12], s hows that the ab ove gen- eralization includes ma ny exa mples of interest (in addition to classical statistica l mechanics and sta nda rd Q M). Notice ag ain that the set of all p o ssible measures satisfying (23a)–(23 c) is conv ex. This o p ens the door to a further generalization o f pro ba bilistic mo dels based o n c onvex sets , that we discuss in the next Section. The states deﬁned in Equations (23a)–(23 c) deﬁne Kolmog orovian proba- bilities when restricted to maximal Bo o lean subalgebras of L . Denote by B to the s et of all p ossible Bo ole a n subalgebr a s of L . It is poss ible to co nsider B as a pas ting of its maximal Boo le an subalgebr a s (see, for exa mple [107] and the discussions p os ed in [20, 108]): L = _ B ∈ B B . (24) The dec o mpo sition r e presented by Equation (2 4) implies that a sta te deﬁned as a measure ov er an orthomo dular lattice can be co nsidered as a pasting of Kolmogo rovian probabilities. If there is only one maximal Bo olean subalg e bra, then the whole L has to b e Bo olea n, and thus we recov er a Kolmo gorovian mo del. In theories that displa y contextualit y , such as standar d QM [10 8, 20, 94], there will b e more than one empirical context, a nd thus the ab ove decompo sition will not b e trivial. The representation of observ ables in this setting c a n b e made as follows (we follow [4 3] here). 24 Deﬁnition 1. A c-morphism is a one to one map α : L 1 − → L 2 b et we en ortho c omplemente d c omplete lattic e s L 1 and L 2 such that α ( _ i a i ) = _ i α ( a i ) (25 a) a ⊥ b = ⇒ α ( a ) ⊥ α ( b ) (2 5 b) α ( 1 1 ) = α ( 1 2 ) (25c) Given a physical system whose ev en t lattice is given b y L , an observable can be deﬁned as a c-mor phis m from a Bo o lean lattice B in to L : Deﬁnition 2 (O bserv able) . An observ able of a physic al system whose event lattic e is L and that takes its values in the outc ome set M wil l b e a c-morphism φ fr om a Bo ole a n algebr a B M of subsets of M , to a Bo ole an sub a lgebr a Σ φ ⊆ L . The following dia g ram illus trates what happens when w e comp o s e a gener a l- ized state (g iven by Equa tions (2 3a)–(23c)) and an observ able (as in Deﬁnition (2)): B M φ   s / / [0 , 1 ] L s ◦ φ < < ② ② ② ② ② The compos ition s ◦ φ : B M − → [0 , 1 ] deﬁnes a Ko lmogorovian probability (i.e., a measure satisfying Eq uations (1a)–(1c)). Let us now co mpare co mpare Equations (5 a) –(5e) and (16a)–(16 e) w ith Def- initions (1) and (2). By lo oking at the deﬁnition of PVM (Equations (16a)– (16e)), it is easy to recognize that a PVM is a c- morphism betw een the set of Borel subsets of B ( R ) a nd the Bo olea n algebra generated by its image pro jec- tions. According to the ab ove de ﬁnitio n of observ able, one can q uickly r ealize that any Bo o le an suba lgebra o f L will determine a n observ able (more pr op erly , a family of observ ables up to rescaling). F or the c lassical c ase, by loo king ag ain at the “imp ortant remark” of Section 2 .2 (Equations (5a)–(5e)), we realize that a cla ssical r andom v ariable als o satisﬁes the general deﬁnitio n o f obse rv a ble given in Deﬁnition 2. 5 Con v ex Op erational Mo dels In the pr evious section, w e demo nstrated that the set of states deﬁned ov er a n arbitrar y orthomo dula r lattice is conv ex. This appro ach contains the quantum and classical s tate spaces as pa rticular cases. Thus, it seems very na tural to attempt to deﬁne generalized pro babilistic mo dels b y app ealing to co nv ex sets. This k ey obser v a tion leads to a gener al appr oach to s tatistical theories based on the study of the geometric al prop erties of convex sets. This is the star ting po int of the Convex O p er ational Mo dels (CO M) approach. In this sectio n, we 25 concentrate on elemen tary notions o f CO M’s, and we refer the reader to [5 7] for