QUANTUM. LOGIC gJJ

DAVID J. FpCLlS , Ulliversily o{ Massa ι'hmells ， Alldlt!rSI. ,Hassacllllsl'llS, L:.5..-\. (5ecs. 1-3)

Rr CHARD J. GREECHIE, Louisial1 a Tech Ulli\'ersil γ ， RU5(Oll, Louisialla, U. 5.A. (5 1' ι'5. 1-3)

MARIA LOUISA DALLA CHIA民主 AND ROBERTO GIUNTINI, Ullivt!rsità di Firω， 1: 1'， Flore l1 ce, llalv (5ecs. 4一 10)

Introduction ...... 229 3.4 Cartesian and T ensor Products 1. A Brief History of Quantum of Orthoalgebras ...... 238 Logic ...... 230 3.5 The Logic of a Physical - 'EA - The Origin of Quantum Svstem ...... 239 Logic ...... 230 3.6 The Canonical Mapping ...... 240 1.2 The Work of Birkhoff and von 3.7 Critique of Quantum Logic ... 240 Neumann ...……-………..….. 230 4. The Logician's Approach .... 241 1.3 The Orthomodular Law ...... 231 S. Algebraic and Possible- 1.4 The Interpretation of Meet World Semantics ...... 242 and Join ...... 232 6. Orthodox Quantum Logic ... 243 2. Standard Quantum Logic .... 232 6.1 Semamic Characterizations of 2.1 The Orthomodular Lattice of QL ...... 243 Projections on a Hilbert An Axiomatization of QL ...... 244 Space ...... 232 6.2 2.2 Observables ...... 233 7. Orthologic and Unsharp 2.3 States ...... …..………...... 234 Quantum Logics ...... 245 2.4 Superposition of States ...... 235 8. Hilbert-Space Models of the 2.5 Dynamics ...... 235 Brouwer-Zadeh Logics ...... 247 2.6 Combinations of Standard 9. Partial Quantum Logics ...... 248 Quantum Logics ...... 236 9.1 Algebraic Semantics for 3. Orthoalgebras 臼 Models for WPaQL ...... 249 a General Quantum Logic '" 236 9.2 An Axiomatization of Partial 3.1 Orthoalgebras ...... 236 Quantum Logics ...... 249 3.2 Compatibility, Conjunction, 10. Critique of Abstract and Disjunction in an Quantum Logics ...... 250 Orthoalgebra ...... 237 3.3 Probability Measures on and Gloss缸-y ...... 250 Supports in an Orthoalgebra . 238 Works Cited ...... 254

INTRODUCTION ing logical formalisms, each with its own domain of applicability. Among these formal­ In forτnulating and studying principles of isms are Boolean-based propositional and valid reasoning, logicians have been guided predicate calcu!i, modal and multivalued log­ not only by introspection and philosophical ics, intuitionistic logic, and quantum logic. reflection, but also by an analysis of various Our purpose in this article is to outline rational procedures commonly employed by the history and present some of the main mathematicians and scientists. Because these ideas of quantum logic. In what follows , it principles have a multitude of disparate wi11 be helpful to keep in mind that there are sources, efforts to consolidate them in a sin­ four levels involved in any exposition of logic gle coherent system have been unsuccessful. and its relation to the experimental sciences. Instead, philosophers, logicians, and mathe­ maticians have created a panoply of compet- 1. Philosophical: Addresses the epistemology

Encyclopedia of Applied Physics , Vol. 15 。 1996 VCH Publishers, Inc. 3.527-28137-1/96/$5.00 + .50 229

飞 230 Quantum Logic

of th.: experiment.:t l sciences. Guides .:t nd tions affiliated with qu.:t ntum-mech.:t nicaJ moti\. .:t t 出 the activities at the remaining entltles. Icvels 认 hile assimibting and coordinating According to von Neumann, a quantum_ the insights gained from these activities. mechanical system c;f is represented mathe­ 2. S:..l/1 tacrÎC: Emphasizes the fOI寸n .:t l struc­ m.:t tically by a separable O.e. , countable di­ ture of a general calculus of experimental mensional) complex Hilbert space 贺， proposlt1ons. obsen'ables for c;f coπespond to self-adjoint 3. SemallfÎc: Focuses on the construction of operators on 坷， and the spectrum of a self­ classes of mathematical models for a log­ adjoint operator is the set of all numerical ical calculus. \'alues that could be obtained by measuring 4. PraglllatÎc: Concentrates on a specific the corresponding observable. Hence, a self­ mathematical model pertinent to a pa 口 ic­ adjoint operator with spectrum consisting at ular branch of experimental science. most of the numbers 0 and 1 can be re­ garded as a qlLantwn-meclzanical propositio J1 For instance, studies regarding the logics by identifying 0 with "false" and 1 with associated with classical physics could be "true." Since a self-adjoint operator has spec­ categorized as follows: trum contained in {O,I} if and only if it is an 1. Philosophical writings extending back at (orthogonal) projection onto a closed linear 况， least to Aristotle. subspace of von Neumann (1 955, p. 253) 2. Propositional and predicate calculi. observed that 3. The class of Boolean algebras. . the relation between the properties σalgebra 4. The Boolean of all Borel sub­ of a physical system on the one hand, sets of the phase space of a mechanical and the projections on the other, system. makes possible a sort of logical calcu­ lus with these. Likewise, for quantum logic, we have

1. Philosophical writings beginning with 1.2 The Work of Birkhoff and Schrödinger, von Neumann, Bohr, Ein­ von Neumann stein. et al. 2. Quantum-logical calculi. In 1936, von Neumann, now in collabora­ 3. The class of orthoalgebras. tion with Garτett Birkho 匠， reconsidered the 4. The lattice of projection operators on a matter of a logical calculus for physical sys­ Hilbert space. tems and proposed an axíomatic foundation for such a calculus. They argued that the ex­ expositorγreasons ， su 凹ey For our proceeds perimental propositions regarding a physical roughly in the order 1, 4, 3, 2. Thus, we give system c;f should band together to fonn a lat­ historγof a brief quantum logic in Sec. 1, tice L (Birkho匠， 1967) in which the meet outline the standard quantum logic of pro­ and join operatíons are fonnal analogs of the jections on a Hilbert space in Sec. 2, intro­ and and or connectives of classical logic (al­ duce orthoalgebras as models for quantum though they admitted that there could be a logic in Sec. 3, and discuss a general quan­ question of the experimental meaning of tum-logical calculus of propositions in these operations). They also argued that L Sec.4. should be equípped with a mapping carτymg each proposition a ε L into its negation a' ε L. In present-day tenninology, they proposed 1. BRlEF HISTORY OF QUANTUM that L fonns an orthocomplemented lattÎce LOGIC with ^. V. and a • a' as meet, join, and or­ thocomplementation, respectively (Kalm­ 1.1 The Origin of Quantum Logic bach, 1983; Pt址 and Pulmannovã, 199 1). Birkho 征 and von Neumann observed that The publication of John von Neumann's the experimental propositions conceming a Mathematische Gnmdlagen der Quantenme­ classical mechanical system c;f can be identi­ chanik (1 932) was the genesis of a novel sys fied with members of a field of subsets of the tem of logical principles based on proposi- phase space for C;f, (01', more accurately, with QuantU l11 Logic 231

dements of a quotient of such a field bv an fail to salísf)' the mudular la 川 E 飞 iJently ， ideal). In any ca妃， for a c1 assical mechanical von Neumann consid..:red this to be a po 吕 SI­ system ':1, they concluded that L forms a ble serious fla 飞 v of the Hilbert- 革 pace formll­ Booleal1 algebra. lation of qtl~lntum mcchanics as proposed in An orthocomplemented lattice L is a Bool­ his own Cnllldlage'l. Much of von Nell­ ean algebra if and only if it satisfies th 巳 dis­ mann's 叭 ork on continuolls geometries tribwive law (1 960) and rings of operators (Mun-ay and von Neumann, 1936) 飞vas nlotl 飞 ated bv his concret 巳 complemented x^(yVz) = (x^y)V(x^z). (1) desire to constluct modlllar lattices can丁ing an "a priori An example in which a ε L denotes the ob­ thermo-dynamic 飞 \'eight of states," that is , a servation of a wave packet on one side of a continuous dimension or trace function. plane, a' ε L its observation on the other ε L side, and b its observation in a state 1.3 The Orthomodular Law symmetric about the plane shows that Although the projection lattice of an infi­ b = b ^ (αVa') ;;é (b ^ a) V (b ^α') 0, nite-dimensional Hilbert space fails to satisfv the modular law in Eq. (2) , it was discovered so that the distributive law of classical logic by Hllsimi (1 937) that it does satisfy the fol­ breaks down even for the simplest of quan­ lowing weaker condition, now called the or­ tum-mechanical svstems. As Birkho旺 and thomodlllar law: von Neumann observed, If Z ::5 x , then x = (x ^ z') V 乙 (3) . whereas logicians have usually as­ sumed that properties of negation were If Eq. (2) holds, then so does Eq. (3) in 札ew the ones least able to withstand a criti­ of the fact that x x ^ (z' V z). The same cal analysis, the study of mechanics condition was rediscovered independently by points to the distributive idel1 tities as Loomis (l955) and Maeda (1 955) in connec­ the weakest link in the algebra of logic. tion with their work on extension of the Invok.ing the desirability of an "a priori Muπav-von Neumann dimension theorv of thermo-dynamic weight of states," Birkho旺 rings of operators to orthocomplemented lat­ and von Neumann argued that L should sat­ tices. An orthocomplemented lattice satisfy­ is 鸟r a weakened version of Eq. (1), called the ing Eq. (3) is called an orthomodular lattice. modular law, and having the following form: ln 1957, Mackey published an expository article on quantum mechanics in Hilbert space based on notes for lectures that he was If z ::5 x , thenx^(yVz) = (x^y)V 乙 (2) then giving at Harvard. These notes 队rere later published in the form of a monograph (Mackey, 1963) in which the basic principles If Eq. (1) holds, then so does Eq. (2) in view of quantum mechanics were introduced in of the fact that z ::5 X implies x ^ z 乙 terms of a function However, Eq. (2) is weaker than Eq. (1) since the projection operators for a Hilbert = Prob(A ，功， E) (4) space of finite dimension 11 ~ 2 fo口n a mod­ p ular, but nondistributive, lattice. Thus, Birk­ ho旺 and von Neumann proposed an ortho­ interpreted as the probability p that a mea­ ψ 肥' complemel1 ted modular lattice as a model for surement of the observable A in state a quantum-mechanical calculus of logic, al­ sults in a value in a set of E of real numbers. though they admitted that it would be satis­ The square A Z of A is then defined by the fying if one could interpret the modular law condition in Eq. (2) "by simpler phenomenological properties of quantum physics." Prob(AZ ， ψ'， E) = Prob(A ，收， F) ， Birkho旺 and von Neumann also gave an example to show that the projection opera­ where F is the set of all real numbers x such tors on an infinite-dimensional Hilbert space that XZ ε E. If A AZ , then A is called a 232 Quantum Logic qutlstioll. Under certain more or \èss reason­ one may regard a sd of compatible measure­ able hypothesès, it can be shown that the sèt ments as a single composite 'measurement'." of all questions forms an orthomodular bt­ ThllS , for compatible propositions, experi­ tice L. O1 ental meaning can be bestowed upon the The generality of Mack队也 formlllation me 巳 t and join by regarding these connecti\'es and the natural way in which Mack町 's ques­ as the conjunction and disjunction in the tions gi 飞已 rise to an 0 口 homodular bttice en­ u孔 lal sense of classical logic. gendered the heady idea of a universal logi­ 川 though the mainstream effort to develop cal calculus for a1l of the experimental a dable quantum logic has concentrated on sciences. Such a calculus would be based on the use of orthomodular Iattices as the basic the class of all orthomodular Iattices-in­ models (Jauch, 1968; Piron, 19ï6; Mittel­ c1 uding the Boolean algebras that would staedt, 1978; Bel trametti and Cassinelli. serve as models for the logics affiliated , with 1981) , altemati 四 models have been intro­ c1 assical mechanical svstems. Would this be duced that avoid the interpretation issue for the realization of Leibniz's dream of a cal­ meet and join by involdng the notion of clllllS ratiocinator? This captivating thought compatibility. Among these are the ortho- helped to motivate an ongoing study of the 1710dular posets introduced in the early 1960s theory of orthomodular lattices by a rela­ (Foulis, 1962) and the orthoalgebras pro­ tively small but devoted group of research­ posed ín th巳 late 19ïOs (Randall and Foulis, ers. An authoritative account of the resulting 19ï8; Hardegree and Frazer, 1981; Lock and the。可 of orthomodular lattices as developed Hardegree, 1984a,b). Thus, the evolution of up to about 1983 can be found in Kalmbach quantum logic from the 1930s to the present (1 983 ). has been the storγof a slow retreat from Boolean-algebra-based logic and the concur­ rent development of more and more generaI 1.4 The Interpretation of Meet and Join mathematical models.

