CC BY
1Логические исследования 2023. Т. 29. № 2. С. 148-162 УДК 510.64
Logical Investigations 2023, Vol. 29, No. 2, pp. 148-162 DOI: 10.21146/2074-1472-2023-29-2-148-162
Vladimir L. Vasyukoy
An axiomatization of quantum computational logic
Vladimir L. Vasyukov
Institute of Philosophy, Russian Academy of Sciences, 12/1 Goncharnaya Str., Moscow, 109240, Russian Federation. E-mail: [email protected]
Abstract: One of the logical proposals that arise from quantum computation is the idea to use the quantum theoretical formalism to represent parallel reasoning. To this aim a quantum computation is considered by means of convenient unitary operators assuming arguments and values in particular sets of qubit systems. Isolating some important unitary operators that have a special role in quantum computation (logical gates or quregisters) we obtain an opportunity to yield the language of Quantum Computational Logic (QCL) (cf. [Cattaneo et al., 2003; Cattaneo et al., 2004; Dalla Chiara et al., 2004]). The basic concept of the semantics of this language is the notion of quantum computational realization such that the meaning associated to any sentence is a quregister. Unlike the semantic of a standard quantum logic QCL-conjunction and QCL-disjunction do not correspond to lattice operations since they are not generally idempotent. Moreover, in QCL the weak distributivity principle breaks down and both the excluded middle and the non contradiction principles are violated. Finally, the axiomatizability of QCL is still an open problem. In the paper an axiomatization of QCL is proposed construing it as a kind of so-called Goldblatt's binary logic. Some metalogical theorems (paraconsistency and completeness) are proved.
Keywords: quantum computation, quantum computational realization, binary logic, para-consistency
For citation: Vasyukov V.L. "An axiomatization of quantum computational logic", Logicheskie Issledovaniya / Logicallnvestigations, 2023, Vol. 29, No. 2, pp. 148-162. DOI: 10.21146/20741472-2023-29-2-148-162
1. Introduction
The pure states of quantum system usually are mathematically represented by unitary vectors in a Hilbert space H, they represent maximal information about the physical system under investigation. Let us suppose the simplest situation, where our Hilbert space H, has dimension 2. In such a case it will have a basis consisting of two unitary elements and thus any vector of the space will be representable as a superposition (or linear combination) of the basis-elements. In Dirac's notation we will deal with the vectors ...;
© Vasyukov V.L., 2023
while the basis-elements will be indicated by |0), |1). For any unitary we have \$) = ao|0) + ai|1) where the coefficients a0, a1 are complex numbers such that their modules |a01, |a1| satisfy the relation |a0|2 + |a1|2 = 1. A quantum pure state assigns to a given physical property only a probability-value, and not a classical truth-value (either True or False). Any state fy) has some certain properties (with probability-value 1), some impossible properties (with probability-value 0) and many indeterminate properties (with probability-value different both from 1 and from 0). If fy) has the form ao|0) + a1|1), then the physical system in state fy) might satisfy with probability |a0|2 those properties that are certain for state |0), and might satisfy with probability |a112, those properties that are certain for state |1). Because fy) is a unitary vector then |ao|2, M2 e [0,1].
If we consider the two-dimensional Hilbert space C2, then any vector fy) should be represented as a pair of complex numbers. Let B = {|0), |1)} be an orthonormal basis for C2. Thus the elements of B are two particular unitary vectors that are mutually orthogonal (i.e. their inner product is 0).
Definition 1 (Qubit). A qubit is any unitary vector fy) of the space C2.
Hence, any qubit will generally have the form a0|0) + a111) where a0, a1 e C and |a0|2 + |a1|2 = 1.
Definition 2 (n-qubit system or n-register). An n-qubit system or n-register is any unitary vector fy) in the product space ®nC2.
Thus, a quantum-logical gate can be described as a special unitary operator, assuming arguments and values in a product Hilbert space C2.
Definition 3 (The Toffoli gate T(1-1-1)). The Toffoli gate T(1-1-1) is the linear operator T(1-1-1) : ®3C2 ^ ®3C2 defined for any element X ® y tg> D of the basis as follows: T(1-1-1)(|x) ® fy) ® |z)) = X ® fy) ® |min(x,y) ©z), where © represents the sum modulo 2.