an excellent presentation of the sub ject. The a pproach based on conv ex sets results as more gener al than the one based in orthomo dular lattices (i.e., the la tter ca n b e included as par ticula r cases of the COM approach). If the state s pa ce of a given probabilistic theory is given by the set S , let us denote b y X to the set of p ossible meas urement outcomes of a n obser v a ble quantit y . T he n, if the system is in a state s , a pro ba bility p ( x, s ) is assigned to any p os sible outcome x ∈ X . This pr obability should b e w ell deﬁned in order that our theor y b e considered as a pr o babilistic one. In this way , we must have a function p : X × S 7→ [0 , 1 ] ( x, s ) → p ( x, s ) (26) T o each outcome x ∈ X and state s ∈ S , this function a ssigns a probabil- it y p ( x, s ) of x to o ccur if the system is in the state s . In this wa y , a tr iplet ( S , p ( · , · ) , X ) is a ssigned for each sys tem of an y probabilistic theory [17]. Think- ing of s as a v aria ble, we o btain a mapping s 7→ p ( · , s ) from S → [0 , 1] X . This implies that all the states of S c an be ide ntiﬁed with maps, which g enerates a cano nical vector space. Their clos ed conv ex hull forms a new set S repre- senting all p ossible pro babilistic mixtures (co nv ex combinations) of s tates in S . Given an ar bitrary α ∈ S a nd a ny outcome x ∈ X , we can deﬁne an aﬃne ev a lua tion-functional f x : S → [0 , 1] in a canonical way by f x ( α ) = α ( x ). More gener ally , we can consider an y aﬃne functional f : S → [0 , 1] as repr e- senting a measurement outcome, and thus use f ( α ) to r epresent the proba bility for that outcome in state α . W e will call A ( S ) to the space of all aﬃne function- als. Due to the fact that QM is also a pr obabilistic theor y , it follows that it can be included in the ge neral framework describe d ab ov e (we denoted its co nv ex set of states b y C in Section 3.2). In QM, aﬃne functiona ls deﬁned a s ab ov e a re called eﬀe cts (and are co incident with the constituen ts of POVM’s as deﬁned in Section 3.4). The generalize d pr obabilistic mo dels deﬁned in Section 4 fall naturally into the scop e of the C O M approa ch, given that their state s paces are conv ex se ts . W e saw that a probability a ( ω ) ∈ [0 , 1] is well deﬁned for an y s tate ω ∈ S and an obser v able a . In the COM approa ch, it is usually assumed that there exists a unitary o bserv able u such that u ( ω ) = 1 for all ω ∈ S . Th us, in analog y with the quantum cas e , the set of all eﬀects will b e encountered in the interv a l [0 , u ] (the order in the gener al case is the canonical one in the space of aﬃne functionals). A (discrete) measurement will be represe nt ed b y a set of eﬀects { a i } such that P i a i = u . S can b e naturally embedded in the dua l space A ( S ) ∗ using the map ω 7→ ˆ ω ˆ ω ( a ) := a ( ω ) (27) Let V ( S ) b e the linea r span of S in A ( S ) ∗ . Then, it is r e a sonable to co nsider S ﬁnite dimensio nal if and only if V (Ω) is ﬁnite dimensional. F or the sake of 26 simplicity , we restrict ours e lves to this case (and to compact spa ces). As is well known, this implies that S can b e expre s sed as the conv ex hull of its extreme po ints. The extr eme po int s will represent pur e states (in the QM case, pure quantum states are indee d the e x treme p oints of C , a nd corresp ond to one dimensional pro jections in the Hilb er t space). It can be shown that, for ﬁnite dimensions d , a system will b e classica l if and only if it is a simplex (a simplex is the convex hull of d + 1 linea r ly indep endent pure states). It is a well known fact that, in a simplex, a point may b e expressed as a unique convex combination of its extreme p oints. This characteristic feature of class ical theories no lo nger holds in q uantum mo de ls . Indeed, in the ca se of QM, there are inﬁnite ways in which one can e xpress a mixed state as a conv ex combination of pure states (for a g r aphical represe ntation, think abo ut the ma ximally mixed state in the Blo ch sphere). Int erestingly eno ugh, there is also a connection b etw een the faces of the conv ex set of states of a given mo del and its lattice o f pro p er ties (in the qua ntum- logical sense ), providing an unexp ected connection b etw een g e o metry , la ttice theory , and statistical theories. F aces of a conv ex s et are deﬁned as subsets that are stable under mixing and puriﬁcatio n. This is means that a con vex subset F is a face if, each time that x = λx 1 + (1 − λ ) x 2 , 0 ≤ λ ≤ 1 , (28) and then x ∈ F if and only if x 1 ∈ F and x 2 ∈ F [91]. The set of faces o f a conv ex set forms a lattice in a cano nical w ay , and it can b e shown that the lattice of faces of a classica l mo de l is a Bo olean one. On the other hand, in Q M, the la ttice of faces of the conv ex set of states C (deﬁned as the set of p ositive trace clas s her mitian op er ators of trace o ne) is iso morphic to the von Neumann lattice of close d subs pa ces P ( H ) [9 1, 37]. F or a g eneral mo del, the la ttice of face s may fail to b e suitably ortho co mplement ed [37] (and th us the COM approach is mor e gene r al than the one based in o rthomo dular la ttices). Let us tur n now to c o mp o und systems. Giv en a co mpo und system, its comp onents will hav e state spa ces S A and S B . L e t us denote the joint s tate space by S AB . It is reaso nable to identify S AB with the linea r span of ( V ( S A ) ⊗ V ( S B )) [57]. Then, a maximal tensor pro duct state space S A ⊗ max S B can b e deﬁned as one that contains a ll bilinear functionals ϕ : A ( S A ) × A ( S B ) − → R such that ϕ ( a, b ) ≥ 0 for all eﬀects a and b and ϕ ( u A , u B ) = 1. The maximal tensor product state space ha s the prop er ty of be ing the larg est set of states in ( A ( S A ) ⊗ A ( S B )) ∗ , whic h as signs probabilities to all pro duct-mea s urements. The minimal tensor pro duct s ta te space S A ⊗ min S B is simply deﬁned by the conv ex hull of all pro duct states . A pro duct sta te will then b e a state of the form ω A ⊗ ω B such that ω A ⊗ ω B ( a, b ) = ω A ( a ) ω B ( b ) , (29) for all pair s ( a, b ) ∈ A ( S A ) × A ( S B ). Given a particular co mp o und system of a general statistical theory , its set o f states S AB —we call it S A ⊗ S B from now on—will sa tisfy S A ⊗ min S B ⊆ S A ⊗ S B ⊆ S A ⊗ max S B (30) 27 As expected, for cla s sical compound s ystems (beca us e of the a bs ence of en tan- gled states ), we ha v e S A ⊗ min S B = S A ⊗ max S B . In the quantum case, we hav e s trict inclusions in (30): S A ⊗ min S B ( S A ⊗ S B ( S A ⊗ max S B . The general deﬁnition of a s eparable state in an arbitrar y CO M is made in analogy with that of [109], i.e., as one that can b e written as a convex co mbination of pro duct sta tes [58 , 62] (see also [64] for a genera lization): Deﬁnition 3. A state ω ∈ S A ⊗ S B wil l b e c al le d separa ble if ther e exist p i ∈ R ≥ 0 , ω i A ∈ S A and ω i B ∈ S B such that ω = X i p i ω i A ⊗ ω i B , X i p i = 1 (31) If ω ∈ S A ⊗ S B is not separa ble, then it will b e re a sonably called entangle d [91, 110].As exp ected, entangled states exist only if S A ⊗ S B is strictly greater than S A ⊗ min S B . The COM approach alrea dy shows that, given an arbitrary statistical the- ory , there is a generaliz ed notion of pro babilities of measurement outcomes. These pr obabilities are enco ded on the states in S . W e have seen tha t ther e a r e many diﬀerence s be t ween classical state spa ces and no n-classica l ones: this is expressed in the g eometrical prop erties of their conv ex sta te s paces and in the correla tions appea ring when comp ound sys tems are considered. Indeed, QM and classica l probability theories are just particular COMs among a v ast family of p oss ibilities . It is imp ortant to remark tha t ma ny informational techniques, such as the MaxEnt method, c a n b e suitably generalized to arbitrary pr obabilistic mod- els [111, 112]. In a similar vein, quantum informatio n theory c o uld be considered as a particular case o f a generalized infor ma tion theor y [1 0 8]. 