ln spite of the appeal of a general scien­ tific logic based on orthomodular lattices, a 2. STANDARD QUANTUM LOGIC nagging question raised in the 1936 paper of Birkhoff and von Neumann was still unre­ 2.1 The Orthomodular Lattice of solved. If, for a quantum-mechanical system, Projections on a Hilbert Space most pairs of observations are incompatible and cannot be made simultaneously, what As a mathematical model for a calculus of experimental meaning can one attach to the quantum logic, the orthomodular lattice L of meet p ^ q of two propositions? Two and a projection operators on a Hílbert space 'iJe is half decades after his initial paper with von called a sta ηdard quantum logic. 飞Vilbur Neumann, Birkhoff retumed to this question (1977) has given a purely lattice-theoretic (Birkho旺， 1961), calling for an autonomous characterizatíon of the standard quantum quantum logic that draws its authority di­ logics. rectly from experiments. (A similar question ln the present section, we sketch the the­ arises in connection with the logic of relativ­ orγof standard quantum logics, considering istic physics where the traditional notion of only the special case in which 就 is a sepa­ simultaneity is meaningless for spatially sep­ rable Hílbert space of dimension at least arated events.) After all, simultaneity is an three over the complex number field C. indispensable constituent of classical propo­ Thus, we leave aside real or quaternionic sitional conjunction. Hilbert spaces as well as the generalized Hil­ An obvious way to avoid the interpreta­ bert spaces of Gross and Keller (Keller, tion issue for p ^ q is to replace the assump­ 1980). We regard 'iJe as the Hilbert space cor­ tion that p ^ q always exists with the weaker responding to a quantum-mechanical system assumption that it exists if p and q are com­ ~. (For the time being, we do not consider patible in the sense that they can be simul­ superselection rules.) taneously tested by means of a single experi­ If A is a bounded operator on 'iJe, we de­ ment. ln this connection, Birkhoff and von note by A* the adjoint of A. Thus, (A tþI φ1 = Neumann were careful to point out that ". . . (IþIA*φ> for all 功， φε 'iJe. A bounded operator Quantum Logic 233

P on Jt is callc: d a projectioll if P P"" p2 , (Pα and 飞 ve defì. ne L L( 页、) to be the set of all (八《优， (P ，J')' is the least lIpp.:r bot l11 d on L of such projection operators. If P ε L and the family (P) , and we 叭Tite V"Pα= (八 ，，( P)') \Consequently ， the standard quan­ .tt = P( 1f) = (P( ψ)1ψε 贸] (5) tum logic L is actually a complete orthomod­ ubr lattice. is the range of P, then At is a c1 0sed linear The inteflJretation of L as a model for a subspace of 开 conversely ， everv closed lin­ logic of quantum mechanics is based on the ear subspace Jt of 就 is of the form given in following premise: Eq. (5) for a uniquely determined P E L. If P and _~t are related as in Eq. (5) , we say that Tlze I I\'o-valtted (tnte/falseJ , experimen­ P is the proj.:ctioll ol1to J l. The zero operator tally leslable propositiolls for the qllall­ D is the projection onto (O) and the identity twn-lIlechal1 ical system :1' are repre­ operator 1 is the projection onto 灾. st!llted by the projections iη tfte standard If P is the projection onto Jtt and Q is the qualltum logic L for the Hilbert space 灾 :1'. projection onto H, we write P :$ Q if and corresponding to only if J饥 is a linear subspace of H. Thus, L Furthe 口口 ore ， if each experimental proposi­ is a partially ordered set (poset) under 三.If tion for :1' is identified with its corresponding is a c10sed linear subspace of 绽， we write M projection P ε L ， it is assumed that Jtu for the set of all vectors in 'iJf that are or­ thogonal to every vector in M. Then M ~ is if P, Q E L, then P 三 Q holds if and again a closed linear subspace of 'iJf. If P is only if P and Q are simultaneotlsly test­ the projection onto Jtt, we write the projec­ able and, whenever they are both tested tion onto A七 as P'. Note that and P is fowzd to be tnle, then Q will also be tme. P' = 1 - P, (P')' = P, 0' = 1, and l' = O.

Furthe 口nore ， if P, Q ε L with P 三 Q ， then 2.2 Observables Q' 三 P'. As is customary, we assume that an ob­ If .tt. and H are c10sed 1inear subspaces of servable or dynamical variable for the quan­ 绽， then so is the set-theoretic intersection Jtt. tum-mechanical system '::f is represented by a n H. If P is the projection onto M and Q is (not necessarily bounded) self-adjoint opera­ the projection onto H, we define P Q to be ^ tor A on the Hilbert space 就 1n particular, the projection onto M n H , noting that P ^ then, each projection operator P E L repre­ Q is the meet of P and Q in the poset L. If sents an observable that, when measured, we define P V Q = (P' Q')' , we find that P ^ can only produce the values 1 (true) or 0 V Q is the join of P and Q in L. Also, P P' ^ (false). As we shall see, the connection be­ and P V P' 1. and so L forms a lat­ o tween general observables and projection ob­ tice that is orthocomplemented by P • P'. servables is e 任'e cted by the celebrated spec­ Furthermore, L satisfies Eq. (3), and hence it tral theorem. is an orthomodular lattice. The smallest collection of subsets of the If P is the projection onto .<<. and Q is the real numbers IR that contains all open inter­ projection onto H, then Jtt is a linear sub­ vals and is c10sed under the fo口ηation of space of H C- if and only if P 三 Q'. If P 三 Q' ， complements and countable unions is called we say that P and Q are orthogonal to each the σfield of real Borel sets. A spectral mea­ other and ....Ti te P 1- Q. 1t can be shown that sure is a mapping E → P E 仕om real Borel P 1- Q if and only if P + Q is again a pro­ sets into projections such that P. 0, P jection operator, in which case, P + Q P R 1, and, for every pairwise disjoint sequence VQ. E" E2' E , … of real Borel sets, If C<<'a) is a family of c1 0sed linear sub­ J spaces of 绽， then the set啕 theoretic intersec­ tion naM" is again a closed linear subspace V PE , PE/UEzUE μ­ of 'iJf. 1f P" is the projection onto Jtt. αfor all α k-l and P is the projection onto naMa, then P is the greatest /ower bound in L of the family If E • PE is a spectral measure ， λεIR ， and J 234 QU :1 ntum Logic

( -::c， λ ]， dè 日 nè P;., = PJ' Thèn, by thè spec­ 2.3 States tral theon:nz there is a one-to-one co 町巳 spon­ A bound巳 d sdf-adjoint opèrator W on 灾 ís dènce bet飞veen observables A and spectral said to be 11O, zl/egati\'e if (W !lA 份主 o for al1 ，卢 measures E • P such that E ε 灾. A nonn巳 gativè operator W belongs to the trace class if the sèríes

tr(W) = 2: (W IjA ψ〉

The projèctions in the family (P E) are called converges for an orthonormal basís B ç 'Jf.. the spectral projectiol1 S for the obsel\lable A. Convergence on any one orthonormal basis We can now be quite explicit about the implies convergence on all orthonormal ba­ connection between obsel\lables in general ses. and projèction observables in particular. A (VOI1 Newnann) density operator on 'X is Suppose that A is an observable and that a bounded, self-adjoint, nonnegative, trace­ (P ) is the corresponding family of spectral E class operator W on 'X such that tr(W) 1. projections. Then Denote by fl ο(花) the set of all density operators on 纪 One of the basic assump­ P E represe l1ls the experimental proposi­ tion asserting that a measurement of the tions of statistical quantum mechanics is the observable A yields a result r that be­ following: longs to the real Borel set E ç IR. There is a one-to-one correspondence be­ tween the possible states of the system ;j In quantum mechanics it is understood and the density operators W E D such that a family of obsel\lables (Aa) is compati­ that, for every experimental proposition ble (that is, jointly or simultaneously observ­ P E L, tr(WP) is the probability that P able) if and only if A，. Ap_三 f包Aa)or all α ， β will be true when tested in the state cor­ (that is, if and only' íf the obsel\lables com­ responding to W. mute with each other). On the basis of this understanding, we can state the following: In accordance with this assumption, we shall identify each possible state of the system ;j A family of projections (P J ç L is com­ with the correspondíng density operator W. patible (that is, simultaneously testable) In particular, if A is an observable wíth spec­ I;'ß Pβ凡 for \主T if and only if Pa = all projec­ tral fami1y (P ) , the probability function. Eq. tions in the famfly. E C L 飞 (4) in Mackey's formulation, is realized as 主工It can be shown that two ob臼rvables com­ Prob(A.W.E) = tr(WP ). (6) 吃 mute with 叫 other liand 百~if their E 3ιspectral projections corñmï:rte---with each Equation (6). one of the fundamental equa­ ζ二寸 other. tions of quantum mechanics. says that It is interesting to note that the question of whether or not two projections commute the probability that a measurement of can be settled in purely lattice-theoretic tenηs. the observable A in the state W yields a Indeed, for P,Q E L, result r in the Borel set E is given by tr(WPJ. PQ QP if and only if = By a countably additive probability mea­ P = (P ^ Q) V (P 八 Q'). sure on the orthomodular lattice L is meant a functionω :L → [0 ， 1] ç IR such that. for The equation stating the condition is a spe­ everγsequence P,. P2 • 町. . . . of pairwise or­ cial case of the distributive law in Eq. (1); thogonal projections in L. hence, in a standard quantum logic, the failure of the distributive law is a direct con­ 叫VJ'k) = 2:叫凡). sequence of the fact that there are incompat­ ible pairs of quantum-mechanical 'observ­ ables. By a celebrated theorem of Gleason (1 957). Quantum Logic 235

ωis a countably additi\'e probability mea队 Ir t: E飞己 I 芋 1\'" is a pUl'e slate, 1hen W is said 10 be on L if and only if there is a (uniqu 巳 lv deter­ a c ο 1rt!1't!l1 t 511p l.nposilioll o[ the stat 巳 sWα. For min 巳 d) density operator W E n sllch that instance. if W is the 日 ctor state determined b~ψ 巳 1 (， each H'αis the vector state deter­ ω (P) = tr(H'P) for all P 巳 L. mined bv 仇 ε 灾， and I卢 differs from a nor­ malized linear combination of the 1/1" by a phase factor, then W is a coherent sup 巳 rposi­