T(1-1-1) transforms any product vector fy) ® fy) tg> D into the product that is obtained by leaving unchanged the first two factors fy) and fy) and by transforming the third factor fy into min(x,y) © z). By means we
can introduce a convenient notion of conjunction. Such conjunction, which will be indicated by AND, is characterized as a function whose arguments are pairs of vectors in C2 and whose values are vectors of the product space ®3C2.
Definition 4 (AND). For any fy) e C2 and any fy) e C2:
AND(|0), fy)) := T(1-1-1)(|0), fy), |0)).
To consider the case where the function AND is applied to arguments that are superpositions of the basis elements in the space C2 let us use the following qubit pair: \0) = ao|0) + a\\1), \p) = b0\0) + 6i|1). By applying the definitions of AND and of T(i'i'i) we obtain:
AND(\0), \p)) = aibi\1,1,1) + aibo\1,0,0) + aobi\0,1,0) + aobo\0,0,0).
By applying the Born rule we will obtain the following interpretation: |aibi\2 represents the probability-value that both the qubit-arguments are equal to |1), and consequently their conjunction is |1). Similarly in the other three cases.
The function NOT can be defined as a unary function assuming arguments in the space C2 and values in the space ®3C2.
Definition 5 (NOT). For any \^)eC2: NOT(\0)) := T(i'i'i)(\0),\1),\1)).
In particular, for the basis-elements |0) and |1) we have: NOT(\1)) = T(i'i'i)(\1) ® \1) ® \1)) = \1) ® \1) ® \0), NOT( \0)) = T(i'i'i)(\0) ® \1) ® \1)) = \0) ® \1) ® \1). In both cases, the first factor corresponds to the classical argument, while the third factor corresponds to the classical value for that argument (|1) is transformed into |0) and the other way round).
The logical gate AND have values in a Hilbert space of the form ®3C2. Nevertheless, the procedure can be easily generalized by defining the Toffoli gate in any Hilbert space having the form (®(n)C2) ® (®(m)C2) ® C2(= ®(n+m+i)C2).
Definition 6 (The Toffoli gate T(n'm'i)). The Toffoli gate T(n'm'i) is the linear operator
T{n,m'i) : (0(«)c2) ® (®(m)C2) ® C2 ^ (®(n)C2) ® (®(m)C2) ® C2, that is defined for any element |x^ ...,xn)®\ yi, ...,ym)®\ z) of the computational basis of ®(n+m+i)C2 as follows:
T (n'm'i)( \ xi, ...,Xn) <8> \ yi, ...,ym) \ z)) = \ xi, ..., Xn) ® \ yi, ...,ym) ® \ Xnym © z), where © represents the sum modulo 2.
On this basis we obtain also generalization of our definition of AND.
Definition 7 (AND(n)). For any \0) e ®nC2 and any \0) e ®mC2 :
AND(n)(\0), \p)) := T(n'm'i)(\0) ® \p) ® \0)).
And the same way we proceed with NOT.
Definition 8 (NOT(n)). The negation-gate is the linear operator NOT(n) that is defined for any element \xi,...,xn) of the computational basis of ®nC2 as follows:
NOT(n) ( \ xi ,...,xn)) = \ xi ,...,xn-i, 1 - xn).
To introduce a disjunction we need firstly define a new logical gate that is called a dual Toffoli gate and will be indicated by Q(1-1-1). Similarly to T(1-1-1), the operator Q(1-1-1) also assumes arguments in the space C2 and values in the space ®3C2. The dual Toffoli gate is obtained by making reversible the classical "or".
Definition 9 (The dual Toffoli gate Q(1-1-1)). The dual Toffoli gate Q(1-1-1) is the linear operator Q(1-1-1) : ®3C2 ^ ®3C2 that is defined for any element X ® fy) ® D of the basis in the following way:
Q(1-1-1)(|x) ® fy) ® D) = X ® y ® max(x,y) ©z), where © represents the sum modulo 2.
Now we can introduce a disjunction function OR assuming as pairs of vectors of C2 and as values vectors of ®3C2.
Definition 10 (OR). For any fy) e C2 and any fy) e C2 OR(|fy), fy)) := Q(1-1-1)№®fy)®|0)).
