CC BY
19V.L.Vasyukov
EFFECTS IN QUANTUM LOGIC OF OBSERVABLES*
Abstract. In the paper a modal and bimodal extension of quantum logic of observables QLO is proposed. The former allows to obtain the syntactic counterpart of D.Mundici's result on embedding of C*-algebra into an MV-algebra while the latter has as its algebraic counterpart quantum MV-algebra of R.Giumtini. The soundness and completeness of both extensions is proved in respect to the set-theoretical semantics developed early for QLO.
1. Introduction
In [4] D.Mundici shown that every approximately dimensional C*-algebra with lattice dimensional group can be embedded into a countable MV algebra. Since such an MV algebra is also a Lindenbaum algebra of Lukasiewicz infinite-valued calculus Lm (the notion of MV algebra was introduced by C.C.Chang in order to provide an algebraic proof of the completeness theorem for L«,) then this result would be treated as a tool for considering properties of quantum systems in the framework of L«,. Needless to say that from the physical point of view in this case we ought to consider an elements of MV algebra as a class of operators whose spectrum is contained in the real interval [0,1].
But the lack of developed interpretation of such operators forces us to approach those as so-called effects of a Hilbert space which are bounded linear operators such that for an every effect E and for all density operators D, 0 < Tr(DE) < 1 (Born probability). It was shown by R.Giuntini [2] that the class of all effects of any Hilbert space turns out to be an instance of an algebraic structure called quantum MV algebras. Those retain some important properties of MV algebras, while violating the crucial axiom of MV algebras: the so-called Lukasiewicz axiom. Quantum MV algebras represent non-idempotent extension of orthomodular lattices just as MV algebras represent non-idem-potent extensions of Boolean algebras.
Thus, in case of transferring Mundici's method onto quantum MV algebra of effects we can interpret those as determining a kind of Born probabilities for quantum observables represented by operators in Hilbert space. In fact, those probabilities would be considered as probabilities for observables to have as the result of measurement a certain
* The work is supported by RFBR grant No 01-06-80006.
magnitude contained in the real line of numbers (as projectors in Hilbert space would be regarding as "yes-no" answering the same question).
2. Quantum Logic of Observables
We obtain the syntactic version of Mundici's result if we have recourse to the so-called quantum logic of observables QLO [5]. QLO is axiomatized by means of the following axiom schemes and the rules: Axl. A ^ A; Ax2. A ^ -—A; Ax3. Aa(BaC) (AaB)aC; Ax4. Av(BvC) (AvB)vC; Ax5. Aa(BvC) (AaB)v( AaC); Ax6. —(Av—A) ^ BaB; Ax7. Aa—A ^ —(Bv—B); Ax8. 1aA A Ax9. JA —(Bv—B); Axl0. JA A; Axll. J„(AaB) JaBAÄ; Axl2. Ja(AvB) JaBvJaA; Axl3. —JaA^Ja—A Axl4. Ja+ßA JaAvJpA; Axl5. JaßA JaJßA
Axl6. — [(AaB)v—(BaA)]2 (Av—A)2a(Bv—B)2 (A2 means AaA).
Rxl. A ^ B
—B ^ —A
Rx2. A ^ B
JaA ^ J^B
Rx3. A ^ B B ^ C
A ^ C
Rx4. A ^ B C^D
AvC ^ BvD
Rx5. AaA ^ B CaC ^ D (AaA) a (CaC) ^ BaD
Here A ^ B means (A,B)eL, where L is some logics, truth-value of JaA is calculated as the result of multiplying truth-value of A on a being a real number.
Let r be a non-empty set of wff. A wff A is said to be QLO-deriv-able from r, T^A, if there exist Bl5..,,B„e r such that
(a) either Biv... vB„^A;
(b) or (B1a...a£„)a(B„a...a£1) ^A;
(c) or J^B^A, i = 1,2,.,n.
If A is QLO-derivable from — (Av—A) then A is QLO-derivable or is a QLO-theorem which writes ^A. r is QLO-consistent if there is at least one wff not QLO-derivable from r, and QLO-inconsistent otherwise (it can be shown that r is QLO-consistent iff for no A do we have both T^A and T^—A). r is QLO-full iff it is QLO-consistent and closed under v, a, J and QLO-derivability, i.e. iff
(1) for some wff A, not T^A;
(2) if AeT and A^B, then BeT:
(3) A,BeT implies AaB, AvBeT;
(4) A e T implies J|aA e T.
If xc O (where O is a set of wff) is QLO-full then
(i) x^A iff A ex;
(ii) — (Av—A)ex, for all wff A.
QLO-full sets and QLO-derivability are linking with the following version of Lindenbaum's Lemma:
T^A iff A belongs to QLO-full extension of T.
It is proved that if x is QLO-full and —Aix, then there exists a QLO-full set y such that Aey, and for all B, either —Bix or Biy.