2.4 Superposition of States tion of the W",

If W" W" W) , '"ε fl is a seqllence of 7- '3 --且 d巳 nsity operators and t" t" t , . . . is a corre­ D‘, u a m cSIM 3 d sponding sequence of nonnegative real num­ bers such that Ltk 1, then W :S, t,W , is Bv d川 ulII zics is meanl a studv of Ihe wav again a density operator, which is refeπed to in which the states (Schrödillger picfllre) or as a mixtllre or an incoherent superpositioll the observables (Heisenberg pictLl re) of a sys­ ofth 巳 states Wk • For instance, W could be re­ lem change or evolve in time. The Schrö• garded as the state of a statistical ens 巳mble dinger and Heisenberg pictures are mathe­ th巳 fraction of systems for which tk is of the matically equivalent. For definiteness, we systems that are in the state U飞. adopt the Schrödinger picture. Thus, if the A state W is called a pllre state if it cannot space n of density operators represents the be obtained as a mixtllre of other states. It is state space of the quantum-mechanical sys­ cllstomary to assume that individual physical tem 'J', then the dynamical evolution of the systems are always in a pure state and that syst巳 m is represented by a function f(t,W) of mixed states apply only to statistical ens巳 m­ the time 巳iR and the state W E fl such bles of systems each of which is in a pure that state, or to physical systems that are inter­ actively coupled with other physical systems. 阳，\1'的巳 β ， f(O ,W) = W and It can be shown that W ερis a pure f(t + s ,W) = f(t ，f(s ，"W丁). (7) state if and only if it is a projection onto a one-dimensional linear subspace Jtt of 绽. Thus, any normalized vector ψε 'iJf. deter­ The understanding in Eq. (7) is that f(t,W) mines a unique pure state, namely the pro­ represents the state of the system after a jection onto the linear subspace of complex time int 巳 rval t if it is in state W at time O. multiples of ψ. Such a state is called a vector The function f is called the dynamicallaw for state, and two normalized vectors deter丁nme the system 'J'. the same vector state if and onJy if each can If the dynamicallaw f in Eq. (7) prese凹es be obtained from the other by multiplying by superpositions, and is continuous in a suita­ a complex number of modulus 1 (a phase ble sense, it can be shown (Mackey, 1963) factor). Every state W is a mixture of pure that there is a family (U,) of unitary opera­ (that is, vector) states. tors continuously indexed by real numbers We define the support of W εβ ， in sym­ such that bols supp(W), to be the set of all Pε L such that tr(WP) 0/= O. This is the same as the set f(t ,W) = U1WU1-1 of al1 Pε L for which WP 0/= D. If W LktkW k is an incoherent superposition of the holds for a11 t ε IR. Hence, by a celebrated sequence (Wk ) , then it is clear that supp(W) representation theorem of Stone (1 932), it is contained in the set-theoretic union follows that there is a self-adjoint operator H UkSUpP(Wk). on 'iJf. such that More generally, if (Wa ) is a family of states, we say that the state W is a s μperposi­ i1H tion of the states W a if and on1y if U1 = e- (8)

supp(\-的 ç Uasupp(Wa) for all t E IR. Equation (8) is the operator form of the Schrδ， dinger equation and H is (Bennett and Foulis, 1990). If W as well as the Hamiltonìan operator for the system. 236 Q LlantLl m Logic

2.6 Combinations of Standard 且已 l~ct~d stat~s are r巳 present~d by density op­ Quantum Logics ~rators W that commute with both P\ and P,.

Suppos~ that 风 and 斑 2 are complex sep­ arable Hilb巳 rt spaces with corresponding 3. ORTHOALGEBRAS AS MODELS FOR standard quantum logics L\ and L2• There 灾\ A GENER<\L QUANTUM LOGIC are two natural ways to combine and 'Jt 2 to form a composite Hilbert space :X with its own standard quantum logic L: We can form 3.1 Orthoalgebras th~ 灾=就\ either direct sum 8;) 'iJf. 2 or the In this section, we present an axiomatic tensor product 'iJf 'iJf.\③就 2 (Foulis, 1989). In neith~r cas巳 is th~ structure of the result. mathematical structure called an orchoalge­ ing standard quantum logic L easy to de­ bra (Foulis et al. , 1992), which generalizes t~ 口ns the standard quantum logics. The idea is scribe in of the structures of L\ and L2 • If :1\ and :1, are quantum-mechanical sys­ to endow a generic orthoalgebra 飞vith an tems represented by corresponding Hilbert absolttte minimwn of mathematical stnlcture spaces 风 and customa可 to so that it becomes possible to investigate the 'X 2, it is regard the tensor product 'X风②'X 2 as the Hil­ meaning and consequences of the special bert space corresponding to the "combined features that distinguish particular orthoal­ gebras-for instance, Boolean algebras, or­ system" :1 = :1\ + :12 (Jauch, 1968). If this is thomodular lattices, or standard quantum so, then in the combination :1\ + :12 the sys­ tems can be tightly correlated, but they can­ logics-as models for a calculus of not exert instantaneous influences on each experimental propositions. other (Kläy et al., 1987). By definition, an orthoalgebra is a set L If W is a state for the combined system :1 containing two special elements 0 and 1 and equipped with a relation J. called orthogonal­ :1\ + :1 , there exist uniquely determined 2 p ， q ε L states 讥'\ ity such that, for each pair with p J. for :1\ and W 2 for :12 such that for all P\ ε L\ and all P E L q, an orthogonal sum p 8;) q is defined in L 2 2 and subject to the following four axioms:

tr(W\P\) = tr(W(P\ ( 1)) and (Commutativity) If p J. q, then q J. P and p tr(W2P2) = tr(W(l( P2)). 8;) q = q 8;) p. (Associativity) If p J. q and (p 8;) q) J. r, then states 讥'\ The and W2 are called reduced q J. r, p J. (q 8;) r) , and p 8;) (q 8;) r) = (p states. In general, W is not determined by 讥rl 8;) q) 8;) r. and 讥'2' but depends on the details of the (Orthocomplementation) For each p E L p' ε L coupling between :1\ and :12 , However, if W there is a unique such that p J. p' either 讥'\ is a pure state and or W 2 is pure, and p 8;) p' 1. then both 讥'\ and 讥'2 are pure and 讥讥'\ (Consistency) If p J. p , then p O. :1\十几 is ( W 2 • Therefore, if :1 in a We note that every orthomodular lattice L pure state and if :1\ and :12 are correlated in becomes an orthoalgebra if we define p 8;) q any way, then neither :1, nor :1 2 can be in a pure state. p V q whenever p 三 q'. In particular, every Boolean algebra and every standard If 'iJf. 'iJf., 8;) 'X2, a superselection ntle (Wick et al., 1952) may be imposed, in which quantum logic is an orthoalgebra. case the quantum logic L associated with 况 If L is an orthoalgebra and p ， q ε L ， we is understood to consist only of projections define p 三 q to mean that there exists r εL with p J. r such that p 8;) r q. It can be that commute with the projections P, and P2 of 'X onto the subspaces 'X, 8;) (0) and (O) 8;) shown that L is partially ordered by 三; 0 sp 三 1 and p = p" hold for all p ε L; and, if p 'X2, respectively. In this case, L is isomorphic 三 q ， then q' 三 p'. Also, if p J. q , then with to the Cartesian product L\ X L 2 of the stan­ dard quantum logics L, and L2' but L is no respect to 三 ， p 8;) q is a minimal upper longer a standard quantum logic. If such a bound for p and q; that is, superselection rule is imposed, it is assumed that the superselected observables are those p ， q 三 p 8;) q and there exists no r E L with spectral projections in L and the super- withp ， q 三 r

, t! However p (Ð q may not be the least upp r ( A = Va A. bound for p and q; that is , th已 conditions rε L and p , q 三 r do not necessarily imply that a 主 cakulated in any such B. If C and D are p (Ð q 三 r. finile orthogonal subst!ls of L , then EÐC .L If x ， y ε L have a lt!ast upp 巳 r bound (rt!­ EÐD it and only if C 门 D ç [0] and C U D is spectively, a greatest lower bound), w巳 write an orthogonal set. in which case (fC EÐ (ÐD it as x V y (respectively, as x ^ y). By defì• = @(C U D). If A = [矶 ， a 2 ). th 巳 n (ÐA a , , ortt/O I η odular posel nition an is an orthoal­ (Ð a,. Thus, if A = [矶，向内， . . . ,a.,), is an or­ gebra L satisfying the condition that p (Ð q thogonal set, we can define p V q whenever p .l.. q. An orthomodular lattice is the same thing as an orthoalgebra a ,(B a,(Ð aj (Ð…(Ð a" = (ÐA in which everγpair of elements x , y has a meet x y and a join x V y. A Boolean al­ ^ without notational conflict. gebra is the same thing as an orthomodular If P.q ε L and both p and q belong to a lattice satis 鸟iÏ ng the condition that x y ^ Boolean subalgebra B of L. then the greatest o only if x .l.. y. lower bound p ^B q and the least upper By a sllbalgebra of the orthoalgebra L, we bound p V B q of p and q as calculated in B mean a subset S ç L such that 0, 1 E S and, may well depend on the choice of B. If P ^B if p , q εS with p .l.. q , then p (Ð q E S and q is independent of the choice of B , we de­ p' εS. Evidently, a subalgebra of an or­ fine the conjunction p & q of P and q by thoalgebra is an orthoalgebra in its own right under the operations inherited 仕om the parent orthoalgebra. If, as an orthoalgebra in p&q =P^Bq. its own right, a subalgebra B of L is a Bool­ p B q ean algebra, we refer to B as a Boolearz 5μ b­ Likewise, if V is independent of the choice of B , we define the disjunctioll p q algebra of L. If p .l.. q in L , then + of P and q by

B = [O , l ,p,q ,p',q',p EÐ q ,(p EÐ q)'J p 十 q =pVBq.

is a Boolean subalgebra of L , so L is a set­ It can be shown that the compatible ele­ theoretic union of Boolean subalgebras. ments p and q have a conjunction if and only if they have a disjunction. Furthe口nore ， 3.2 Compatibillty, Conjunction, and if p & q and p + q exist, then p & q is a Disjunction in an Orthoalgebra maximal lower bound and p + q is a mini­ mal upper bound for p and q in L. If p and q We say that a subset of an orthoalgebra L are compatible and at least one of p ^ q or p is a compatible set if it is contained in a V q exists in L , then p & q and p + q exist, Boolean subalgebra of L. A compatible set of P & q = p ^ q , and p + q = P V q. If p & pair飞vise orthogonal elements is called an or­ q exists, then so do p' & q' and p' + q', and thogorzal subset of L. If L is a standard quan­ we have p + q = (p' & q')' and p & q = (p' tum logic, then a subset of L is compatible if + q')'. If p .l.. q , then p + q = P (Ð q and p and only if the projections in the subset &q = O. commute with one another. If L is an orthomodular poset, then any t 叭'0 compatible elements p , q ε L have a Let A [a"a 2.a 3 , . . . ,anl be a finÏte or- thogonal subset of L. Then, it can be shown conjunction p & q = p ^ q and a disjunction that the least upper bound p + q = P V q; however, there are orthoal­ gebras containing compatible pairs of ele­ ments that do not admit conjunctions or VBA = a , V Ba 2 V Ba 3 V B… VBan disjunctions. There are non-Boolean orthoal­ gebras in which everγpair of elements forms as calculated in any Boolean subalgebra B of a compatible set. There exist orthomodular L that contains A is independent of the posets containing three elements that are choice of B. Thus, we define the orthogonal pairwise compatible, but that do not form a sllm compatible set; however, in an orthomodular 2 quQu Q YL tL a n'EL EL m O 仔 户」 E po 吕 et ， e\' è' l 芋 pairwise olthogonal subsè'l is an It S = supp(ω) ， then IεS anc1, for all p, q 011 hogonal sub 吕巳t. Th巳 r巳 exist orthoalgebras εL 认 ith p ..L q , containing three elements that are painvise

(q CU r-- .d pbEEA S 011 hogon::l l , but do not form an orthogonal p· d- E a n 0 n 、.- p E Or σA ε na se t. (9)

3.3 Probabi1ity Measures on and A subset S of L such that Eq. (9) holds is Supports in an Orthoalgebra calkd a SllppOrc in L. In general. there ar巳 suppo口 s S ç L that are not of thc:: fonη By a probability l11 easure on an 0 口 hoalge­ supp(ω) for ωξ n; those that are of this bra L , we mean a mappingω :L → [0 ，门 ç R fonηar c:: callc:: d slOclzastic sllpports. Thc:: sc:: t­ suc h tha t, for p ,q ε L , 飞飞'ith p ..L q , theoretic union of suppo 口 s is again a sup­ po 口， and it follows that the c01lection of all suppo口 s ω(p (f) q) = ω(p) + ω (q) ， (8) in L forms a complete lattice under s c:: t-theoretic inclusion.