One of the most exotic quantum gates we need for the further considerations is the squareroot of the negation NOT, which is indicated by VNOT. The characteristic property of the gate VNOT is the following: for any quregister fy)
VNOT(V NOTfy)) = NOTfy). In other words: applying the squareroot of the negation twice "means" negating. The general definition of VNOT looks like that.
Definition 11 (a/Not). The square root of the negation on ®(n)C is the linear
operator such that, for every element |x1, ...,xn) of the computational
basis,
VNOTn(|x1,...,Xn)) = X1,..,Xn-1)®( + Xn) + —11 - Xn)) (where, of course, i2 = -1).
2. QCL semantics
In [Cattaneo et al., 2003; Cattaneo et al., 2004; Dalla Chiara et al., 2004] a particular semantics of Quantum Computability Logic (QCL) is described in a following way. A sentential language L of QCL contains the following connectives: the negation (—), the conjunction (a) and the square root of the negation (y—). The notion of sentence (or formula) of L is defined standardly. Let FormL represent the set of all sentences of L. As usual, the metavariables p, q, r, ...will range over atomic sentences, while a, 3, y, ...will range over sentences. The disjunction (y) is defined via de Morgan's law: y := -(- a. - ).
The basic concept of the semantics is the notion of quantum computational realization which is given by an interpretation of the language L, such that the meaning associated to any sentence is a quregister (qubit-register) — either a qubit or an n-qubit system (any unit vector \p) in the product space ®nC2). This determines that the space of the meanings corresponds not to a unique Hilbert space, but to varying Hilbert spaces, each one of the form ®nC2. The formal definition is the following.
Definition 12. A quantum computational realization of L is a function Qub associating to any sentence a quregister in a Hilbert space ®nC2 (where n depends on the linguistic form of a):
Qub : FormL ^ [J ®n C2.
n
Hereafter we will write \a) instead of Qub(a); and we will call \a) the information value of a. The following conditions are required.
(i) \p) is a qubit (in particular, \ 1) = \ 1), \0) = \0));
(ii) Let \0) e ®nC2. Then, \-0) = NOT(\p)) e ®nC2;
(iii) Let \0) e ®nc2, \y) e ®mC2.
Then, \ 0 x y) = AND(\0), \y)) e (®nC2) ® (®mC2) ® C2;
(iv) Let \0) e ®nC2. Then, \/-0) = /NOT( \0)) e ®nc2.
Now let Qub be a quantum computational realization and let a be any sentence with associated meaning \a). Since the computational basis of ®nC2 can be labelled by binary strings \011...10) which represents, on the other hand,
n—times
a natural number j e [0; 2n — 1] in binary notation, then any unit vector of
2n — i
®nC2 can be shortly expressed as \a) = ^ aj \ j). If we distinguish a particular
j=o
set of coefficients that occur in the superposition-vector a
C + \a) = {aj : 1 < j < 2n — 1 and j is odd} then like all quregisters a) will have a probability-value
Prob( \a)) =def Y^ \ aj \2
aj EC + \a)
which we identify with the probability-value of any sentence of our language setting Prob(a) = Prob( \a)).
For the probability-values the following properties hold [Dalla Chiara et al., 2004, p. 261]:
(i) Prob(AND(3,y)) = Prob(3)Prob(Y);
(ii) Prob(NOT(a)) = 1—Prob(a);
(iii) Prob(OR(3, y)) = Prob(3)+ Prob(Y) — Prob(3)Prob(Y) ;
(iv) Prob(^NOT(a)) = E | 2(1 — i)aj-1 + 2(1 + i)aj| 2
aj £C+ \a)
(v) Prob(VNOTNOT(a))=Prob(NOTVNOT(a))
= £ | 1 (1 + i)aj-1 + 2 (1 — i)aj |2
aj eC+\a)
(vi) Prob(VNOTAND(3,Y)) = 2.
Finally, we define the notions of truth, logical truth, consequence and logical consequence.
Definition 13. A sentence a is true in a realization Qub (indicated by =Qub a) iff Prob( a) = 1.
Definition 14. a is a logical truth (indicated by |= a) iff for any realization Qub, =Qub a.
Definition 15. 3 is a consequence of a in the realization Qub (indicated by a =Qub 3) iff Prob(a) < Prob(3).