QLO have some peculiarities featuring quantum orthologic. Both in QLO and quantum orthologic the proof of Lindenbaum's Lemma does not require such power tools as, for example, Zorn's Lemma, which was in case of orthologic regarded as unprecedented for logical systems. As to the QLO-full sets, then from topological point of view they are, in fact, proper filters and not the ultrafilters. This, in turn, leads that for both quantum orthologic and QLO there is not need in some version of an axiom of choice which is required to prove an existence of ultrafilters.
It is easy to see that an algebra corresponding to QLO be an algebra of observables satisfying the axioms of algebraic approach in [1]. If we define an equivalency of formulas A and B, A ~ B as + A B then denoting the set A/~ as [A] we obtain [A]+[B] = [AvB],
[A]°[B] = [AaB],
-[A]= [—A], 0 = [—(Av—A)],
1 = [1], a [A] = [JaA].
A structure F= (F,+,°,-,a,0,1) (where F = {P/~: P is a formula}, aeR) is an algebra (of observables) while E = <F,+,-,a,0) be a vector (linear) space, 0 is a unit relative to +, and 1 is a unit relative to °.
3. Modal Quantum Logic of Effects
Let us modify our formulation of QLO by replacing Axl with Axl'. —(AvB) —Av—B
The following theorems of QLO will be used in the sequel: Bxl. —(Av—A) vB B
It is easy to see that this modification does not lead to any change of QLO. As to the Axl then A ^ A can be proved from Ax2 by means of Rx3. Bxl is proved by means of Ax9, AxlO, Ax14.
To introduce effects into QLO we enrich the language of QLO with a unary operator Q and axiomatics of QLO with the following axiom schemes and the rule: Axl7. QA^ QQA Axl8. Q—A lv—QA Axl9. Ql l Ax20. Q(AvB) (QAvQB) Ax2l. —(Bv—B) ^ QA ^ l Ax22. l lvQA
Rx6. A ^ B QA ^ QB
Let us denote the system QLO + {Axl7-Ax23, Rx6} as QLO-MV (with Axl'). In order to prove that QLO-MV really describes the effects let us firstly recall the algebraic structure responsible for those. According to P.Mangani [3] MV algebras can be defined in the following way:
(MVl) (a©b) ©c = a©(b ©c)
(MV2) a©0 = a
(MV3) a©b = b©a
(MV4) a©l = l
(MV5) (a*)* = a
(MV6) 0* = l
(MV7) a©a* = l
(MV8) (a*©b)*©b = (a©b*)*©a (Lukasiewicz axiom)
As in QLO we define [A] © [B] = [QAvQB] and [A]* = [—QA].
Theorem 3.1. A structure F = (F, ©,*,0,1) where F = {P~: P is a formula prefixed with Q}. 0 = [—(Av—A)], 1 = [l] is an MV algebra.
Proof. Associativity of © for (MVl) follows from the definition of © and associativity of v in QLO as well as commutativity for (MV3). (MV2) is fulfilled since [A]©0 is defined by QAvQ—(Av—A) Q(Av—(Av—A)) and then by Bxl it will be equivalent to QA which
under the definition of P gives us [A]. In case of (MV4) we have QAvQi ^ QAvi by Ax19. As to (MV5) then [A]** is determined by Q—Q—A and by Ax18, Ax17 it gives us Q—Q—A ^ 1v—Q—A ^ 1v—(Q—A) ^ 1 v—11 vQA. But we obtain 1v—1 ^ ——1v———1 ^ —(—1v——1) ^ —(1v—1) with the help of Ax2, Ax1'. So, by Bx1 we obtain (1v—1)v QA QA. (MV6) follows from Q——(Av—A) Q(Av—A) QAvQ—A QAv1v—QA^ 1.
In order to obtain (MV7) we have QAvQ—A by the definition and Ax17. Then like in case of (MV6) we get QAvQ—A 1.
In case of Lukasiewicz axiom for the left part we have Q—(Q—AvQB)vQB by the definitions and Ax17. Now by Ax18 and Ax17 we obtain Q—(Q—AvQB)vQB Q—Q—AvQ—QBvQB 1v—Q—AvQ—QBvQB 1v—(1v—QA)v1v—QBvQB. By Ax1', Ax2, Ax1' we have 1v—(1v—QA)v1v—QBvQB ^ QAv1 ^ 1. For the right part we likewise obtain Q—(Q—BvQA)vQA QBv1 1 and this determines that Lukasiewicz axiom will be satisfied. □
In the sequel under wff we always mean a wff prefixed with Q.
Definition 3.2. Let r be a non-empty set of wff. A wff A is said to be QLO-MV-derivable from r, T^A, if there exist B1,.,Bne r such that
(a) either B1v. vBn^A;
(b) or (B1A.ABn)A(BnA.AB1) ^A;
(c) or JaiBj^A, i = 1,2,...,n;
(d) or QB^A, i = 1,2,.,n.