It is possible to defìne G'-complete orthoalge­ bras and cmmtab抄 additive probability mea­ 3.4 Cartesian and Tensor Products sures thereon, and thus extend Eq, (8) to se­ of Orthoalgebras quences in L, but we do not do so here. The set of a1l probability measures on L is de­ If L , and L 2 are orthoalgebras, the Carte­ noted by n = n(L), Evidently, n is a convex sian product L , X L 2 becomes an orthoalge­ subs巳 t of the vector space of a11 real-valued bra under the obvious componentwise oper­ functions on L. ations. If L , is identifìed with L , x [O} and L 2 If the eIements of L are regarded as rep­ is identi且ed with (O} x L 2 in LI X L2' then resenting two-valued experimental proposi­ every element in L , X L 2 can be written tions conceming a physical system '::1, then a uniquely in the form p (f) q with p ε L ， and probability measure ωεοmay be inter­ q E L 2• This construction generalizes the su­ preted in any of the following ways: perselected direct sum of standard quantum logics. Just as is the case for standard quan­ (Freqllency) ωis a complete stochastic l1lodel tum logics, I2(L X L ) is isomorphic in a ω(p) 1 2 for '::1 in the sense that is the "long­ natural way to the convex hull of n(L,) and run relative frequency" with which the n(L,). proposition p ε L will be true when re­ A construction for the tensor product LI peatedly tested (D'Espagnat, 1971). @ L of orthoalgebras based on Foulis and (Subjective) ωis 2 a model for coherent belief Randall (1 98 1) can be found in Lock (1 981). encoding all of our current information The factors L , and L 2 are embedded in the about the system '::1. Thus, if p ε L ， then p → p ② orthoalgebra LI @ L 2 by mappings ω(p) measures our current "degree of be­ 1 and q • 1 ( q for p εL p q ε L 2 in such scale 仕om lief," on a 0 to 1, in the truth a way that p @ 1 and 1 @ q are compatible of the proposition p (Jaynes, 1989). and have a conjunction (p @ 1) & (1( q) (Propens ity) ω(p) is a measure on a scale p ( q , Furthe口口ore ， elements of the form 仕om 0 to 1 of the "propensity" of the sys­ p ( q generate Ll ( L2' If αεο(L ，) and β tem '::1 to produce the outcome 1 (= true) ε n(L 2 ) ， there is a uniqueγ=αβε [2(L 1 ② when the proposition p is tested (Popper, that 说 p = α(p) β(q) L2 ) such @ q) for all p 1959) , ε L" q ε Lz. A probability measure on L ， ② (Mathematica l) ωis a mathematícal artifact L2 of the form aß is said to be factorizable, that may be of use in making inferences and a convex combination of factorizable about '::1 using data secured by making probability measures is said to be separable measurements on '::1 (Kolmogorov, 1933), (Kläy, 1988). The existence of probability

measures on L , ( L 2 that are not separable ωε n ， de且 ne ωby For we the Sllpport of seems to be a characteristic feature of the tensor product of non-Boolean orthoalge­ supp(ω) = [p E LI ω(p) > O}. bras. Qua 口 tum Logic 239

3.5 The Logic of a Physical System In thc stochastic postU !a lè , thc probabilil 兀 me川 ureωρcan be in tt: rpreted in anv 0 1' the We are now in a position to summarize four 叭'ays (frequency, subj c: cti 飞飞 propensity, the quα nt l/l1/ logic approach to the studv of mathematical) suggested in Sec. 3.3. In 飞\'hat physícal s\"ste /1lS (quantum-mechanical or follows , we denote by ~ the subset of n(L) not). The basic posllllate o( qlla 门 tlllll logic for consisting of all probability measures of the a physical system :1 is as follo 叭's fo 口口 ωψfor 功 εψ" and we refer to each ω ，ρ ε~. as a probability state for the system :1. It (Logic postlllate) The set L o{ all two­ is customa 巧! to identify the state ψwi th the vallled, experi l1Z entα llv testable proposi­ corτ巴 sponding probability stat 巳 ωψand to tions {or 'J has the structllre o{ al1 speak of the elements in ~ as states for Y. :-\1- ortlzoalgebra sllch that ever.. simultal1 e­ though this custom can lead to philosophical ollsly testable set o{ propositiOlls fOnlZSil and math巳 matical difficulties (what ifφ，收 ε compatible sllbset o( L and every flllite 1[/， φ 芦功， and yetωφ=ω ，þ?)' 飞 ve shall follo \V compatible sllbset of L is a si/11 11ltane­ it in the interests of simplicity. ollsly testable set of propositiolls. 扩 p ， q Let ωE ~ be a state and let p ε L be an εL with p J.. q, and i(p, q, and p æ q experimental proposition for the physical are tested simulta 门 eously ， then at most system 'J'. We say that p is possible, impossi­ one of the propositions p ,q w il/ be trlle, ble, or certain in the stateωif p εsupp(ω) ， p and p æ q wil/ be tnle if and only if eÌ• EE supp(ω) ， or p' EE supp(ω) ， respectively. If ther p or q is true. both p and p' are possible, we say that p is contingent in the state ω. The state space ~ is We refer to L as the logic of the system 'J. said to be w !Ïtal if every nonzero p ε L is It is customarγto assume that there is a certain in at least one state ωE I. state space ψassociated with the physical If 11 C I is a set of states, then a state ω system 'J. The elementsψεψare ca11ed εI is said to be a superposition of the states states, and, at any given moment, 'J is pre­ in 11 if sumed to be in one and only one state ψε ψ'. A state is supposed to encode a11 available supp(ω) ç U(supp(λ)1λε A}. inforrnation about the consequences of per­ forrning tests or making measurements on 'J The sllperpositioll c/os μ re of /1 is defined to when 'J is in that state. be the set .1>.0 of all superpositions of states Whereas the truth or falsity of an experi­ in J1. If A Np , then 11 is called sllperposi­ mental proposition p ε L can be deterrnined tion c/osed. A state ωis pure if the set (ω} is by a suitable test, it may or may not be pos­ superposition closed. Ifωis a pure state, 11 sible to deterrnine the curτent state ψεψof is a set of pure states, and ωε A' P ， then ωis 'J by a test or measurement; however, it may a colzerent superposition of the states in A. In be possible to bring 'J into a state ψby what fo11ows , we denote by ;;e the set of a11 means of a suitable state-preparation proce­ superposition-closed subsets of I. Note that dure. The state of the system 'J can change ;;e is closed under set-theoretic intersection, under the action of a dynamical law, under a and hence, it forrns a complete lattice under state collapse when an observer tests a prop­ set-theoretic inclusion. osition or measures an observable, because a It has long been a tenet of natural philos­ state-preparation procedure is executed, or ophy that a国liated with a physical system 'J' simply by virtue of a spontaneous state tran­ is a class s1 of attributes or propeη ies. At any SltlOl1. given moment, some of these attributes may A connection between the state space 'i' be actιlal， while the others are only potential. for 'J and its logic L is effected as fo11ows: The attributes of 'J that are always actual are its intrinsic attributes; those that can be ei­ (Stochastic poswlate) Each state ψεv ther actual or potential are its accidental at­ determines a corresponding probability tributes. The charge of an electron is one of measure ωψ GrZ L irz such a way that, for its intrinsic attributes, whereas the attribute pεL ， ω. ， (p) is the probability that the "spin up in the z direction" is accidental. proposition p is true when tested with To each attribute Aε s1 there corre­ tlze system 'J in the state 中. sponds a set A" ç I consisting precisely of 240 Quantum Logic

lhose staks ωsuch that A is actual 飞\'hene 、1: 1 properties (or attributesl 飞 vith the projections .'1 is in the st 且 teω. A heuristic ar2: umcnt. in the st:mdal-d quanlum logic L affiliated which we omit here, indicates thal ,\, shou\d with the quantum-mechanical system ::1. be sup 巳 rposition closed, so that .\， ε :f. 1n the more general situation under dis­ Similar arguments suggest that ever芋巳 le­ cussìon , it is also possible to relate Proposi­ ment of :;e corresponds in this \Vay to an at­ tions p ε L and propèrtiès (i 已， attributes) !\ tribute, and thus lead us to our third postu­ ε :1. For p ε L ， dehne late: [p] = (ωε {21 ω(p) = 1), (Attribllte postzt!ate) Eac/z attribwe A detennùzes a corresponding superposi­ tion-closed subset ^A o( the state space 三 We claim that [p] is superposition closed. 1n­ d巳 ed ， αE α 0, p' ε supp(α) ， p' εsupp(ω) ωε more, every ^ε 二e has the fonn AA for and so for some then ， ω (p') ω(p) some Aε 51. [p]. But, > 0, and so < 1. contradicting ωε [p]. Thus p • [p] provides Just as we identified states with probabil­ a mapping from experimental propositions p ity states, we propose to identify elements A ε L to attributes [p] E :;E. We refer to p • of the complete lattice :;e with attributes of [p] as the canonical mapping (Foulis et al. , the system '3'. (Note that, as a perhaps unde­ 1983)_ sirable consequence, all of the intrinsic at­ An attribute of the forτη [p] is called a tributes of :f become identified with the su­ principal attribute; the principal attributes perposition-closed subset 1: itself.) Thus, we are those that can be identi且ed with experi­ shall refer to the complete lattice :;E as the mental propositions as von Neumann did. It attribute lattice for the system :f. is not di 伍cult to show that every attribute is If A ， r ε :;E are attributes of '3', then A ç r an intersection (i.e. , a conjunction) of (pos­ if and onlv if r is actual whenever 11 is ac­ sibly in且 nit eJ y many) principal attributes. tual. Furtherrnore, the attribute A n r εg The state space 1: is unital if and only if [p] corresponds to a bona fide conjunction of o implies that p O. the attributes A and r in the sense that A n Evidently, p ， q ε L withp 三 q implies that r is actual if and only if both A and r are [p] ç [q]. If the converse holds, so that [p] actual. However, the least upper bound of A ç [q] implies that p 三 q ， then '3' is said to and r in :;e is (A U r)'p , and it can be actual have a fitll set of states. If 'J' has a full set of in states in which neither A nor r is actual. states and every attribute is principal, then Following Aerts (1 982), we say that the at the logic L is isomorphic to the attribute lat­ tributes A and r are separated by a superse­ tice :;E一 this is precisely what happens for a lection rule if A U r is superposition closed, standard quantum logic and it accounts for so that the least upper bound of A and r in von Neumann's identification of projections :;E corresponds to a bona fide disjunction of and properties. the attributes A and r.