6 Co x ’s Metho d App lied T o Ph ysics Now, w e review a relatively r ecent appr oach to the probabilities app earing in QM that uses distr ibutive lattices. A nov el deriv a tio n of F eynma n’s rules for quantum mechanics was presented in [2 9, 34]. Ther e, a n exp erimental lo gic of pr o c esses for quantum systems is pr esented, and this is done in s uch a wa y that the resulting lattice is a distributive one. This is a ma jor diﬀerence with the approach describ ed in Section 3.2 b ecause the lattice of pro jections in a Hilber t space is non-distributive. The logic of pro cesses is c o nstructed a s follows. Given a sequenc e o f measur e- men ts M 1 , . . . , M n on a q uantum system, yielding results m 1 , m 2 , . . . , m n , a par- ticular pr o cess is repr e sented as a me asuring se quenc e A = [ m 1 , m 2 , . . . , m n ]. Next, conditional (logical) prop os itio ns [ m 2 , . . . , m n | m 1 ] a re intro duced. Us- ing them, a probability is naturally asso ciated to a sequence A with the formula P ( A ) = P r ( m n , . . . , m 2 | m 1 ) (32) representing the pro bability of obtaining o utco mes m 2 , . . . , m n conditional up on obtaining m 1 . 28 Let us see how this works with a co ncrete ex ample in which the m i ’s have t wo p o s sible v a lues, 0 and 1. Then, A 1 = [0 , 1 , 1 ] and A 2 = [0 , 0 , 1 , 1 ] represent the measurement sequences of three and four measurements, resp ectively . Here, P ( A 1 ) = P r (1 , 1 | 0) represents the probability of obtaining outcomes m 2 = 1 and m 3 = 1 co nditio na l up on obtaining m 1 = 0. Measurements can b e coarse gra ined as follows. Supp ose that we wan t to coarse gr ain M 2 . The n, w e can unite the t w o o utco mes 0 and 1 in a joint outco me (0 , 1 ). Then, a new mea surement f M 2 is created. Th us, a po s sible sequence obtained by the replac e men t of M 2 by f M 2 could b e [1 , (0 , 1) , 1]. Analogous constructions can be done for other mea surements. In this w ay , a n op eration can b e deﬁned for the sequences: [ m 1 , . . . , ( m i , m ′ i ) , . . . , m n ] := [ m 1 , . . . , m i , . . . , m n ] ∨ [ m 1 , . . . , m ′ i , . . . , m n ] (33) Another o p er ation c an b e deﬁned re ﬂec ting the fact that sequences can b e comp ounded a s follows [ m 1 , . . . , m j , . . . , m n ] := [ m 1 , . . . , m j ] · [ m j , . . . , m n ] (34) Notice that, in the a b ov e equation, the la st measurement and outcome o f the ﬁrst sequence m ust be the same as the ﬁr st measurement and outcome of the second. With these oper ations at hand, it is easy to show that, if A , B , and C are measur ing s equences, then A ∨ B = B ∨ A (35a) ( A ∨ B ) ∨ C = A ∨ ( B ∨ C ) (35b) ( A · B ) · C = A · ( B · C ) (35c) ( A ∨ B ) · C = ( A · C ) ∨ ( B · C ) (35 d) C · ( A ∨ B ) = ( C · A ) ∨ ( C · B ) , (35e) Equations (3 5a)–(35e) show explicitly that “ ∨ ” is commutativ e and asso cia- tive, “ · ” is ass o ciative, a nd that there is right- and left-distr ibutivit y of “ · ” ov er “ ∨ ”. Equations (35 a)–(35e) deﬁne the a lgebraic “symmetries” of the exper iment al logic of pro ces ses. As in the approach of Cox to class ic a l probability , these symmetries are used to derive F eynman’s rules [34]. How ev er, a t this step, a crucial a ssumption is made: eac h measuring sequence will be represented by a pair of r eal num bers [34]. This assumption is justiﬁed in [34] by app