If A is QLO-MV-derivable from 1 then A is QLO-MV-derivable or is a QLO-MV-theorem which writes ^A. T is QLO-MV-consistent if there is at least one wff not QLO-MV-derivable from T, and QLO-MV-inconsistent otherwise (it can be shown that T is QLO-MV-consistent iff for no A do we have both T^A and T^Q—4). T is QLO-MV-full iff it is QLO-MV-consistent and closed under v, a, J, Q and QLO-MV-derivability, i.e. iff
(1) for some wff A, not T^A;
(2) if AeT and A^B, then BeT;
(3) A,BeT implies AaB, AvBeT;
(4) AeT implies J^A eT;
(5) AeT implies QAeT.
Lemma 3.3. fxcO (where O is a set of wff) is QLO-MV-full, then
(i) x^A iff A ex;
(ii) Q—(Av—A)ex, for all wffs A prefixed with Q.
Proof. (i) Since in QLO-MV A^A, sufficiency follows from the definition of QLO-MV-deivability. Necessity follows from 3.2(2), (3), (4), (5).
(ii) By definition x is non-empty, thus there exists Bex. But by 3.2(4) J0Bex, so by Ax9and 3.2(2) we have — (Av—A)ex. The result follows under Ax8 and 3.2(2). □
QLO-MV-full sets and QLO-MV-derivability are connected with the following version of Lindenbaum's Lemma : Theorem 3.4. T=A iff A belongs to every QLO-MV-full extension of T.
Proof. If T=A then there are Bl,...,BneT such that either Blv. vBn=A or (Bla. ABn)A(BnA. aB1) =A, or Jafi=A or QB=A (l < i < n). If x is QLO-MV-full and Tc x, then Bl,!..,Bnex. Applying 3.2(3),(4),(5) and then 3.2(2), we obtain Aex.
The other way round, suppose A is not QLO-MV-derivable from T. We put x = {B: T=B}. By Axl we have Tc x, and by hypothesis Agx. The proof will be accomplished if we can show that x is QLO-MV-full. Suppose Bex and B=C, then there exist Bl,.,BneT such that either Blv.vBn=B, or (BlA.ABn)A(BnA. aB1) =B, or J|a|B;^B, or QB=A (l < i < n). So by Rx3 we obtain either Blv.vBn=C, or (BlA.ABn) A (BnA.ABl) =C, or J^fi^C, or QB=A, hence, T=C, i.e. C e x.
On the other side, if B,Cex, then there exist Bl,.,Bn,Cl,.,CneT, such that Blv.vBn=B and Clv.vCn^C. Then by Rx4 we obtain Blv. vBnvClv. vCn=BvC. So we have T=BvC and thus BvCex.
Furthermore, if Bex, then there exists B 'eT such that J|a |B '=B. But by Rx2 J|P|J|a'|B '=J|p|B and by Axl5 we obtain Jjpa '|B '=J|P|B, thus, T=J|P|Bex.
Again, let B,Cex. Then there exist Bl,...,Bn, Cl,.,CneT such that (B1a.ABn)A(BnA...aB1)=B and (C1a.ACn)A(CnA.ACl) =C. But then by Rx5 we obtain (B1a.ABnAClA.ACn)A(CnA.AClABnA ... aB1) = BaC. Hence, T=BaC and BACex.
If Be x, then there exists B e T such that QB = B. Since by Rx6 QQB '=QB then under Axl7 we have QB '=QB. Thus, T=QBex.
This shows that x is closed under QLO-MV-derivability, conjunction, disjunction, J- and Q-operators. Since Agx then A is not QLO-MV-derivable from x, therefore x is QLO-MV-consistent. □ Theorem 3.5. If x is QLO-MV-full and Q—Agx, then there exists QLO-MV-full sety such that Aey, and for all B, either Q—Bgx or Bgy.
Proof. Lety = {B: A=B}. By Axl Aey. Now let —Bex. Then Bgy, or else A=B, whence — B = —A by Rxl, and so, in turn, by Rx6 and by 3.2(2), Q—Aex contrary to hypothesis. By 3.2(ii) we have Q—(Av—A) ex. According to what we just proved, it follows that Av—A gy. Proceeding in a similar manner to 3.4 we can show that y is closed under v,a;J,Q and QLO-MV-derivability. Then since Av—A is
not QLO-MV-derivable from y, i.e. y is QLO-MV-consistent, y be QLO-MV-full as required. □
Thus, in turn, for QLO-MV there is also not need in some version of an axiom of choice which is required to prove an existence of ultrafilters.
4. Semantics of QLO-MV
Since our logic is an extension of QLO then we adduce main definitions of QLO-semantics modifying them as may be necessary for QLO-MV.
Definition 4.1. QLO-MV-model is a 4-tuple M = <X,±,^,v>, where
(a) X is a non-empty set;
(b) ± is an orthogonality relation on X;
(c) is a non-empty collection of ±-closed subsets of X closed under set-theoretic intersection and the operation * (7* is defined as {x: x±y});
(d) v is a function assigning to each propositional variable and formula of QLO-MV recursively in every point (every element) of X a real number, i.e. v: ^иФ) x X ^R where S is a set of propositional variables and Ф is a set of wffs.