3.7 Critique of Quantum Logic 3.6 The Canonical 即lapping Quantum logic is a relatively young sub­ We continue our discussion of the physi­ ject, it is still under vigorous development, cal system '3' subject to the logic, stochastic, and many consequences of the epistemologi­ and attribute postulates of Sec. 3.5. cal and mathematical insights that it has al­ Von Neumann (1 955, p. 249) writes, ready provided have yet to be exploited. Quantum-Iogical techniques involving the Apart 仕om the physical quantities . tensor product have already cast some light there exists another category of con­ on the well-known Einstein-Podolsky-Rosen cepts that are important objects of paradox (K1 äy, 1988), and it is hoped that physics-namely the properties of the they will also clarify some of the other clas­ states of the system '3'. sical paradoxes (Wigr町 's 缸 end ， Schrödin• Furtherrnore, he goes on to identify these ger's cat, etc.). The problem of hidden vari- Quantum Logic 241 ables can be forrnulaled, lInderslood, and dard qUWlllllll [ogic is identitìed \、 ith the studied rigorousJy in terms of qllanlU ll1 log­ complete olthomodular lattice of the projeι­ ics (Greechie and Gudder, 1973). Quantum­ tions on a separable Hilbert space of dimen­ logical techniques have enhanced our llnder­ sion at least three o\'er the complex nllmber standing of group-theoretic imprimitivity field. Thus standard quantum logic is a par­ methods and the role of superselection rules ticular kind of semantic model for a forrn of (Piron, 1976), and ideas related to quantllm abstract qllantum logic. Gen巳 rally ， a logic L logic ha飞 e been used to help llnravel the c3n be determined as a triple (FL. ←，仨)， con­ measurernent problem (Busch ec al., 1991) sisting of a fornlal language FL , a prool二 tlze­ There is a strong possibility that llnre­ oretic cO l/sequence relation, and a semantic­ stricted orthoalgebras are too general to (or l11odel-theoretic) cOllseqllellæ relation. For serve as viable models for quantum logic. the s3ke of simplicity. \\巳叭.il\ consider only Some orth03lgebras are extremely "patholog­ sentential languages, generated by an alpha­ ical" and thus mav be suitable onlv for the bet containing construction of counterexamples. It seems likely that only an appropriately specialized 1. a d 巳 nurnerably infinite sequence of atomic c1 ass of orthoalgebras, e.g. , unital orthoalge­ sentel1 ces (i.e. , sentences whose proper bras, might prove to be adequate as models parts are not sentences). for a general logic of experirnental proposi­ 2. a finite sequence of primitive logical con­ tíons. nectlves The rnain drawback of quantum logic is The set of the sentel1ces of the language FL is already evident in the standard quanturn the small 巳 st set that contains the atomic sen­ logic L of a Hilbert space 灾 In the passage tences and is c1 0sed under the logical con­ from the wave functions 功 in 沈阳 the pro­ nectIves. jections P ε L ， all phase inforτnation is lost. The proof-theoretic concept of conse­ The lost inforτnation becomes critical when quence 1-- for L is defined by referring to a sequential measurements--e.g., iterated calcultls (a set ofaxioms and of mles) that, Stern-Gehrlach spin resolutions (Wright, in turn, determines a notion of proof 斤onl a 1978)-are to be performed. There are at set of premises to a conclusion. A sentence β least two ways to restore the lost inforτna­ is called a proof二 theoretic cO l1 seqllence of a tion, both of which are currently being stud­ sentenceα(α ← β) if and only if (hereafter ied. One can introduce complex-valued am­ abbreviated as iff) there is a proof where ωis plit ι lde funccions on the logic L (Gudder, the prernise and βthe conclusion. The se­ 1988), or one can introduce a general math­ rnantic-consequence relation 1= refers to a ematical infrastructure called a manual or c1 ass of possible interpretations (models) of test space (Randall and Foulis, 1973; Foulis, the language. which render any sentence 1989) that can carry phase info口nation and "more or less" tme or false. A sentence ß is that gives rise to orthoalgebras as derived called a semanric conseqllence of α(α 』 β) structures in rnuch the same wav that HiI­ i 旺 in any possible rnodel of the language, ß bert spaces give rise to the standard quan­ is at least as true asα. turn logics. The two consequence relations • and 1= are reciprocally adequate 旺 they are equiva­ lent. In other words: for any sentences α ， β: 4. THE LOGICIAN'S APPROACH

We now present an approach to quanturn α I-- ß 旺 α 』 β. logic more c10sely aligned with that of stan­ dard logical techniques. In the preceding sec­ The "if arrow" represents the soundness tion. we gave an axiomatic approach to or­ prope口y of the logic, whereas the "only if ar­ thoalgebras, the most general rnathematical row" is the semantic completeness prope口y. structures currently used as models for Naturally, a logíc can be characterized by quantum logic. This section deals with quan­ different consequence relations that turn out tum logics by using the methods of logical to be equivalent. A logic L is called axioma­ tradition. In so doing, we will speak of ab­ tizable i 旺 it admits a proof-theoretic relation, stract quantum logics. As we have seen. stan- where the notion of proof is decidable. Fur- 少』A 少』 nv TL 『 u a n+··· u m O UD C ther, L is called decidable iff the proof-theo­ where the meanings of th 巳句 mbols are as retlc-consequence r巳 lation ← is decidable. follo 飞vs.

1. 1 is a nonempty set of possible worlds possibly cOITelated by r巳 lations in the se­ 5. ALGEBRAIC AND POSSIBLE- 飞VORLD quence R, and operations in the sequence SEMANTlCS Oj' 1n most cases, we ha飞 e only one rela­ tion R, called the accessibility relation. 1n the logical tradition, logics can be 2. n is a set of sets of possible worlds, rep­ generally characterized by means of two resenting possible propositions of sen­ privileged kinds of semantics: an algebraic tences. Any proposition and the total set semantics, or a possible-world semantics of propositions n must satisfy convenient (caHed also Kripkean semantics). closure conditions that depend on the These semantics give di 旺erent answers to particular logic. the question: What does it mean to interpret 3. v transforms sentences into propositions a formal language? 1n the algebraic seman­ preserving the logical form. tics, the basic idea is that interpreting a lan­ guage essentially means associating to any A world i is said to verify a sentence α (i Þ sentence an abstract tnah value or, more α) iff iεv(α). generally, an abstract meaning: an element of On this basis, the Kripkean semantic-con­ an algebraic structure. Hence, generally, an sequence relation is defined as follows: algebraic model for a logic L will have the fo口口 DEFINITION 5.2-ß is a semantic consequence of α(αÞβ) iff for any model Wè m = (d ， 外， (/， 风 'Oj ，Il，v) and for any world i E 1,

Þαthen Þ 自 where $1. is an algebraic structure belonging ifi i to a class .sl of structures satisfying a given set of conditions and v transforms sentences (in other words: whenever αis verified, also into elements of .sd, preserving the logical βis verified). form (in other words, logical constants are 1n both semantics, a sentence αis called a interpreted as operations of the structure). logical truth (Þα) i 旺 a is the consequence of We wiIl consider only structures where a bi­ any sentence ß. nary relation 三 (possibly a partial order) is An interesting varian时lt of Kripkean seman­ defined. On this basis, the semantÎc-conse­ tic臼s 怡i s represented by the ma臼an盯1。ηy quence relation is defined as follows: ble斗110旷rld sem饲1ant归ics口s ， founded on a generaliza­ tion of the notion of proposition. As we have seen, in the standard possible-world seman­ DEFINITION 5.1 一βis a semantic consequence tics, the proposition of a sentence αis a set of α(αÞ ß) iff for any model m 俐，时， of worlds: the worlds whereαholds. This v(α) s v(β) (in other words, the abstract automaticalIy determines the set of the meaning of αprecedes the abstract meaning worlds where αdoes not hold (the "mean­ of ß). ing" of the negation of α). 1ntermediate truth values are not considered. 1n the many-val­ 1n the possible-world semantics, instead, ued possible-world semantics, instead, one one assumes that interpreting a language es­ fixes, at the verγbeginning ， a set of truth sentially means associating to any sentence α values V ç;; [0,1] and any proposition is rep­ the set of the possible worlds (or situations) resented as a function X that assocìates to where αholds: This set, that represents the any truth value r ε[0 ， 1] a convenient set of extensional meaning of α ， is called the prop­ possible worlds (the worlds where our prop­ osition associated to α(simply ， the proposi­ osition holds with truth value r). As a conse­ α). tion of Hence, generally, a Kripkean quence, the total set of propositions Il tums model for a logic L will have the form: out to behave like a family of fuzzy subsets of 1 (see THEORY AND ApPLICATIONS OF Fuzzy m = (1 ， 良，0尸H川， LOGIC). Quantum Logic 243

Classical logic (CL) can be characterized 1. "J 仙，三， \1.0) is an ol'thomodubr lat­ both in the algebraic and in the Kripkean se­ tlce; mantics. Algebraically, it is determined by 2. t' (the intell)['etatÌon function) interprets the class of all algebraic structures C"'Z 川， the conn 巳 cti\'e -, as the operation the where s1 is a Boolean algebra and v inter­ connective G as the bttice-meet ^: prets the classical connectives (n巳 gation ， a. t'(α)εA for any atomic sentence α. conjunction. disjunction) as the correspond b. t"(-， β) = t"( β)'. ing Boolean operations (complement, meet, C. v( βGγ) = v( β) ^ v( γ) . join). In the framework of Kripkean seman­ tics, instead. CL is characterized by the class DEFINITION 6.1.1-A sentence αis called Ime of all models (I， R ，n. 叫， where in a modd ("J.l') iff v( α 1. Accordinglv, 1. the accessibility relation R is the identity 飞 e \Vill have that βis a cO l1 seqllellCe of α111 Ul (α 七=古Lβ) relation (in other words, anv \Vorld is ac­ the algebraic sell lltics of QL iff cessible only to itself); v( α) 三 v( β) in any modd (01.v) based on an 2. the set of the possible propositions n is orthomodular lattice sl. Further ， αis a quan­ the set of all subsets of 1; tWIl-logical tnlllz in the algebraic semantics 3. v interprets the classical connectives as (仨缸 α) 旺 αis true in any algebraic model the corresponding set-theoretic opera­ of Q L. t \Ons. As a consequence of the orthomodular prope口 y. a semantic version of a "deduction lemma" can be proved:

6. ORTHODOX QUANTUM LOGIC LEMMA 6. 1. 1 一αÞðLβi旺仨缸 α → β. In In the abstract quantum-logical universe, other words,• represents a "good" condi­ a privileged element is represented by ortho­ tional connective:α → βis logically true i 旺 P dox quantwn logic (QL), first described "as a is a consequence of α. logic" by Birkhoff and von Neumann (Birk­ hoff and von Neumann, 1936). QL is a sin­ DEFINITION 6. 1. 3-A Kripkean model of QL gular point in the class of all logics that are has the form 9)( (I， R. n.川. where the fol­ weaker than classical logic. Many logical and lowing conditions held: metalogical problems concerning QL have been solved. However, some questions seem 1. The accessibility relation R is reflexive to be stubbornly resistant to being resolved. and symmetric [we will also write L j for Rij; and i .L j for not Rij. Moreover. if 6.1 Semantic Characterizations of QL X ç;; 1. we will write i 一L X for VjεX (i.L j)]. Similarly to classical logic, QL can be A possible proposition of 幻 is a m a..x:i­ characterized both in the algebraic and in mal set X of worlds. which contains all the Kripkean semantics. The language of QL and only those worlds whose accessible --, contains the two primitive connectives \Vorlds are accessible to at least one ele­ (not), @ (and). Disjunction is supposed to be ment of x. In other words. iεX iff Vj L metalinguistically defined via De Morgan's i. 3k L j with k E X. law: For any X ç;; 1. let X~ : = (i ε I1 i .L X). One can prove that X'( is a possible prop­ αQ β. = -，(-， α@ --，β). osition for any X ç;; 1; X is a possible proposition i旺 X = ,XUlQ;); ø and 1 are pos­ A conditional connective can be defined as sible propositions; if X, Y are possible the "Sasaki hook": propositions. then X n Y is a possible proposition. α → β-αQ(α0 日). 2. n is a set of possible propositions closed under. 1. Q;). n. DEFINITION 6. 1. 1-An algebraic model of QL 3. n is orthomodular: X n (X n (X 门盯Q;))。 is a pair 9Jè = (刻，时. where ç;; Y. for any X, Y ε n. 244 QU :lntum Logic

4. a. v( α) 巳 rT. for anv atomic senl<: m:e cr; logic (HQL) the logic that is semanticalh b. v( ....， 卢) == v( 卢)之 characterized bv the class of all Hilbertian C. v(βGγ) == v(β)nγ(γ) modds. Apparently, HQL is stronger than QL. Hence. abstract quantum logic turns out DEFI :"J ITlO:-'; 6.1 .4-A sentenc 巳 αis caIled t l"lle to be ddinitdy more general with respect to in a modd ~l) i (l.R.n.I') iffαis veri lÌ.ed b\ its phvsical and historical origin. Th 巳 axiom. anv 飞I.' orld i 巳 1. atizability of HQL is still an open problem.