ealing to Bohr’s co mplemen tarity principle. If mea suring sequenc e s A , B , etc. induces pairs of real num ber s a , b , etc., then, due to Equations (35a)–(35e), the asso ciated pairs should satisfy a ∨ b = b ∨ a (36a) ( a ∨ b ) ∨ c = a ∨ ( b ∨ c ) (36b) 29 ( a · b ) · c = a · ( b · c ) (36c) ( a ∨ b ) · c = ( a · c ) ∨ ( b · c ) (36d) c · ( a ∨ b ) = ( c · a ) ∨ ( c · b ) (36e) The reader can easily v erify that Equations (36a)–(36e) ar e sa tisﬁed by the ﬁeld of complex num bers (provided that the op er ations are interpreted as sum and pro duct of complex num bers). How ca n w e b e assur e d that complex n um- ber s a re the o nly ﬁeld that satisﬁes Equations (36a)–(36e)? In o rder to sing le out co mplex num bers amo ng o ther p oss ible ﬁelds, a dditional assumptions must be added, namely , pair symmetry , additivity , and symmetric bias (see [3 4, 2 9] for details ). O nce these conditions are assumed, the path is clear to derive F eyn- man’s rules b y a pply ing a deduction similar to that of Cox, to the exp erimental logic deﬁned by Equatio ns (35a)–(35e). 7 Generalization of Co x’s Metho d As we hav e seen in previous sections, there are tw o versions of CP , namely , the a pproach of R. T. Cox [25, 24] and the o ne of A. N. K olmogor ov [3]. The Kol- mogorovian appr oach can b e generalized in or de r to include non-Bo olean mo d- els, a s we hav e shown in Section 4. In wha t follows, we will see that Cox’s metho d c a n also b e genera lized to non-distributive lattices, and thus the no n- commutativ e character of QP c an be captured in this framework [19, 27]. Generalized P robabilit y Calculus Using Cox’s Metho d As we hav e seen in Section 2, Cox studied the functions deﬁned ov er a distribu- tive lattices and derived classical probabilities. In [27], it is s hown that if the lattice is assumed to b e non-distr ibutiv e, the pr op erties of Q P describ ed in Sec- tion 3.2 can b e derived b y applying a v ar ia nt of Co x’s method as follows (see [2 7] for details). Supp ose that the prop ositions of o ur system a re represented by the lattice of elementary tests of QM, i.e., the lattice o f pro jections P ( H ) of the Hilber t space H . The goal is to show tha t the “deg ree o f implication” mea- sure s ( · · · ) demanded b y Cox’s metho d satisﬁes Equations (17a)–(17c). This means that we ar e loo king for a function to the real num bers s , such that it is non-negative and s ( P ) ≤ s ( Q ) whenever P ≤ Q . The op er a tion “ ∨ ” in P ( H ) is asso ciative. Then, if P and Q are o r thogonal pro jections, the relatio ns hip betw een s ( P ), s ( Q ) , a nd s ( P ∨ Q ) must b e of the form s ( P ∨ Q ) = F ( s ( P ) , s ( Q )) , (37) with F a function to be determined. If a third prop os ition R is added, following a similar pro cedure to that o f Cox, we obtain for “ P ∨ P ∨ R ” the following functional equa tion F ( F ( s ( P ) , s ( Q )) , s ( R )) = F ( s ( P ) , F ( s ( Q ) , s ( R ))) . (38) 30 The above equation can b e solved up to rescaling [113, 30, 31, 33], and w e ﬁnd s ( P ∨ Q ) = s ( P ) + s ( Q ) . (39) whenever P ⊥ Q . It can b e shown that, for any ﬁnite family of or thogonal pro jections P j , 1 ≤ j ≤ n [27 ]: s ( ∞ _ j =1 P j ) = ∞ X j =1 s ( P j ) , (40) and we r ecov er condition (23c) of the axio ms of quantum probability . B y ex- ploiting the pr op erties of the or thogonal complement acting on subspa ces, it can a lso b e shown [2 7] that s ( P ⊥ ) = 1 − s ( P ) , (41) On the other hand, as 0 = 0 ∨ 0 and 0 ⊥ 0 , then s ( 0 ) = s ( 0 ) + s ( 0 ), and thus, s ( 0 ) = 0, which is condition (2 3b). In this w ay , it follows that Cox’s method applied to the no n-distributive lattice P ( H ) yields the same probability theory as the one provided by Eq uations (17a)–(17 c) for the q uantum cas e. What happ ens if Cox’s metho d is