Denoting the set {xeX: v(A,x) = a} as \\A\\a we define recursively the value of a wff in a QLO-MV-model as follows:
(1) \N\a = {xeX: v(pi,x) = a}e^;
(2) \\AvB\\fl = {xeX: xe\\A\\bn\\B\\c & a = b + c};
(3) \\AaB\1 = {xeX: xe\\A\\bn\\B\\c & a = bc};
(4) \\ A\\a = {xeX: x±\\A\\.fl & v(-A,x) = a };
(5) \\J]aA\\a = {xeX: xe\\A\\b & a = ab};
(6) \\l\\i = Xт.е. v(1,x) = 1 for all xeX;
(7) \\qA\\a = {xeX: xe\\A\\b & a = q(b)} where q: v ^ [0,1] such that
(i) q(q(v(A))) = q(v(A));
(ii) q(v(—A)) = 1 - q(v(A));
(iii) q(v(1)) = 1 and q(v(-(Av-A)) = 0;
(iv) q(v(Q(AvB))) = max{q(v(QA)) + q(v(QB)),1}.
If Г is a non-empty set of wffs then we say that Г implies А at x in M, M: Г |=x A iff VBeT(v(B,x) < v(A,x)), Г M-implies А, M: Г |= A iff either ЗBeГ(xg||B||(_)), i.e. when B is not verified at x (verification but not truthfulness since we deal with many-valued logical matrix), or Г implies A at all x in M. If we define 3 = <X,±£> be QLO-MV-frame then Г 3-implies A iff 3: Г |= A for all QLO-MV-models M on 3. If p is a class of QLO-MV-frames then Г p-implies A, p: 3:Г |= A iff 3:
T |= A for all 3ep. A class p is said to determine QLO iff for all
A,Be O, A=B iff p: A |= B. p strongly determines QLO iff for all TA, iff p: T|= A.
If we define a range of values of a formula A as ||A|| = U aeR IK1\\a then extending this definition on 4.l(l)-(7) hereafter we denote as Ip^l, ||AvB||, ||AaB||, ||—A||, |J|aAII, ||l||, 11QA11 the ranges of respective formulas while ||A||(_) means an arbitrary value of respective formula. Lemma 4.2. For any QLO-MV-model M and any AeO, ||A||(_)e^.
Proof. By induction on the length of A, exploiting 4.l. □ Theorem 4.3. (Soundness of QLO-MV). if ©: T |= A, where © is a class of all QLO-MV-frames.
Proof. The proof, by induction on QLO-MV-derivability, proceeds by showing that the result holds for all Axl-Ax22 and is preserved by application of Rxl-Rx6. We consider only the less obvious cases.
Ax2. Let xe||A||a. Then if ye||—A||_fl, by 4.l(4) ylx and hence (symmetry) xly. 4.l(4) again gives xe||——A||a.
Now let xe||——A||a. Then ye ||—A||-a only if xly, i.e. yl||A||a only if xly. But ||A||a is l-closed by 4.2 and thus xe||A||a.
Ax6. It is easy to make sure that v(—(Av—A),x) = 0 at any point xeX and likewise v(Av—A,x) = 0. But if xe ||—i (Av—A)||0, then ye||Av—A||0 just in case of xly. By 4.l(2)ye||A||6n||—A||c and b + c = 0, i.e. c = -b. But then by 4.l(4) yly contrary to the irreflexivity of l. Hence, there is no y in any M for which we have yl||Av—A||0, whence it follows by the definition that xe||—(Av—A)||0 for any x. Besides, for all
B, by 4.l(3), v(BaB,x) > 0.
Ax7. Let xe||lAA||a. Then ye||l||ln||A||a by 4.l(3) and v(1aA,x) = v(l,x) ° v(A,x). But v(l,x) = l at any point xeX in virtue of the definition of (see 4.l(6)). So v(1aA,x) = v(A,x) and thus M: 1aA |=A and M: A |= 1aA for any A.
Axll. Let xe||Ja(AAB)||a. Then by 4.1(5) xe||AAB||b and a= ab. By 4.1(5) xe||A||cn||B||d and b = cd. Hence, a = adc. But by 4.1(5) xe||JaA||aCn||B||d and by 5.3.3(3) xe||JaAAB||acd=fl.
Axl3. Let xe||A||a. Then by 4.1(5) xe||JaA||aa and by 4.1(4) ye ||— JaA||-aa just in case of xly. But then ye||—A||-a because of xly, and by 4.1(5) ye||JaA|U
Axl7. Let xe||QQA||a. Then by 4.1(7) xe||QA||6 and a = q(b). Again, by 4.1(7) this implies xe||A||c and b = q(c). We have q(b) = q(q(c)) = q(c) by the property of q and thus a = q(c).