αis Accordingly ，飞ve will have that βis a COII­ DEFI:--

R6 α 卡 -,-.tY (\、;eak double n 巴巳 ation) tradict Ot丁 andιY川 Il plt!f e set T'"' s l1 ch that for R7 -，-， α 卡 α(strong doubl 巴 negation ). an飞出:ntenceα. eitherαE三 r" or -， αεr". α~β The set T : = (-， (α → (β → α))) (which con­ R8 一一一一一一 (contraposit 「β 卡「α tains the negation 0 1" the a fortiori principlèl R9α 。-， (α 。-，(αO β)) ← β(0口 homodu­ represents an example of a noncontradictory larit 飞) set that cannot b巴巴 xtended to a noncontra­ dictory and completc' set. The Sc' t T is non­ DEFtNITION 6.2.1-A proof is a finik se­ contradictory, because in some models (,-:1,\-'): quence of con 且 gurationsα ← βwhere an:­ v( -， (α → (β → α))) r' O. For instance, take el 巳 ment of the sequence is either an im­ (刀 1. 1') based on the orthomodular lattice of prop 巴 r rule or the conclusion of a proper the closed subspaces of íR', where v( α) and rule 叭:hose premises are previous elements 1"( β) are two rionorthogonal unidimensional of the sequence. Sl1 bSpacð. Ho\\ 巴飞己 r ， one can easilv check that \'(-.(α → (β → α))) 1 is impossible Hence, ....， (α → (β → α)) cannot belong to a DEFINITION 6.2.2 一βis a proof二 tlzeoretic COll­ noncontradictorγand complete set T去， which sequence of α(or provable 仕omα) (α ← QLβ) would trivially admit a model (5'l, v) such that i 旺 there is a proof whose last configuration v(β1 ， βεT气 From isα ← β. for any an intuitive point of view, the failure of the Lindenbaum property represents a very strong incom­ DEFINITION 6.2.3一βis a proof二 theoretic con­ pleteness result. The tertillm non datllr prin­ sequence of a set of sentences T (or provable ciple breaks down at the verγdeep level: 仕om T) (T ~QLβ) iff T includes a finite sub­ There are theories that are intrinsicallv in­ [α1" thatαI QS)…。 αη 卡 QL set . . ,a..l súch complete, even in mellte De i. β. Among the questions that are still un­ solved, let us mention at least the following: DEFlNlTION 6.2 .4-A set of sentences T is called contradictory if T ← QLβ QS)-，βfor 1. Is QL decidable? some sentence β noncontradictory ， other­ 2. Does QL admit the 卢 nite-model property? wise. A sentence αis contradictorγif [α) is In other words, if a sentence is not a quantum-logical truth, is there any finite contradict。可 noncontradictory ， othef\vise. model 叭.here our sentence is not verified? A positive answer to the 自 nite-model The proof-theoretic and the semantic-con­ property would automatically provide a sequence relations tum out to be equivalent. positive answer to the decidability ques­ Namely, a soundness and a completeness tion, but not vice versa. theorem can be proved: 3. Is the set of all possible propositions in the K.ripkean canonical model of QL or­ THEOREM 6.2.1-Sollndness thomodular? (The worlds of the canonical model are all the noncontradictory and If α~QLβ then αÞ QLβ. deductively closed sets of sentences T, whereas two worlds T and T' are accessi­ THEOREM 6.2.2-Completeness ble i 旺 whenever T contains a sentence α ， T' does not contain its negation -， α.) This If αÞ QLβ then α 卡 QLβ. problem is correlated to the critical ques­ tion whether any orthomodular lattice is As a consequence, one obtains the result that embeddable into a complete orthomodu­ a sentence is noncontradictory iff it is se­ lar lattice. Only partial answers are mantically consistent. known. A characteristic 飞 nomaly" of QL is the vi­ olation of a metalogical condition, which is 7. ORTHOLOGIC AND UNSHARP satisfied not only by CL but also by a large QUANTUM LOGICS class of nonclassical logics. This condition is represented by the Lindenbaum prope汀y， ac­ By dropping the orthomodular condition cording to which any noncontradictory set of both in the algebraic and in the Kripkean se­ sentences T can be extended to a noncon- mantics. one can characterize a weaker fonn 246 QU~lntum Logic

guag~J \、 ith t\、'0 of quantum logic , which is usually called 0 1'­ primitive negations: ..., rep. tl /O lοgic or /l I ÌlzÎmal q /lalltll1l/ logic (MQ L) röc: nts thc: fuzzy "not." whereas - is the This logic tums out to be more μtractab lt:" intuitionistic "not." On this basis, a necessitv 仕om a metalogical point of view: It satis 日出 operator can be definc: d in terms of the two the finite-modeI property; consequently, it is negatlons: decidable (Goldblatt, 1974). A calcllllls that represents an adeqllate axiomatization fO I" Lα: = --， α. MQL can be, natllrally, obtained by r已 plac­ ing the orthomodlllar rule R9 of Ollr QL cal­ In other words: 、 ecessar i1 yα" means the in­ clllllS with the weaker Duns Scotus ruleα tuitionistic negation of the fuzzy nc: gation of 。「 α ← β (ex absurdo seqttit ι lr qllodlibet: α. A possibi1 ity operator is then defined in Any sentence is a conseqllence of a contra­ terms of L and ...,: diction). A less investigated form of qllantllm logic Mα'= ...， L ...， α. is represented by paraconsistelzt qllantum logic (PQL) (Dalla Chiara and Gillntini, We will consider two forr口 s of Brouwer­ 1989), which is a weak example of an lln­ Zadeh logics: BZL (weak Brouwer-Zadeh sharp qllantum logic, possibly violating the logic) and BZU, which represents a fo口n of noncontradiction and the exc\uded-middle th自己 -valued quantum logic. Both logics ad­ principles. As we will see, unsharp quantum mit of Hilbert-space exemplifications. Alge­ logics represent natural abstractions 仕om braically, BZL is characterized by the class the unsharp approaches to quantum the。可­ of al1 models JH (.>1 .v) , where SIl. Algebraically, PQL is characterized by the 悦，三气 1.0) is a Brouwer-Zadeh lattice (sim­ class of all models based on an involutive ply a BZ lattice). In other words: lattice 仙，三:， 1 ， 0) ， with smallest element 0 (A ， 三:， 1.0) and largest element 1. Equivalently, in the 1. a. is an involutive lattice with Kripkean semantics. PQL is characterized by smallest element 0 and largest element 1. the class of all models (I ， R. n.吟， where R is a b. - behaves like an intuitionistic comple­ symmetric, not necessarily reflexive. relation. ment: and n behaves like in the MQL case. Di旺er­ entiy 仕om QL and MQL. a world i of a PQL a ^ a- O. model may verify a contradiction. Since R is generally not reflexive. it may happen that i a 三 a--. εν(卢) and i .1 v(β). Hence: i Þ ß 0) ...， β. An adequate axiomatization for PQL can be ob­ If a 三 b. then b- 三 a\ tained by dropping the orthomodular rule R9 in our QL calculus. Like MQL. also PQL c. The following relation holds between satisfies the finite-model prope口y and conse­ the fuzzy and the intuitionistic comple­ quently is decidable. ment: Interesting unsharp extensions of PQL are the Brouwer-Zadeh logics first investigated by a 、 a Cattaneo and Nisticò (1 989). A characteristic of these logics is a sp1itting of the connective d. The regularity condition holds: "not" into two fo 口ns of negation: a fuzzylike negation, which gives rise to a paraconsis­ a^a' 三 bVb'. tent behavior. and an intuitionisticlike nega­ tion. The fuzzy "not" represents a weak ne­ 2. v interprets the fuzzy negation ..., as the gation. which inverts the truth values truth fuzzy complement " and the intuitionistic and falsity, satisfies the double-negation negation ~ as the intuitionistic comple­ principle. but generally violates the noncon­ ment -. tradiction and the excluded-middle princi­ ples. The second "not" is a stronger negation. The logic BZL. which can be equivalently a kind of necessitation of the fuzzy "not." As characterized also by a Kripkean semantics, a conseqllence. the language of the Brouwer­ is axiomatizable and decidable (Giuntini, Zadeh logics is an extension of the QL lan- 1991). The modal operators of BZL behave Quantum Logic 247

clo 吕 ed similarly to the corresponding op巳 rators 0 1' 3. f7 is a set of possible propositions under 八- 0 例 .1). the famous modal system 5 ,. For instance, and LL αis equivalent to Lct; and LHαis equiva­ 4. \. (the interpretation function) maps sen­ lent to Mα tences into propositions in IT and inter­ The three-valued BZL' can be naturally, prets the connectives 0 , -, -. as the cor. characterized by a kind of man盯1ηv-可-叩va剖lu巳ed poωs- responding operations. 5剖ible巳 The other basic semantic definitions are be sk巳 tched as follows: On 巳 suppo 吕 es that in 啕 like in the algebraic semantics. One can t忧er叩pr陀eting a language m口1 巳 ans associa山ting to show that in any ortho-pair model the set of any sentence two domail1 s of certail1ty: the propositions has the structure of a BZ lat­ domain of possible worlds where the sen­ tice. As a consequence, the logic BZL3 is at tence holds, and the domain of possible least as strong as BZL. In fact , one can prove worlds where the sentence does not hold: All that BZ13 is properly stronger than BZL. As the other worlds are supposed to associate a counterexample, let us consider an in­ an interrnediate truth value (indetennined) to stance of the fuzzy excluded middle and an our sentence. The models of this semantics instance of the intuitionistic excluded middle 飞飞Ii!l be called models with positive and rzega­ applied to the same sentence α. One can eas­ tive domaills (shortly, ortho-pair models). ily check: Briefly, an ortho-pair model has the forrn j<<. (I ，R. IT， 时， where = αQ-.α Þ= BZLJαQJ-αand 1. 1 is a nonempty set of worlds and R (the α <9 -α 1= BZL)αQJ-.α. accessibility relation) is reflexive and sym­ metric (like in the Kripkean characteriza­ However, generally tion of QL). αQJ-.α 际 BZLαQJ-α. The possible propositions (in the sense of our definition of the Kripkean model for QL) Also BZ13 is axiomatizable (Cattaneo et al., will be here ca!led simple propositiorzs. The 1993) and can be characterized by means of set 1: of all simple propositions gives rise to an algebraic semantics. an ortholattice; let us indicate by #, I寸， U the lattice operations defined on 1:

2. A possible proposition of J<<. is any pair 8. HILBERT-SPACE MODELS OF THE (凡.xo) ， where X" Xo are simple proposi­ BROUWER-ZADEH LOGICS tions such that X ç; )曰~ (in other words: , Hilbert-space models of both BZL and X" X are orthogonal). The fo!lowing op­ o BZ13 can be obtained in the 仕amework of erations and relations are defined on the the unsharp (or operational) approach to set of all possible propositions: quantum theory that was first proposed by a. the fuzzy complement Ludwig (1 954) and developed (among others) by Kraus (1 983), Davies (1 976), Busch et al. (X,.xo)' = (Xo.x,); (1 991), and Cattaneo and Laudisa (1 994). One of the basic ideas of this approach is a b. the intuitionistic complement "liberalization" of the mathematical counter­ pa口 for the intuitive notion of "experimen­ (X,,xo)- =仅卢 .xð); ta!ly testable proposition." As we have seen, in orthodox Hilbert-space quantum mechan­ c. the propositional conjunction ics, experimental propositions are mathemat­ ica!ly represented as projections P on the Hilbert space 'ðe corresponding to the physi­ (Y" 凡) (X ， 内瓦， Xo (X,,xo) ^ = U YO); cal system ~ under investigation. If P is a projection representing a proposition and W d. the order relation is a density operator representing a state of ~， the number tr(WP) represents the proba­ (X ，，xo) 三 (Y"ì飞) iff X, ç; Y, and Yo ç; Xo. bility value that the system ~ in the state W 叫 A QU n) T ζ 且『 u a n+』 u m O σ .1户』 丁 咀『 』 口