Definition 16. 3 is a logical consequence of a (indicated by a = 3) iff for any Qub: a =Qub 3.
Specific examples of logical consequences that hold in QCL are the following [Dalla Chiara et al., 2004, p. 264]:
• a = ——a, ——a = a; (double negation)
• y—y—a = —a, —a = y—y—a
• a a. 3= 3 a a, a t 3= 3 t a (commutativity)
• a a (3 a y) = (a a 3) a y, (a a 3) a y = a a (3 a y) (associativity)
• a t (3 t y) = (a y 3) y y, (a t 3) t y = a t (3 t y) (associativity)
• —(a y 3) = —a a —3, —a a —3 = —(a y 3) (de Morgan)
• —(a a 3) = —a y —3, —a y —3 = —(a a 3) (de Morgan)
• a x a\= a (semiidempotence 1)
• a x (0 y y) |= (a x 0) y (a x y) (distributivity 1).
Besides, some logical consequences and some logical truths are violated in QCL:
• a ¥ a x a (semiidempotence 2)
• ¥ a y —a (excluded middle)
• ¥ —(a x —a) (non contradiction)
• (a x 0) y (a x y) ¥ a x (0 y y) (distributivity 2).
If we compare QCL with a standard form of quantum logic then we can quickly conclude that it turns out to be a non standard quantum logic. Firstly, conjunction and disjunction do not correspond to lattice operations since they are not generally idempotent. Secondly, differently from the usual (sharp and unsharp) quantum logics, in QCL the weak distributivity principle (a x 0) y (a x y) = a x (0 y y) breaks down while the strong distributivity a x (0 y y) \= (a x 0) y (a x y), that is violated in orthodox quantum logic, is here valid. Finally, both the excluded middle and the non contradiction principles are violated and as a consequence QCL to be an example of an unsharp quantum logic.
The mostly intriguing in the situation with QCL is that the axiomatizability of QCL is still an open problem [Dalla Chiara et al., 2004, p. 266]. Below we will try to fill up this gap taking into account all peculiarities of the semantics of QCL.
3. QCL axioms
Following R. Goldblatt [Goldblatt, 1974], we will conceive a quantum computational logic not as a set of wffs, but as a collection L of ordered pairs of wffs that satisfies certain closure conditions, the idea being that the presence of the pair (a, 0) in L indicates that 0 can be inferred from a in L. Logics of this kind usually are called binary logics, we will write a h 0 in place of (a, 0) e L. We also enrich sentential language L of QCL with the constants 0 and 1. Schemes of axioms of QCL:
A1. a HI—i—a
A2. a x (0 x y) HI (a x 0) x y
A3. a x (0 y y) h (a x 0) y (a x y)
A4. 0 I 0 x 0
A5. —0 Hh 1
A6. a h 1 a a
A7. a a 3 h a
A8. a a 3 h 3
A9. /—/—a Hh —a
A10. v——a Hh —/—a
All./—(a a 3) Hh —/—(a a 3).
Rules of QCL: ah3
Rl. R2. R3.
—3 h —a a h 3 3 h y
a h y a h 3 y h 5
a a y h 3 a 5
Definition 17. Let QCL be a quantum computability logic and r a non-empty set of wff. A wff a is said to be QCL-derivable from r (indicated by r h a), if there exist 31,..., 3n G r such that (31 a ... a 3n) a (3n a ... a 31) h a. If a is QCL-derivable from 1 then A is QCL-derivable or is an QCL-theorem which writes hQCL a. r is QCL-paraconsistent if there is at least one wff not QCL-derivable from r, and QCL-inconsistent otherwise. r is QCL-full iff it is QCL-paraconsistent and closed under a and QCL-derivability, i.e. iff
(1) for some wff, not r h a;
(2) if a G r and a h 3 then 3 G r;
(3) a,3 G r only if a a 3 G r.
Lemma 1. If x C FormL (where FormL is a set of wff) is QCL-full, then
(i) a a 3 G x iff a G x and 3 G x,
(ii) x h a iff a G x;
(iii) 1 G x.
Proof. (i) The 'if' part is 17(3), and the converse follows from A7, A8 by 17(2).
(ii) Since by A6,A7 a h a, sufficiency following from the definition of QCL-derivability. Necessity uses 17(2),(3).