Axl8. Let xe ||A||b. Then by 4.1(4) ye||—A||_6 only if xly, i.e. yl||A|b only if xly.Furthemore, by 4.1(7) ye||Q—A||a and a = 1 - q(b) according to the properties of q. But it is easy to check that the result will be the same for the right side of Axl8, i.e. ye||lv—QA||a and a = l-q(b).
Rx1. Suppose M: A = B and let xe||— B||_a. Then ye||A||fe only if ye||B||a (by inductive hypothesis), only if x±y. This shows that xe||—A||-b.
The rest is obvious. □
Definition 4.4. Let L be a modal quantum logic of effects. The canonical QLO-MV-model of L is the structure ML = (XL,±L,^L,vL), where;
(1) XL = {x c O: x is a QLO-MV-full set};
(2) x±iy iff there is a wff A such that Q—4ex, Aey;
(3) ^ = {|A|l: AeO }, where |A|L = {xeXL Aex};
(4) Vl: (SuO) x Xl ^ R.
Denoting {xeXL: v(A,x) = a} as ||A||La we come to the definition of the value of formula and ranges of valuation in canonical model ML analogously to 4.1(1)-(7).
Lemma 4.5. 3L = <XL,±L,^L,vL) is a QLO-MV-frame.
Proof. Let xeXL. Then for any A neither Q—A,Aex nor x is QLO-MV-inconsistent (by Ax7). Hence, x±Lx does not take place. If x±Ly, then for some wff A we have Q—Aex, Aey. By means of Ax2 we come to the conclusion that Q—Bey, Bex, where B = Q—4. Thus x±Ly and ±L is an orthogonality relation. To check whether ||A||L be ±L-closed suppose that xi||A||L, i.e. Aix. By Ax2 ——Aix and so by 3.5 there is yeXL such that x±Ly fails and —Aey. Meanwhile if ze||A||L then Aez and, hence, y±Lz. Thus, y±L||A||L as it was required. Clearly, will be closed under intersection (by virtue of properties QLO-MV-derivability and QLO-MV-fullness). □
Theorem 4.6. (Fundamental theorem for QLO-MV). For all A and all
xeX, xe||A||L iff Aex.
Proof. By induction on the length of A. In case of A = BvC, A=BaC, A = JaB and A = QB it is easy to see that B,Cex follows from BvC, BaC, JaB, QAex. It will suffice to use 4.1(2),(3),(5),(7). Conversion follows from 3.2(3),(4),(5).
Suppose that A = — B and for B the theorem is true. Let Q—Bex. If ye||B|| (_) then by inductive hypothesis Bey and hence x±Ly. By 4.1(4) it follows that xe||B||L(_). Again, if Q—Bix, then according to 3.5 there is yeXL such that Bey and thus by inductive hypothesis ye||B||L(_) but x±Ly fails. By 4.1(4) we come to the conclusion that xi||B||L(_). □
Corollary 4.7. T^A iff MT 1= A.
Proof. If T^A then there are Bi,...,BneГ such that either B1v.vBn^A or (BiA...ABn)A(BnA...ABi)^A, or J\a.\B{^A, or QBi ^A
(1 < i < n). If xe||B||L(_) for all BeT, then by 4.6 Bl,...,B„ex. By 4.1(2)-(5) it follows that Aex and thus xe ||A||L(_).
The other way round, if A is not QLO-MV-derivable from T, then by 3.4 there exists xeXL such that Tc x and Agx. Then by 4.6 xe||B||L(_) for all BeT, but xg||A||L(_). □
Theorem 4.8. T=A iff |= A.
Proof. Let M be an arbitrary QLO-MV-model on 3L. For every i<ra, INfe^L there is Bi such that |N|M(.) = |Bi|L(-) (|Bi|L(_) is defined as in 4.4) and |Bi|(.) = ||B,||L(_). For any wff C let C' is the result of uniformly replacing each pi, occurring in C, with Bi. Clearly, there are in T such A\,...,An that either Aiv...vA„=A or (AiA...aA„)a(A„a.aA1)=A, or JIaA=A, or QB=A (1 < i < n) and so we have A\v...vA '„ =A ' etc. Tl^n by 4.7 either ML: A W...vA „ |= A, or M: (A \a...aA „) a (A „a. .. aA 1) |= A, or Ml: J^A M= A, or ML: QA i |= A. But a simple induction shows that ||C||M = ||C'|| L and so either Ml: Aiv... vA„ |= A, or Ml: (AiA...aA„)a(A„a...aAi) |= A, or Ml: J^A |= A, or Ml: QAi |= A whence it follows that M: T |= A. Since this holds for all models M on 3L, we conclude 3L: T |= A. □
Corollary 4.9. (Strong completeness for QLO-MV). ©: T |= A only if T = A.