applied to an arbitr ary a to mic or thomo d- ular complete la ttice L ? Now, we m ust deﬁne a function s : L − → R , such that it is alwa ys non-negative s ( a ) ≥ 0 ∀ a ∈ L and is also o r der pr eserving a ≤ b − → s ( a ) ≤ s ( b ). In [27], it is shown that, under thes e r ather general assumptions, in any atomic o rthomo dular la ttice and for any orthog o nal de nu- merable family { a i } i ∈ N , s must satisfy (up to rescaling ) s ( _ { a i } i ∈ N ) = ∞ X i =1 s ( a i ) (42a) s ( ¬ a ) = 1 − s ( a ) (42b) s ( 0 ) = 0 . (42c) In this wa y , a gener alized proba bility theor y is derived (as in (17a)–(17c)). Equations (42a)–(42 c) deﬁne non-classica l (non-Ko lmogorovian) probability mea - sures, due to the fact that, in a ny non- distributive o rthomo dular lattice, there alwa ys exist elements a and b such tha t ( a ∧ b ) ∨ ( a ∧ ¬ b ) < a, (43) How ev er, in any class ic a l pr o bability theory , s ( a ∧ ¬ b ) + s ( a ∧ b ) = s ( a ) is alwa ys satisﬁed. In the non-B o olean setting of QM, von Neuma nn’s entropy (VNE) plays a similar r ole to tha t of Shannon’s in Cox a pproach [20]. This allows us to int erpret the VNE as a natur al measures of infor mation for an expe r imenter who dea ls with a co ntextual e vent str uctur e. 31 8 Conclusions W e present ed a new approach for pro babilities app e a ring in QM. While there exist (at least) t w o a lternative formalisms to CP (the Kolmogor ovian and the one due to R. T. Cox), w e have also shown that these tw o approaches can b e extended to the non-co mmut ative case. In this wa y , we ﬁnd that CP are a particular case of a more gener al mathema tical framework in which the lattice is distributive. QP is also a particular case of a v as t family of theories for which the prop ositiona l lattice is non-distr ibutive. Thus, we hav e a precise for ma l expression of the notion of QP . These formal frameworks do not exhaust the philosophical debate around the existence or not of a well-deﬁned notion of QP; notwithstanding, the ex- tension of Cox’s metho d to the non-distributive ca se, as well as the p os sibility of including a description o f the pr obabilities in QM in it, constitutes a precise step tow ards under standing the no tion of QP , o ﬀering a new point of view o f this no tion. According to this in terpretation, a ra tional a g ent is confronted with a par ticular ev ent structure. T o ﬁx ideas, suppos e that the ag ent is confronted with a physical system, and that the a gent has to p erfor m ex pe r iments and determine deg r ees of b elief ab out their p ossible outcomes. • If the lattice o f even ts that the age nt is facing is Bo o lean (as in Cox’s approach), then, the measures of degree of b elief will ob ey laws eq uiv alent to those of Kolmog orov. • O n the contrary , if the state of a ﬀa ir s that the agent must face presents contextualit y (as in standard quantum mec hanics), the measures inv olved m ust be non-Kolmo gorovian [27]. • Rando m v ar iables and information mea sures [20] will b e the na tur al g ener- alizations of the cla ssical case if the ev en t str ucture is not cla ssical. A s imi- lar observ ation ho lds for the application of the MaxE nt metho d [11 1, 11 2]. Our appro ach allows for a natural justiﬁca tio n of the p eculiarities a rising in quantum phenomena from the standp oint of a B ayesian a pproach. In particu- lar, quantum information theory could be considered as a non-K o lmogor ovian extension o f Shannon’s theor y [108]. Our appro a ch can b e cons idered as an alternative step to addr ess Hilb ert’s problem fo r the case of probability theor y in QM: the development of a n axioma tiza tion endo w ed with a clear a nd natural int erpretation o f the notions in volv ed. This work w as partially suppo rted by the grants PIP N ◦ 6461/ 05 amd 1177 (CONICET). In additio n, b y the pr o jects FIS200 8-007 81/FIS (MICINN)— FEDER (EU) (Spain, EU). F.H. was partially funded by the pr o ject “Per un’estensione semantica della Logica Computazio nale Quantistica-Impatto teor ico e ricadute implementativ e”, Regione Autonoma della Sa rdegna, (RAS: RASSR4034 1), L.R. 7/201 7, annualit` a 20 1 7—F ondo di Sviluppo e Co esione (FSC) 201 4 -202 0 and the Pro ject P ICT-201 9 -012 7 2. 