、 erihes P (Bom probability). Ho\Vever. proj 己心 can sho\V that any BZ posd can be embed­ tions are not the only operators for \、 hich a ded into a complete BZ lattice [for the Bom probability can be de fì. ned. Let us con­ MacNeille colllplelioll (Birkhoff. 1967) of a sider the c\ass 't;( 灾) of al\ linear bounded op­ BZ poset is a complete BZ lattice (Giuntini. erators D such that for any density operator 199 1)]. As a consequence. the MacNeil\e W. completions of the effect-BZ posets represent natural Hilbert-space models for the logic tr(WD) ε[0.1]. BZ L. As to BZLJ. Hilbert-space models (I ， R. JI. ν) It tums out that 宅(灾) properly inc\udes the in the ortho-pair semantics can be con­ structed as fol\o\Vs: set L( 灭) of al\ projections on 就. In a sense. 宅(灾) the elements of represent a "maxima\" 1. 1 and R are de且 ned like in the Kripkean possible notion of experimental proposition. Hilbertian models of QL. The simple in agreement with the probabilistic rules of propositions tum out to be in one-to-one quantum theory. In the framework of the un­ correspondence to the set of the projec­ sharp approach, the elements of 'íß(就) have tions of 况 e所 cts. been cal\ed An important difference 2. Il is the set of all possible propositions. between projections and proper effects is the Anve任ect D can be transformed into a fo l\owing: Projections can be associated to proposition f(D) (X? Xg) , where sharp propositions having the fo口口 "the value for the observable A lies in the exact 砰:= [ψε I1 tr(凡 D) = 1) and Borel set F," whereas e任ects may represent xg:= [ ψε I1 tr(Pψ D) = 0) also fit::;:y propositions like "the value of the observable A lies in the fuzzy Borel set F." As (Pψis the projection onto the unidimen­ a consequence, there are e旺écts D that are sional subspace spanned by the vector ψ). different from the null projection 0 and In other words. Xf and xg represent re­ that are verifì.ed with certainty by no state spectively the positive and the negative [for any W, tr(WD) '" 1]. A limit case is rep­ domain of D (in a sense, the extensional resented by the semitransparent effect ~ 1 meaning of D in the model). The map f (where 1 is the identity operator), to which tums out to preserve the order relation any state W assigns probability value ~ and the two complements. The class of all e 旺écts of OX gives rise to a 3. The interpretation function v follo \Vs the structure 何 (iJ() • 三:，-， 1 ， 0) which is a BZ poset intuitive physical meaning of the atomic (not a BZ lattice!). ln other words ，三 is a sentences. partial order with largest element 1 and smallest element 0, while the fuzzy and the intuitionistic complement (' and -) behave like in the BZ lattices. The relation and the 9. PARTIAL QUANTUM LOGICS operations of the effect-structure are defined exampl 巳 s as follows: So far we have considered only of abstract quantum logics, where conjunc­ 1. D ， 三 D 2 iff for any density operator W, tions and disjunctions are supposed to be al­ ways de且ned. However, as we have seen, the tr(WD,) $ tr(WD2 ). 2. 1 1. experimental and the probabilistic meaning 3. D' = 1 - D. of conjunctions of incompatible propositions the。可 has 4. D- is the projection PK

(SPaQL). Are WPaQL and SPaQL axiomatiz­ The rules of 认'PaQL are as follo\\'s: able? Rlα ← α (i dentity). R2 α 卡- ß β ← γ(transitivitv) 9.1 Algebraic Semantics for WPaQL α ← γ 飞 rans lt1 vlty) The language of WPaQL contains two R3α ←-，-， α(weak double negation). -，-， α ← α(strong primitive connectives: the negation -', and R4 double negation). the exclusive disjunction ffi (allt). A conjunc­ α 1- β R5 一一一一一一 (contraposi tion). tion is metalinguistically defined, via De 「β ←「 α Morgan's law: R6β ← α 曰「 α(excluded middle). α ←「βαB -， α 卡-α 囚 β R7 _. I D -- - r- (unicity of α 曰 β -，(-， α 田「 β). 「α ← β negation). The intuitive idea underlying our semantics α 1--，βα 1- α1α ，1- αβ 1- β1β ，1- β Di到 unctions R8 for WPaQL is the following: α 田 β ← α ， ffiβ1 and conjunctions are considered "legitimate" (weak substituthity). 仕om a mere linguistic point of view. How­ α ←「β ever, semantically, a disjunctionα 田 βwill R9 (weak ∞ mmutativitv). 田 β ← β 田 α Jf have the intended meaning only in the "well­ β ←「γα ←-， (β 田 γ) behaved cases" (where the values of αand β RI0 are orthogonal in the corresponding orthoal­ α ←「β β ←「γα ←-， (β 田 γ) gebra). Otherwise ， α 田 βwill have any Rll meaning whatsoever (generally not con­ α 曰 β ←「 γ αand β). β ←「γα 1- -， (β 曰 γ) nected with the meanings of A R12 similar semantic "trick" is used in some stan­ α 囚 (β 田 γ) ← (α 曰 β) 田 γ. dard treatments of the description operator , β ←「γα ←-， (β 囚 γ) R13 ("the unique individual that satisfies a given (αffi ß) 回 γ) ← α 曰 (β 田 γr property") in classical model the。可· (RI0-R13 require a weak associativity.) DEF1 NITION 9. 1. 1-An algebraic model of The other basic proof-theoretic definitions WPaQL is a pair J (.t = (s1，吟， where are given like in the QL case. Some derivable rules of the calculus are the following: 1. s1 (A,8:), l ,O) is an orthoalgebra. 2.ν(the interpretation function) satisfies the α 卡- Dl ß following conditions: β 1- α 曰-，(α 曰「β) ν(α)ε A ， for any atomic sentence α; α 卡- D2 ß ν( -，β) =ν(β)' ， where ' is the orthocom­ α tE -， (α 曰 -，ß) ← β plement operation that is defined in s1; D3 ν叫(卢曰 γ吵0) = α ←「γβ ←「γα 田 γ ← β 曰 γβ 田 γ ← αE γ {~俨ν叫咿(伊卢削附附)肋@φν圳( 讪忧， 叫)肋@φν叫白ω(仆ωγ讪训)川i叫m时叫din α 卡-ß anyelement otherwise α 1- -，ß α 1- γβ 1- γγ 1- α 田 β D4 Accordingly, we will have that α 田 ß I- γ

αÞWPaQL ß 旺 in any WPaQL model The proof-theoretic and the semantic-con­ sequence relations for the logic WPaQL are M = (s1，ν) ， ν(α) 三 ν(剖， reciprocally adequate. Namely, a soundness and a completeness theorem can be proved. where :S is the partial order relation defined in s1. THEOREM 9.2.1 一-Sollndness

9.2 An Axiomatization of Partial If α 1- WPaQLβ then αÞWPaQLß. Quantum Logics The logic WPaQL is axiomatizable. We THEOREM 9.2.2-Completeness present here a calculus that is obtained as a natural transforrnation of our QL calculus. If αÞ WPaQL ß then α 卡 WPaQLβ. nJ phJnu ny YL ，、 eal nt u m O-OE C 』 ‘

As to strong partial quantum logic quantum logic is not unique. Besides onho­ (SPaQL), an axiomatization can be obtained dox quantum logic , different forms oE partial by adding to our WPaQL calculus the rollow­ and unsharp quantum logics have been de­ ing rule: veloped. In this situation, one can wonder 飞vhether it is still reasonable to look for the R14α ←「自 α ← γβ ← γ most adequate abstract logic that should α 田 β ← γ faithfully represent the structures arising in the quantum 认 orld. A question that has been often discussed Semantically, the models of SPaQL will concerns the compatibility between quantum be based on orthoalgebras .71 仙，(fl， 1 ，酌， logic and the mathematical formalism oE satisfying the following condition: If defìned , quantum theo巾， based on classical logic. Is a EÐ b is the sup of a and b. As we have seen the quantum physicist bound to a kind of in Sec. 3.1 , this condition is necessarv and "logical schizophrenia"? At 且 rst sight, the sufficient in order to make the orthoposet in 司 compresence of different logics in one and duced by the orthoalgebra stl. an orthomodu­ the same the。可 may give a sense of uneasi­ lar poset. The soundness and the complete 萨 ness. However, the splitting of the basic log­ ness theorems for SPaQL (with respect to ical operations (negation, conjunction, dis­ this semantics) can be proved similarly to junction, • • .) into different connectives with the case of WPaQL. different meanings and uses is now a well­ accepted logical phenomenon that admits consistent descriptions. As we ha 飞 e seen, 10. CRITIQUE OF ABSTRACT classical and quantum logic turn out to ap­ QUANTUM LOGICS ply to different sublanguages of quantum the。可 that must be sharply distinguished. Do abstract quantum logics represent "real" logics or should they rather be re­ garded as mere extrapolations from particu­ lar algebraic structures that arise in the GLOSSARY mathematical formalism of quantum me­ chanics? Different answers to this question Abstract Quantum Logic: A logic (血，←， have been given in the historγof the logi­ þ), where the proof-theoretic and the se­ coalgebraic approach to quantum theo巧r. Ac­ mantic-consequence relations violate some cording to our analysis, the logical status of characteristic classical principles like the dis­ abstract quantum logics can be hardly put in tributivity of conjunction and disjunction. question. These logics turn out to satisfy all Adjoint: If A is a bounded operator on a the canonical conditions that the present Hilbert space, then the adjoint of A is the community of logicians require in order to unique bounded operator A 食 that satisfies call a given abstract object a logic: s)ntacti­ (A~ φ) = (ψIA* φ) for all vectors ψ， φin the cal and semantical descriptions, proofs of space. soundness and completeness theorems, and Algebraic Model of a Language: A pair so on. (stl., v) consisting of an algebraic structure stl. Has the quantum-logical research defi­ and of an interpretation function v that nitely shown that "logic is empirical"? At the transforms the sentences of the language ve巧 beginning of the history of quantum into elements of stl., preserving the logical logic, the thesis according to which the form. choice of the "right" logic to be used in a Algebraic Semantics: The basic idea is given theoretic situation may depend also on that interpreting a formal language means experimental data appeared a kind of ex­ associating to any sentence an element oE an tremistic view, in contrast with the tradi­ algebraic structure. tional description of logic as "an a priori and Attribute: One of a class of properties af­ analytical science." These days, an empirical filiated with a physical system. At any given position in logic is no more regarded as a moment some of the attributes of the system "daring heresy." At the same time, we are may be actual. while others are only poten­ facing a new difficulty: As we have seen, tial. Quantum Logic 251

A."{iomatizable Logic: A logic is a:düma• ha\'e a disjunction p 十 q if there is a Bool­ tizable when its concept of proof is d l.'cida­ ean subalgebra B of L \vith p , q E B and the ble. join p V 8 q of p and q in B is independent of Boolean Algebra: An orthocomplemented the choice of B , in \\'hich case p + q bttice L that satisfìes the distributive bw p p\/8 q. ^ (q V r) = (p ^ q) V (p ^ r) for all p , q , r Dynamics: The e\'ülution in time of the E L. state of a physical system. Borel Set: A (real) Borel set is a set that Effect: A linear bounded operalor A of a belongs to the smallest collection of subsets Hilbert space such that for any density op­ of the real numbers IR that contains all open erator W , tr(WA)ε[0 ，门. In the unsharp ap­ inten;als and is closed under the formation proach to quantum mechanics, effects repre­ of complements and countable unions. sent possible experimental propositions. Brouwer-Zadeh Lattice (or Poset): A Experimental Proposition: A proposition lattice (or poset) L with smallest element 1 飞vhose truth value can be determined by con­ and largest element 0 , equipped with a reg­ ducting an experiment. ular involution ' (a fuzzylike complement), Greatest vs Max缸nal: If L is a partially and an intuitionisticlike complement 丁 sub­ ordered set (poset) and X ç L , then an ele­ ject to the following conditions for all p ， q ε ment b εX is a greatest element of X if x :s L: (i) p 三 q 司 q- 三 P 飞(ii) p ^ p- = 0, (iii) b for all x εX. An element b εX is a max­ p 三 p →， (iv) p-' = p 二. imal element of X if there exists no element Brouwer-Zadeh Logic: A logic that is xεX with b < x. characterized by the class of all models lnvolution: A mapping p • p' on a poset based on Brouwer-Zadeh lattices. L satisfying the following conditions for all Compatible: A set C of elements in an or­ p , q ε L: (i) p 三 q 司 q' 主三 p', (ii) p" = p. thoalgebra L is a compatible set if there is a Involutive Lattice (or Poset): A lattice Boolean subalgebra B of L such that C ç B. (or poset) with smallest element 0 and larg­ Complete Lattice: A lattice in which est element 1, equipped with an involution. everγsubset has a least upper bound, or Join: If L is a partially ordered set (poset) join, and a greatest lower bound, or meet. and p , q ε L ， then the join (or least upper Completeness: A logic (FL ，'t-， 乍 is (se­ bound) of p and q in L , denoted by p V q if mantically) complete when all the semantic it exists, is the unique element of L satisfying consequences are proof-theoretic conse­ the following conditions: (i) p , q 三 p V q and quences (i fα 乍 ß then α~ ß). (ii) r ε L with p , q 主三 r 司 p V q :s r. Conjunction (in an Orthoalgebra): If L Kripkean Model of a Language: A sys­ is an orthoalgebra and p , q ε L ， then p and tem (I,R". . . ,Rno ,o". . . ,o",ll,v) consisting of a q have a conjunction p & q ε L if there is a set 1 of possible worlds, a (possibly empty) Boolean subalgebra B of L with p , q ε B ， sequence of world relations R" . . . , Rm and and the meet P ^B q of p and q in B is in­ of world operations 0" . . . , 0 月， a family n of dependent of the choice of B , in which case subsets of 1 (called the propositions), and an p & q = P ^B q. interpretation function v that transforms the Decidable Logic: A logic is decidable sentences of the language into propositions, when its proof-theoretic consequence rela­ preserving the logical form. tion is decidable. Kripkean Semantics: The basic idea is Density Operator: A self-adjoint, nonneg­ that interpreting a formal language means ative, trace-class operator W on a Hilbert associating to any sentence the set of the space, such that tr(W) 1. possible worlds where the sentence holds. Direct Sum (or Cartesian Product) of This set is called also the proposition associ­ Hilbert Spaces: The direct sum of the Hil­ ated to the sentence. be 口 spaces 况 and 页 is the Hilbert space 'iJf. Lattice: A partially ordered set (poset) in EÐ 'J[ consisting of all ordered pairs (x ,y) with which everγpair of elements p , q has a least x ε 绽 ， y ε 页， and with coordinatewise vec­ upper bound, or join, p V q and a greatest tor operations. The inner product is defined lower bound, or meet, p ^ q. by ((x"y,)I(xz ,yz)) = (x ， L~z) + (y,lyz). Least vs Minimal: If L is a partially or­ Disjunction (in an Orthoalgebra): If L is dered set (poset) and X ç L , then an element an orthoalgebra and P, q ε L ， then p and q G εX is a least element of X if a 三 x for all 252 Quantum Logic