(iii) By definition x is non-empty, so there exists 3 G x. But by A7 3 h 1a 3 so the result follows by 17(2) and 17(3). ■
QCL-full sets and QCL-derivability are linking with the following version of Lindenbaum's Lemma.
Theorem 1. r h a iff a belongs to every QCL-full extension of r.
Proof. If r h a then there exist 0i,..., 0n e r such that 0i x ... x 0n h a. If x is QCL-full and r C x, we have 0i,..., 0n e x. Applying 17(3) and then 17(2) we obtain a x.
For the converse, suppose a is not QCL-derivable from r. Let x = {0 : r h 0}. From a h a we have r C x and by hypothesis a / x. Our proof will therefore be completed if we can show that x is QCL-full. Now if 0 ex and 0 h y then there exists 0i,..., 0n e r such that 0i x ... x 0n h 0, hence by R2 0i x ... x 0n h y and so r h y i.e. Y e x. If on the other hand 0,Y ex then there exist 0i,..., 0n, y1, ■■■,Ym e r such that 0i x ... x 0n h 0 and Yi x ... x Ym h Y. Letting 5 = (01 x ... x 0n) x (y1 x ... x Ym) we have by A7, A8 and R2 that 5 h 0,5 h y and so by R3, 5 x 5 h 0 x y. Thus, r h 0 x y and therefore 0 x y x.
This shows that x is closed under QCL-derivability and conjunction. Hence, since a e x, a is not QCL-derivable from x, and x is QCL-paraconsistent. ■
Theorem 2. If x is QCL-full and —a e x, then there exists an QCL-full set y such that a e y, and for all 0, either —0 ex or 0 /y.
Proof. Let y = {0 : a h 0}. Since by A6,A7 a h a then a e y. Now let —0 e x. Then 0 e y, or else a h 0, whence —0 I—<a and so by 17(2), —a e x, contrary to the hypotheses. Hence, —0 e x.By 7(iii) 1 e y, —0 e y. According to what we just proved it follows that 0 /y.
Proceeding in a similar manner to 8 we can show that y is closed under conjunction and QCL-derivability, and hence that 0 is not QCL-derivable from y i.e. y is QCL-paraconsistent, and therefore y is QCL-full as required. ■
Theorem 3. If x is QCL-full,(a x 0) h a, then there exists an QCL-full set y such that a e y and —a e y, and for all 0, either 0 / x or —0 e y.
Proof. Let y = {0 : a h 0}. Since by A6,A7 a h a then a e y. From y—(a x 0) h a by R1, A11 we get —a h a and thus —a e y. Now let —0 e y.
We have a h 0, whence —0 I--<a and so by 17(2), —a e y, contrary to the
previous result. Hence, —0/y. If 0 ex then y—"(a x 0) h 0 by R2 and since \—(a x 0) h a we get —a h 0 which contradicts to a h 0 by hypothesis. Hence,
0 e x.
By 7(iii) 1 ex, 0 e x by A4,y—(a x 0) h 1 by A6. According to what we just proved it follows that 0 / y. Proceeding in a similar manner to 8 we can show that y is closed under conjunction and QCL-derivability, and hence that 0 is not QCL-derivable from y i.e. y is QCL-paraconsistent, and therefore y is QCL-full as required. ■
4. The Characterization of QCL
If r is a non-empty set of wffs, then we say r implies a in a realization Qub, r \=QUb a, iff there exist 31,...,3n G r such that
(31 a ... a 3n) a (3n a ... a 31) \=Qub a.
Theorem 4 (Correctness Theorem for QLC). r h a only if r \=Qub a
Proof. The proof, by induction on QCL-derivability, proceeds by showing that the result holds for A1,..., A11 and is preserved by applications of R1, R2, R3.
A1. We have Prob(NOT(NOT(a))) = 1-Prob(NOT(a)) = 1 - 1 + Prob(a) = Prob(a). This means that ——a =Qub a and a =Qub ——a.
A2. We have Prob(a a (3 a y)) = Prob(a)(Prob(3 a y)) = Prob(a)(Prob(3)Prob(Y)) =Prob(a)(Prob(3))Prob(Y) =Prob((a a 3) a y), hence a a (3 a y) \=Qub (a a 3) a y and (a a 3) a y \=Qub a a (3 a y) \=Qub (a a 3) a y.