Proof. Since by 4.5 3L is QLO-MV-frame, then © contains 3L as its element. The rest is obvious. □
Thus corollary 4.9 shows that QLO-MV is strongly determined by the class of all QLO-MV-frames.
5. Bimodal Quantum Logic of Effects
Regarding effects of a Hilbert space as bounded linear operators E such that for all density operators D, 0 < Tr(DE) < 1, we can define over the class E(H) of all effects a partial ordering relation < in the following way [2, p.397]. For any E,HeE(H):
E < H iff for all density operators D: Tr(DE) < Tr(DF).
The class of all effects coincides with the class of all bounded linear operators between 0 and 1. Clearly, E(H) contains the class of all (with A,e[0,l]) where for any state vector ^eH (A,1)^ := A,^. Now we define for any E,HeE(H):
f E + F if E + FeE(H)
E © F := \
I 1, otherwise
where + the usual operator-sum,
E* := 1 - E.
It is easy to see that
E 0 F = E + F iff E + F < 1.
Likewise one can easily check that the structure E(H) = (E(H),0,*,1,O) violates Lukasiewicz axiom of MV-algebra. Actually, let us consider two non-trivial effects E,F such that it's not the case that E < F and it's not the case that F < E. Then, by definition of 0 we have E 0 F* = 1 and F0 E* = 1. Hence, (E*0 F)*0 F = 0 0 F = F*E = 0 0 E = (E 0 F*)* 0 E. Thus, Lukasiewicz axiom is violated in the structure E(H).
As it was mentioned above R.Giuntini [2] showed that the class of all effects (determines by Born probability) of any Hilbert space turns out to be an instance of an algebraic structure called quantum MV algebra (QMV algebra). The latter is a structure M = <M, 0,*,1,O) where M is non-empty set, 0 and 1 are constant elements of M, 0 is a binary operation and * is a unary operation satisfying the following axioms (where a®b := (a*0 b*)*, anb := (a0b*)®b and al_lb := (a®b*)0b):
(QMV1) (a0b) 0c = a0(b0c)
(QMV2) a0O = a
(QMV3) a0b = b0a
(QMV4) a01 = 1
(QMV5) (a*)* = a
(QMV6) 0* = 1
(QMV7) a0a* = 1
(QMV8) a_(bna) = a
(QMV9) (anb)nc = (anb)n(bnc)
(QMV10) a0 (bn(a0c)*) = (a0b)n(a0 (a 0c)*)
(QMV11) a0 (a* nb) = a0b
(QMV12) a0(a*0b) _(b*0a) = 1
It seems possible to yield logic of effects in QLO framework corresponding quantum MV algebra. To this end we will enrich the language of QLO with the help of a binary modal operator 0 and unary modal operator * and enlarge the list of QLO axiom with the following inference rules:
Rx7. —(Cv—C) ^ A ^ 1 —(Cv—C) ^ B ^ 1 AvB ^ 1
A0B AvB
Rx8. 1 ^ A0B 1 A0B Rx9 — (Av—A) ^ A ^ 1
1v —A ^ A*
(the double line means an inference in both directions).
Let us denote a system QLO + {Rx7-Rx9} as QLO-QMV (with Ax1'). As in QLO we define [A] © [5] = [A©B] and [A]* = [A*].
Theorem 5.1. A structure F = (F, ©,*,0,1) where F = {P/_: P is a
formula and —(Av—A) ^ P ^ 1}, 0 = [—(Av—A)], 1 = [1], представляет собой QMV алгебру.
Proof. It is easy to see that satisfiability of (QMV1) and (QMV3) is a consequence of associativity and commutativity of v. (QMV2) will take place in virtue of Вх2. We have (QMV4) because from —(Av—A)^ B (by Rx7) one get —(Av—A)©1 ^ B©1 (by Rx4), and since by Вх1 — (Av—A)©1«1 then by Rx8 1 « B©1. In case of (QMV5) by Rx11 we have 1 v —A « A*, then again implementing Rx9 we obtain 1v—(1v —A) « A**. But by Вх2 this reduces to 1v—1v ——A) « A**, which in view of 1v—1 « ——1v———1 « —(—1v——1) « —(1v—1) (by Ax2, Ax1') and Ax2, Ax1' reduces, in turn, to A « A**. Analogous manipulations allow to ascertain the satisfiability of (QMV6) and (QMV7).
In order to check the satisfiability of the remainder axioms we define A®B « (A*© B*)* « AvBv—1,
A®B« (A*© B*)*,
AnB « (A©B*)®B and AuB « (A®B*)©B.
Moreover, we obtain that
fA, if A ^B fA, if B ^A
AnB « i AuB « i
IB, otherwise IB, otherwise
Actually, by the definition AnB « (A©B*)®B « (A©B*)vBv—1. If A^B then A©B*^B©B*«1 and by virtue of Rx7 and Rx9 A©B*«Av—Bv1, and thus AnB «-A. Otherwise by Rx11 A©B*«1 and AnB «B.