工 ε X. An dement a ε X is a minimal 己 le­ L: (i) P 1. q 司 q 上 p aml p e q = q 8 p , (ii) ment of λ~ if there exists no 巳 lement 工 εX P 1. q and (p 8j q) 1. r 司 q .L r , p .L (q e r) with x < a. and p (B (q (B r) = (p 8 q) ( r , (iii) p E L Lindenbaum Property: A \ogic satis 且由 ~ there is a unique p' ε L such that p .L p' the Lindenbaum proper1y when any noncon­ and p 8 p' 1. and (iv) p 1. P 司 p = O. tradictorv set of sentences T can be 巳 xtended Orthocomplementation: A mapping p • to a noncontradictory and cO l1lplete set T' p' on a poset L \Vith smallest element 0 and (such that T' contains, for anv s 巳 ntence o[ largest element 1 satisfying the following th己 language ， either the sentence or its ne 噜 conditions for all p , q 巳 L:(i) pVp' 1, gation). Abstract quantum \ogics generally (ii) P ^ p' 0 , (iii) p 三 q 斗 q' 三 p' ， and viü\ate the Lindenbaum prope 口y. (Ï\') p" = p. Logic: According to the tradition of logi­ Orthocomplemented Lattice (or Poset): 0 口 ho­ ca\ methods , a \ogic can be describcd as a A lattice (or poset) equipped \Vith an system (FLJ- ,Þ) consisting of a (omral la l1- complementation p • p'. guage, a proortlzeoretic-conseqllence re\ation Orthodox Quantum Logic: A logic that is 1- (based on a notion of proon, and a se­ semantically characterized by the c\ass of a11 lat 町 mantic-cOlzseqllence relation Þ (based on a algebraic models based on orthomodular notion of model and of tnah). tices. Standard quantum logic is a particular MacNeille Completion of a Brouwer­ model of orthodox quantum logic. Zadeh Lattice: Let L be a Brouwer-Zadeh Orthogonal: If L is an orthocomple­ lattice (or poset). For X ç L. let X' [p E mented poset and p , q E L , then p is orthog­ onal to q , in symbols p .L q , if P 三 q'. LI'r:fq E X.p 三 q'). X- = [p E LI 'r:f q ε X ， p 三 Orthomodular Lattice: An orthocomple­ q-), and P(L) [X ç LIX = X'). The struc- mented lattice L satis 鸟ring the orthomodular ture 7f' (L) (P(L),Ç: ,-,[ )",L) is called the law: For all p ,q E L , P 三 q ~ q p V (q MacNeille completion of L. 7f' (L) is a com­ = ^ p'). plete Brouwer-Zadeh lattice and L is em­ Orthomodular Poset: An orthoalgebra L beddable into 7f'(L) via the mapping p • (pJ. such that, for all p ,q E L , p .L q 导 p8q= where (p] = [q E Llq 三 p). p V q. Meet: If L is a partially ordered set (po­ Paraconsistent Quantum Logic: A logic set) and p , q E L , then the meet (or greatest that is semanticallv characterized bv the 10屿'er bound) of p and q , denoted by p q if ^ c\ass of all models based on involutive lat­ it exists, is the unique element of L satisfying tices with sma11est element 0 and largest el­ q 三 p ， the following conditions: (i) p ^ q and ement 1. r ε L r 三 p ， =丰 r (ii) with q :S P ^ q. Partially Ordered Set (or Poset): A set L Minimal Quantum Logic: A logic that is equipped with a relation 三 satisfying the fol­ semantically characterized by the class of all lowing conditions for all p ， q ， r ε L: (i) p 三 p ， models based on orthocomp\emented lat­ (ii) P 三 q and q 三 p ==> p = q, (iii) P 三 q and tices. q :S r 司 p 三 r. Modular Lattice: A lattice L satisfying Probab出ty Measure: A functionω :L → p 三 r 萄 p the modular law: V (q ^ r) = (p [0, 1] ç IR on an orthoalgebra L such that p ， q ， r εL V q) ^ r for all 叫 0) = 0 ， ω (1) = 1, and, for all p ,q E L with Nonnegative Operator: A self-adjoint op­ P .L q ， ω(p ⑥ q) = 叫 p) + ω (q). erator A on a Hilbert space 'iJe such that Projection: An operator P on a Hilbert (A Iþj ψ〉三 o for alI vectors φε 'iJe. space that is self-adjoint (P P"') and idem- Observable or Dynamical Variable: A potent (P = P2). numerical variable associated with a physical Pure State: A state ψis pure if the set [的 svstem the value of which can be determined consisting only of that state is superposition by conducting a test, a measurement, or an c\osed. In Hilbert-space quantum mechanics, experiment on the system. the pure states are precisely the vector Orthoalgebra: A mathematical system states. consisting of a set L with two special ele­ Quantum Logic: The study of the formal ments 0 , 1 and equipped with a relation .L structure of experimental propositions affili­ such that, for each pair p ， q 巳 L with p .L q, ated with a quantum physical system, or any an orthogonal Sum p 8j q E L is defined sub­ mathematical model (e.g. , an orthoalgebra) ject to the following conditions for all p ,q ,r E representing such a structure. Quantum Logic 253

Regular Involution: An involution ' on a that comrnute \\'ith a certain s 巳 t or pair飞、 ise poset L thM satist1 es the regubrity condition o 口 hogolul prokctions (i.e. , projections onto For all p ， q ε L: p 三 p' and q 三 q' => p 三 q'. superposition sectors) repr巳 S 巳 nt possible If L is a lattice, then an involution is regular states 0 1' the svstem吕. iff it satisfies the Kleene condition: For all Support: If W is a d巳 nsitv op 巳 rator on a p ， q ε L:p 八 p' 三 q V q'. Hilb 巳 rt space Jf, then the suppo 口 of W is d 巳­ Soundness: A logic (FL ,• J= ì is sound fined to be th 巳 set supp(Wl 0 1' all projection when all the proof-theor巳 ticιonsequ巳 nces operators P on :J{ such that tr(WP) "" O. Mor巳 are semantic consequences (i fα ← βthenα generally, ifωis a probability measure on an 仨 β) . Orihoalg巳 bra L , then S l1 pp(ω) = (p εLIω (p) Spectral Measure: A mapping from real "" 0). Borel sets into projection operators on a Hil­ Superposition Closed: A s 巳 t of states 吕 be 口 spac 巳 that maps the empty set into [), sup 巳 rpo吕 ition clo且已 d if it contains all of its maps IR into 1, and maps th 巳 union of a dis­ own superpositions joint sequence of real Borel sets into the Superposition in Hilbert Space: For a least upper bound Uoin) of the correspond­ Hilberτspace 灭， if (Wu) is a family of vector mg proJection operators. stat 臼 det巳口口 ined by the corresponding fam­ 绽， ifψ Spectral Theorem: The theorem estab­ ily (wu ) of normalized vectors in and lishing a one-to-one correspondence between is a normalized Iinear combination of the (not necessarily bounded) self-adjoint opera­ vectors in this family, then the vector state tors A on a Hilbert space 就 and spectral W dete口口 ined by ψis a (coherent) superposi­ measures E • PE on 'iJC such that, if PA tion of the family (W,). If (Wa ) is an arbi­ P{ _~AJ for all λε !R， then A = f:~ λdP ，、. tra巧 family of density operators on 就 and Spectrum: If A is a (not necessarily (ta) is a corresponding family of nonnegative bounded) self-adjoint operator on a Hilbert real numbers such that Lata 1, then W space and E • PE is the corresponding spec­ LataWa is an (incoherent) superposition (or λbelongs tral measure, then a real number mixtures) of the family (Wa ). to the spectrum of A if P{A-~A+ 的笋 o for all E Superposition in an Orthoalgebra: A > O. probability measure ωon an orthoalgebra L Standard Quantum Logic: The complete is a superposition of a family (ωα) of proba­ supp(ωç orthomodular lattice L of all projection op­ bility measures on L if U a erators on a Hilbert space. For P, Q ε L ， P supp(ωα) ， 三 Q is defined to mean that P = PQ and the Tensor Product of Hilbert Spaces: If灾 orthocomplement of P is de 自 ned by P' and J{ are Hilbert spaces, then the tensor - P. product 觉②'j[ is a Hilbert space together State: The state of a physical system en­ with a mapping (x ， y) → x@y εcx @ 'j{ for codes all information conceming the results x ε 页 ， Y E 'j{ that is separately Iinear in of conducting tests or measuring observables each argument and has the property that if on the system. It is usually assumed that, (向) is an orthonorτnal basis for 'iJC and (4)) is corresponding to each state ψof the system, an orthono口口 al basis for 页， th巳 n (向 @φ，) is there is a probability measure ωψon the an orihonormal basis for 灾②沉. logic L of the system such thatωψ(p) is the Trace: The trace of an operator A on a probability that the experimental proposition Hilbert space 'iJC is de自 ned by tr(A) L 但 B pε L is true when the system is in the state (Aiþl ψì ， where B is an orthonor刀1al basis for 仇 况， provided that the series converges. State Space: The set of all possible states Trace Class: An operator A on a Hilbert of the physical system. space 灾 belongs to the trace class if tr(A) Strong Partial Quantum Logic: A logic 立 ψε B (A ljJlljJì converges absoll1 tely, where B is that is semantically characterized by the an orihono口口 al basis for X. c1 ass of al1 models based on orthomodular Unsharp Quantum Logics: Examples of posets. paraconsistent logics where the noncontra­ Superselection Rule: A rule that deter­ diction principle is general1y violated. mines the possible states of a physical sys­ Vector State: A probability measure on tem. The usual quantum-mechanical super­ the standard quantum logic L of a Hilbert selection rules state that only vector states space 贸 determined by a normalized vector 254 QU:.ì ntum Logic t卢 E 'X、 and assigning to each projectíon op­ Foulis, D. (υ19 且 9). Fο tμt川F川山tμd. erator P E L the probability (P ,þl ,þ). Phys. 7, 905-922. Weak Partial Quantum Logic: A logic Foulis, D.. Randall. C. (1981), "Empirical Logic a~zd that is semanticallv characterized bv the and Tensor Products," in: lI11erpretatÍOIzs FowzdatÍolls o( Qua l1t wll ηzeory ， Grundlagen der c \ass of all models based on orthoalgebras. Exakt Naturwissenschaften, Vol. 5, Mannheim/ 飞，\i ien: Bibliographisches Institut

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