A3. For the left side we get Prob(a)Prob(3 t y) = Prob(a)(Prob(3)+ Prob(Y)-Prob(3)Prob(Y)). For the right side we obtain Prob(a)Prob(3)+ Prob(a)Prob(Y)—Prob(a)Prob(3)Prob(a)Prob(Y)=Prob(a)(Prob(3 )+Prob(Y) — Prob(a)Prob(3)Prob(Y)). Since Prob(a)Prob(3 )Prob(a)Prob(Y) < Prob(a)Prob(3)Prob(Y) then we obtain a a (3 t y) \=Qub (a a 3) t (a a y). A4. Since Prob(0) = 0 then 0 |=Qub 3 a 3.
A5. Taking into account that Prob(1) = 1 we get Prob(—0) = 1 — Prob(0) = 1 — 0 = 1 and thus —0 \=Qub 1 and 1 \=Qub —0.
A6. Here Prob(1 a a) = Prob(1)Prob(a) = Prob(a) and a \=Qub 1 a a will take place.
A7-A8. Prob(a a 3) = Prob(a)Prob(3) < Prob(a) and analogously Prob(aa3) = Prob(a)Prob(3) < Prob(3). Thus aa3 i=Qub a and aa3 i=Qub 3. A9. Early it was noted that y/—y/—a \= —a, —a \= y/—y/—a, thus
\=Qub —a, —a \=Qub yZ—yV—a.
A10. As stated above Prob(VNOTNOT(a)) = Prob(NOTVNOT(a)) hence y——■a \=Qub —y/—a and —yf—a \=Qub V——.
A11. Since Prob(VNOT(AND(a,3))) = 1 then Prob(NOT/NOT(AND(a,3))) = 2. Hence /—(a a 3) \=Qub —/—(a a 3) and
R1. Suppose that a \=Qub 3. Then 1—Prob(3) < 1—Prob(a) and Prob(NOT(3)) <Prob(NOT(a)) and thus —3 \=Qvb —a.
R2. Let a \=Qub 3 and 3 \=Qub Y. Then because of Prob(a) < Prob(3) and Prob(3) < Prob(Y) we obtain Prob(a) < Prob(Y). Thus a \=Qub Y-
R3. Suppose a \=Qub 3 and y \=Qub 5. This gives us Prob(a) < Prob(3) and Prob(Y) < Prob(5). But then we have Prob(a)Prob(Y) < Prob(3)Prob(5) and a a y \=Qub 3 a 5. ■
A non-empty subset F of n-qubit systems is said to be a filter if the folowing conditions are fulfilled:
(1) \a)\0) e F only if \a x 0} e F;
(2) if \a) e F,a\= 0 then \0) belongs to F.
A filter F is a proper filter if \0) does not belong to F. Denoting the set {F : F is a proper filter and \a) e F} as F\a) we define the operations on filters in the following manner:
Definiti°n 18. F\a) x Fll3) = FlaX!3)
F\a) y F\?) = F\aY?)
—F\a) = F\-a) V—F\a) = F\^-a) F\a) < F\i) iff a \= 0
Now we will introduce an associated quantum computational realization.
Definition 19. An associated quantum computational realization is a function Qub* assigning to any \a) a set of proper filters to which \a) belongs
Qub* : FormL — [J F\a).
As the next step we redefine the notions of truth, logical truth, consequence and logical consequence.
Definition 20. A sentence a is true in a realization Qub* (indicated by \=QUb* a) iff Qub*( a)= U F\a).
Definition 21. a is a logical truth (indicated by \=* a) iff for any realization
Qub*, \=QV,b* a.
Definition 22. 0 is a consequence of a in the realization Qub* (indicated by a \=Qvb* 0) iff Qub* (a) < Qub*(0).
Definition 23. 0 is a logical consequence of a (indicaterd by a \=* 0) iff for any Qub* we have a \=Qvb* 0.
As above, if r is a non-empty set of wffs, then we say r implies a in a realization Qub* (indicated by r !=Qvb* a), iff there exist 01,...,0n e r such that (0i x ... x 0n) x (0n x ... x 0i) \=Qub* a.