Further, by the definition we have AuB « (A®B*)©B. If B^A, then B®B*^A®B*, which leads to —(Bv—B) ^A®B*. This gives us an opportunity to exploit Rx7 for calculating (A®B*)©B, which gives (A®B*)©B « (A®B*)vB « AvB*vB v—1 « Av1v—BvBv—1« A. Otherwise we get A®B* ^ —(Bv—B). But by Rx7 we obtain that from A©B^—(Bv—B) it follows A©B«—(Bv—B), and thus A®B*«—(Bv—B) and (A®B*) © B « B.
In case of (QMV8) if A^B then Au(BnA)« AuB«A. If it is not the case that A^B, then Au(BnA)« AuA«A.
For (QMV9) we need that (AnB)nC « (AnB)n(BnC). Two cases are possible:
1) B ^C,
2) it is not the case that B ^C. Case 1). If A^B then by virtue Rx3 A^C. Then (AnB)nC « AnC«A«AnB« (AnB)n(BnC). If it is not the case that A^B, then (AnB)n(BnC) « BnB « B «BnC « (AnB)nC. Case 2). Since it is not the case that B ^C, then we get BnC «C. Hence, (AnB)n(BnC) «(AnB)nC.
In order that (QMV10) is satisfied we need A0(Bn(A0C)*) « (A0B)n(A0 (A0C)*). Two cases are possible:
1) A0C«1,
2) it is not the case that A0C« 1.
Case 1). A0(Bn(A0C)*) « A0(Bn—(Av—A)) «A and (A0B)n(A0 (A0C)*) «(A0B)n(A0—(Av—A)) « ((A0B) 0 A*)®A« ((B0(A 0 B*))®A« (B 0 1))®A« A. Case 2) has two subcases:
a) B^(A0C)*,
b) it is not the case that B ^ (A0C)*.
Subcase a). By hypothesis, A0(Bn(A0C)*)«A0B and (A0B)n(A0 (A0C)*) « (A0B)n(A0 (AvC)*). If (A0(AvC)*) «1 then we succeed. Therefore we can suppose that (A0(AvC)*) «1 is not the case. Then (A0(AvC)*) « (Av(AvC)*) « C*. Thus (A0B)n(A0 (AvC)*) « (A0B)nC*. By hypothesis, B^(A0C)* « 1v—Av—C, hence, AvB^C*. Finally, (A0B)nC*«A0B.
Subcase b). By hypothesis, we have that A0(Bn(A0C)*) « A0(Bn(AvC)*) «A0(AvC)* « Av(AvC)* « C*. Now, (A0B) n(A0 (A0C)*) « (A0B)nC*. By hypothesis, it is not the case that B ^ (A0C)*. Then it is not the case that C ^ (A0B)*, hence it is not the case that (A0B) ^C*. Thus, (A0B)nC* « C*.
Cases of (QMV11) and (QMV12) are easily verified. □ In the sequel under wff we always mean wff P, for which —(Av—A) ^ P ^ 1 is true.
Definition 5.2. Let T be a non-empty set of wffs. A wff A is said to be QLO-QMV-derivable from T, T^A, if A is QLO-derivable from T and there exist B1,.,Bne T, such that
(a) B10...0B„ ^A. The notions of QLO-QMV-derivability, QLO-QMV-consistency etc. are defined in the same way as in case of QLO-MV (it can be shown that T is QLO-QMV-consistent iff for no A do we have both T^A and T^A*). T is QLO-QMV-full iff it is QLO-full and A,BeT implies A0BeT.
Lemma 5.3. If xcO (where O is a set of wff) is QLO-QMV-full, then
(1) x^A iff Aex;
(2) 1ex for all wff A.
Proof. (ii) By definition x is non-empty, thus there exists Be x. But by 5.2 J0Bex, so by Ax9 and 3.2(2) we have — (Av—A)ex. But since for wff P always will be true that — (Av—A) ^ P ^ 1, then by 5.2 we obtain the desired result. □
QLO-QMV-full sets and QLO-QMV-derivability are connected with the following version of Lindenbaum's Lemma : Theorem 5.4. T^A iff A belongs to every QLO-QMV-full extension of T.
Proof. We need verify only the cases of B10.0B„^A and B0Cex. □
Theorem 5.5. If x is QLO-QMV-full and A*ix, then there exists QLO-QMV-full sety such that Aey and for all B, either B*ix or Biy.
Proof. Let y = {B: A^B}. By Ax1 Aey. Now let B*ex, that implies —Bex by Rx9. Then Biy, or else A^B and — B ^ —A by Rx1, —A ex and then A*ex contrary to hypothesis. Further, by 5.3 we have 1ex. According to what we just proved, by Rx9 we obtain that 1 v—11 iy. Proceeding in a similar manner to 5.4 we can show that y is closed under QLO- and QLO-QMV-derivability. Then since 1v—1 is not QLO-QMV-derivable from y, i.e. y is QLO-QMV-consistent, y be QLO-QMV-full as required. □
6. Semantics of QLO-QMV
Since our system is an extension of QLO then for its description we will exploit the definitions of QLO-semantics modifying them as required to convey specificity of QLO-QMV.