32 A Lattice Theory Lattices can b e deﬁned by using equations, i.e., they can be characterized a s algebraic structures satisfying certain ax iomatic identities. A set L endow ed with t w o o p er ations ∧ a nd ∨ will b e called a lattic e , if, for all x, y , z ∈ L , the following equa tions a re s a tisﬁed. x ∨ x = x , x ∧ x = x (idemp otence) (44a) x ∨ y = y ∨ x , x ∧ y = y ∧ x (comm utativit y) (44b) x ∨ ( y ∨ z ) = ( x ∨ y ) ∨ z , x ∧ ( y ∧ z ) = ( x ∧ y ) ∧ z (assoc iativity) (44c) x ∨ ( x ∧ y ) = x ∧ ( x ∨ y ) = x (absortion) (44d) If the extra relationships x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) (distributivit y 1) (45a ) x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ ( x ∨ z ) (distributivit y 2) (45b) are sa tis ﬁed, the lattice is called distributive . Lattice theor y can also b e s tudied using p artial ly or der e d sets ( p oset ). A p oset is a set X endowed with a pa rtial ordering rela tion “ < ” satisfying • F or all x, y ∈ X , if x < y and y < x , then x = y . • F or all x, y , z ∈ X , if x < y and y < z , then x < z . W e use the notation “ x ≤ y ” for the ca se “ x < y ” or “ x = y ”. A la ttice L will b e a poset for whic h any tw o e le men ts x and y hav e a unique supremum and a unique inﬁmu m with resp ect to the or der structure. The least upp e r b ound of t wo g iven elements “ x ∨ y ” is called the “join”, and their grea test lower b o und “ x ∧ y ” is called their “meet”. A la ttice for which all its subsets ha ve b oth a supr emu m a nd an inﬁm um is called a c o mplete lattic e . If, furthermor e, there ex ists a grea test element 1 and a least e le men t 0 , the lattice is calle d b ounde d . They ar e usually called the maximum and the minimu m , respe ctively . An y lattice can b e extended into a bo unded lattice by adding a gr eatest and a lea st element. Every non-empty ﬁnite la ttice is b ounded. Complete lattices are alwa ys b ounded. An or tho com- plement ation in a bounded p oset P is a unary op er ation “ ¬ ” satisfying: ¬ ( ¬ ( a )) = a (46a ) a ≤ b − → ¬ b ≤ ¬ a (46b) a ∨ ¬ a and a ∧ ¬ a exist a nd a ∨ ¬ a = 1 (46c) a ∧ ¬ a = 0 (46d) hold. 33 A b ounded p os et with o rtho complementation will b e called an orthop oset . An ortholattic e , will b e a n orthop oset, which is also a lattice. F or a, b ∈ L (an ortholattice or or thop oset), we say that a is or thogonal to b ( a ⊥ b ) if a ≤ ¬ b . F ollowing [71], we deﬁne an orthomo du lar lattic e a s an or tholattice s atisfying the o rthomo dular la w: a ≤ b = ⇒ a ∨ ( ¬ a ∧ b ) = b (47) A mo dular lattic e , is an ortholattice satisfying the stro nger condition (mo dular law) a ≤ b = ⇒ a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) , (48) and ﬁnally a Bo o le an lattic e will be an o rtholattice satisfying the still s tronger condition (distr ibutive la w) a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) (49) Thu s, a Bo ole a n lattic e is a compleme n ted di s tributiv e lattice . W e use the terms Bo ole an lattic e and Bo ole an algebr a interc hangeably . If L has a null element 0 , then an element x of L is an atom if 0 < x a nd there exis ts no element y of L suc h tha t 0 < y < x . L is said to b e: • Atomic , if, for every no nz e ro elemen t x of L , there exists an atom a of L such that a ≤ x . • Atomistic, if every element of L is a supre mum o f atoms. 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