Theorem 5 (Paraconsistency Theorem for QLC). r h a only if r \=Qub* a.
Proof. The proof, by induction on QCL-derivability, proceeds by showing that the result holds for A1,..., A11 and is preserved by applications of R1, R2, R3.
A1. We have —F\a) = F—a) and then ——F\a) = F—-a). This means that ——a \=Qub* a and a \=Qub* ——a.
A2. We have F^^ay)) = F^a^a-), hence a a (3 a y) =Qub* (a a 3) a y and (a a 3) a y =Qub* a a (3 a y).
A3. For the left side we get Prob(a) Prob(3 t y) = Prob(a)( Prob(3)+ Prob(Y)— Prob(3)Prob(Y)). For the right side we obtain Prob(a)Prob(3)+ Prob(a)Prob(Y)— Prob(a)Prob(3)Prob(a)Prob(Y) = Prob(a)(Prob(3)+ Prob(Y)— Prob(a)Prob(3)Prob(Y)). Since Prob(a)Prob(3)Prob(a)Prob(Y) < Prob(a)Prob(3)Prob(Y)) then we obtain a a (3 t y) < (a a 3) t (a a y) and thus F\aA(fiTY)) < F\(aAi3)T(aAj)) whence a a (3 t y) =Qub* (a a 3) t (a a y). A4. Since | 0) < \ 3 a 3) then F|o) < FfAf) and 0 \=Qub* 3 a 3. A5. Taking into account that \ 1) = \—0) we get Fix) = F—0) and thus
—0 \=Qub* 1 and 1 \=Qub* —0.
A6. Here Prob(1 a a) = Prob(1)Prob(a) = Prob(a) and thus \a) < \ 1 a a) and F\a) < FilAa) whence a \=Qub* 1 a a will take place.
A7-A8. Prob(a a 3) = Prob(a)Prob(3) < Prob(a) and hence \ a a 3) < \a). Analogously Prob(a a 3) = Prob(a)Prob (3) < Prob(3) and \a a 3) < \3). Then we have FiaAf) < F\a) and FiaAf) < Ff) . Thus a a 3 \=Qub* a and a a 3 \\Qub* 3.
A9. Early it was noted that -sf—yf—a \= —a, —a \= yf—yf—a, thus
\ <\—a), \—a) < \ V-V-a), and FiiV-V-V < Fi—a), Fi-a) < FilV-V-a)
from which we get a =Qub* —a, —a \\Qub* V—V—a.
A10. As stated above \ yj——a) = \—yj—a) and thus Fi^--a) < F—^-a) . Hence y——a \=Qub* —yf—a and —yf—a \=Qub* V——.
A11. Since Prob(/—(AND(a,3))) = 1 then Prob(NOT/NOT(AND(a,3))) = 1. Hence \/—(a a 3)) = \—/—(a a 3)),
FiiV-(VAf)) = Fii-V-(aAf)) and finally we get V^(a a 3) \=Qub* —V—(a a 3) and (a a 3) \=Qub* V— (a a 3).
R1. Suppose that a \=Qub 3.Then 1— Prob(3) < 1— Prob(a), Prob(NOT(3)) < Prob(NOT(a)) and \ —3) < \ —a). Thus F-f) < F-a) and —3 \=Qub* —a.
R2. Let a \=Qub 3 and 3 \=Qub Y. Then because of Prob(a) < Prob(3) and Prob(3) < Prob(Y) we obtain Prob(a) < Prob(Y). Thus \ a) < \y),F\a) < F\Y) and it implies a \=Qub* Y-
R3. Suppose a \=Qub 3 and y \=Qub 5. This gives us Prob(a) < Prob(3) and Prob(Y) < Prob(5). But then we have Prob(a)Prob(Y) < Prob(3)Prob(5)
that gives us \a x y ) <\pP x 5) and F\aXj) < F\pX$) respectively. Finally, we get a x y \= Qvb* 0 x 5. ■
Definition 24. If L is a quantum computational logic then a canonical realization of L is
Qubc : FormL — XL, such that Qubc(a) = {x e XL : a e x},
where XL = {x C FormL : x is QCL- full set of wffs}. Lemma 2. Qubc is indeed a quantum computational realization.