Definition 6.1. QLO-QMV-models are QLO-models enriched with the following two points in recursive definition of the value of wff:
(1) ||A0B||a = {xeX: xe||A||bO||B||c & a = min(1,b + c)};
(2) ||A*||a = {xeX: xl^U-a & v(A*,x) = 1-a }.
Lemma 6.2. For any QLO-QMV-model M and any AeO, HA^e^.
Proof. By induction on the length of A, exploiting 6.1. □ Theorem 6.3. (Soundness of QLO-MV). T^A if ©: T |= A, where © is a class of all QLO-QMV-frames.
Proof. The proof, by induction on QLO-QMV-derivability, proceeds by showing that the result holds for all Ax1-Ax16 and is preserved by application of Rx1-Rx5, Rx7-Rx9. We consider only the cases of Rx7-Rx9 (accounting of the proof for QLO).
Rx7. By hypothesis, M — (Cv—C) |=A, M A |=1, M — (Cv—C) |=B, MB |=1, M:AvB |=1. According to the definition M — (Cv—C) |=XA iff v(—(Cv—C),x) < v(A,x), i.e. 0 < v(A,x), and MA =1 iff v(A,x) < v(1,x),
i.e. v(A,x)<l. The same is true for B. Besides, we have v(AvB,x) < v(l,x). Then by 4.1(2), 6.2(1) we obtain (AvB,x) = v(A,x) + v(B,x) = v(A©B,x). In the reverse direction the proof is obvious.
Rx8. By hypothesis, M:1 |=A©B. Then we have M:1 |=xA©B iff l<v(A©B,x). But since v(A©B,x) = min(l,v(A,x) + v(B,x)), then, clearly, 1 = v(A©B,x).
Rx9. Let M — (Cv—C) |=A, M A |= 1. Then M — (Cv—C) |=XA iff 0< v(A,x), and M A |=xl iff a = v(A,x) < 1. Further, ye||—A||_a only if xly. By 4.1(2) we getye||lv—A||c iffye||l||ln||—A||.fl and c = 1 - a. But by 6.2(2) we have ye||A*||c. □
Definition 6.4. The canonical QLO-QMV-model of L (bimodal quantum logic of effects) is the structure Ml = (Xl,1l,^l,vl) where:
(1) Xl = {x c O: x is a QLO-QMV-full set};
(2) xl,y iff there is a wff A such that A*ex, Aey;
(3) ^ = {|A|L: AeO }, where |A|L = {xeX, Aex};
(4) Vl: (SuO) x Xl ^ R.
Lemma 6.5. 3L = (XL,lL,^L,vL) is a QLO-QMV-frame.
Proof. Argumentation is similar to the case of QLO-MV-frame. □ Theorem 6.6. (Fundamental theorem for QLO-QMV). For all A and all xeX, xe||A||L iff Aex.
Proof. By induction on the length of A. In case of A = B©C it is easy to see that B,Cex follows from B©Cex. Suffice to use 6.1(1). □ Corollary 6.7. T^A iff MlT |= A.
Proof. If T^A then we need to consider an additional case of Bj©.©B„^A. If xe||B||L(-) for all BeT, then by 6.6 Bj,.,B„ex. By 6.1(1) it follows that Aex and thus xe||A||L(_).
The other way round, if A is not QLO-QMV-derivable from T, then by 5.4 there exists xeXL such that Tc x and Agx. Then by 6.6 xe||B||L(_) for all BeT, but xg||A||L(_). □
Theorem 6.8. T^A iff 3l:T |= A.
Proof. We need to consider an additional case of Aj©...©A„^A for T. □
Corollary 6.9. (Strong completeness for QLO-QMV). ©: T |= A only if T^A.
Proof. Since by 6.5 3L is QLO-QMV-frame, then © contains 3L as its element. The rest is obvious. □
Thus corollary 6.9 shows that QLO-MV is strongly determined by the class of all QLO-MV-frames.
REFERENCES
1. Emch G.G. Algebraic Methods in Statistical Mechanics and Quantum Field Theory. N.-Y., 1972.
2. Giuntini R. Quantum MV Algebras // Studia Logica. Vol. 56. 1996. P.393-417.
3. Mangani P. Su certe algebra connesse con logiche a più valore // Bollettino dell'Unione Matematica Italiana. Vol. 8. 1973. P.68-78.
4. Mundici D. Interpretation of AF C*-Algebras in Lukasiewicz Sentential Calculus // Journal of Functional Analysis. Vol. 65. 1986. P.15-63.
5. Vasyukov V.L. Quantum Logic of Observables // V.A.Smirnov (ed.). Syntactic and Semantic Studies of Non-Extensional Logics. Moscow, 1989. P.120-169 (in Russian).