Proof. Let x e XL. Since x is QCL-full then x is QCL-paraconsistent and by theorem 8 0 does not belong to x. Being QCL-full x is closed under x and QCL-derivability. Since QCL-derivability means that if a e r and a h 0 then 0 e r and taking into account that a h 0 according to correctness theorem only if a \=Qvb 0, we conclude that a h 0 only if \a) ^ \0) and thus 0 belongs to x only if \0) belongs to respective filter. As to closeness under x then by lemma 7 we have a x 0 ex iff a ex and 0 ex, which similarly lead us to \a),\0) e G only if \a x 0) e G (where G is a filter). So, finally we introduce operations on Qubc[Formc] (denoting Qubc(a) = {x e XL : a e x} as xa) in the following way:
xa x xl — xaXl
xa y xl = xaYl —xa = x—a
V—xa =
xa < xp iff a h 0. ■
Theorem 6 (Completeness Theorem for QLC). r h a iff r \=Qvbc a.
Proof. If r h a then there exist 01,..,0n e r such that (01 x ... x 0n) x (0n x ... x 01) \=Qvb* a. If x is QCL-full and r C x, we have 01,..., 0n e x. Applying 17(3) and then 17(2) we obtain a ex and hence x e xa. Conversely, suppose a is not QCL-derivable from r. Then by 8 there exists x e XL, such that r C x and a /x. For all 0 e r we have 0 e x and hence x e xp, but not a ex. Thus r ^Qvbc a. ■
References
Cattaneo et al., 2003 - Cattaneo, G., Dalla Chiara, M.L., Giuntini, R. "An Unsharp Quantum Logic from Quantum Computation", in: Alternative Logics. Do Sciences Need Them?, ed. by P. Weingartner, Springer Verlag, Berlin — Heidelberg — New York, 2003, pp. 323-338.
Cattaneo et al., 2004 - Cattaneo, G., Dalla Chiara, M.L., Giuntini, R. and Leporini, R. "An unsharp logic from quantum computation", International Journal of Theoretical Physics, 2004, Vol. 43, Is. 7-8, pp. 1803-1817.
Dalla Chiara et al., 2004 - Dalla Chiara, M.L., Giuntini, R., Greechie, R. Reasoning in Quantum Theory. Sharp and Unsharp Quantum Logic, 2004, Vol. 3, No. 1-2, pp. 240-266.
Goldblatt, 1974 - Goldblatt, R. "Semantic Analysis of Orthologic", Journal of Philosophical Logic, 1974, Vol. 3, No. 1-2, pp. 19-35.
В.Л. Влсюков
Аксиоматизация логики квантовых вычислений
Владимир Леонидович Васюков
Институт философии РАН
Российская Федерация, 109240, г. Москва, ул. Гончарная, д. 12, стр. 1. E-mail: [email protected]
Аннотация: Одной из логических рекомендаций, касающихся квантовых вычислений, является идея использовать квантовый теоретический формализм для представления параллельных рассуждений. С этой целью квантовое вычисление представляется с помощью соответствующих унитарных операторов, предполагающих аргументы и значения в конкретных множествах систем кубитов. Выделив некоторые важные унитарные операторы, играющие особую роль в квантовых вычислениях (логические вентили, или quregisters), мы получаем возможность построить язык квантовой вычислительной логики (QCL) (ср. [Cattaneo et al., 2003; Cattaneo et al., 2004; Dalla Chiara et al., 2004]). Основным понятием семантики этого языка является понятие квантовой вычислительной реализации, когда значением, ассоциированным с любым предложением, является логический вентиль. В отличие от семантики стандартной квантовой логики, QCL-конъюнкция и QCL-дизъюнкция не соответствуют решеточным операциям, так как они, как правило, не являются идемпотентными. Более того, в QCL нарушается принцип слабой дистрибутивности, и нарушаются как принцип исключенного третьего, так и принцип непротиворечия. Наконец, аксиоматизируемость QCL все еще остается открытой проблемой. В статье предлагается аксиоматизация QCL, трактующая ее как разновидность так называемой бинарной логики Гольдблатта. Доказаны некоторые металогические теоремы (паранепротиворечивость и полнота).
Ключевые слова: квантовое вычисление, квантовая вычислительная реализация, бинарная логика, паранепротиворечивость
