ar X iv :1 50 4. 06 83 8v 2 [ qu an tph ] 9 N ov 2 01 5 Quantum Set Theory 1 Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory∗ MASANAO OZAWA Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan ozawa@is.nagoya-u.ac.jp Abstract The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness. Keywords: quantum logic, quantum set theory, quantum theory, quantum measurements, von Neumann algebras 1 Introduction Set theory provides foundations of mathematics; all the mathematical notions like numbers, functions, relations, and structures are defined in the axiomatic set theory, ZFC (Zermelo-Fraenkel set theory with the axiom of choice), and all the mathematical theorems are required to be provable in ZFC. Quantum set theory, instituted by Takeuti [32] and developed by the present author [24], naturally extends the logical basis of set ∗An extended abstract of this paper was presented in the 11th International Workshop on Quantum Physics and Logic (QPL 2014), Kyoto University, June 4–6, 2014 and appeared as Ref. [27]. 2 Masanao Ozawa theory from classical logic to quantum logic to explore mathematics based on quantum logic. Despite remarkable success in axiomatic foundations of quantum mechanics [35, 12], the quantum logic approach to quantum foundations has not been considered powerful enough to solve interpretational problems [29, 9]. However, this weakness is considered to be mainly due to the fact that the conventional study of quantum logic has been limited to propositional logic. Since quantum set theory extends the underlying logic from propositional logic to predicate logic, and provides set theoretical constructions of mathematical objects such as numbers, functions, relations, and structures based on quantum logic, we can expect that quantum set theory will provide much more systematic interpretation of quantum theory than the conventional quantum logic approach. This paper represents the first step towards establishing systematic interpretation of quantum theory based on quantum set theory, and naturally focusses on the most fundamental notion in mathematics, namely, equality. The notion of equality between quantum observables will play many important roles in foundations of quantum theory, in particular, in the theory of measurement and disturbance [22, 23]. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables [36]. In this paper, quantum set theory is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state based on the fact that real numbers defined in quantum set theory exactly corresponds to quantum observables [32, 24]. It is shown that every observational proposition on a quantum system corresponds to a statement in quantum set theory with the same projection-valued truth value and the same probability in any state. In particular, equality between real numbers in quantum set theory naturally provides a state-dependent notion of equality between quantum mechanical observables. It has been broadly accepted that we cannot speak of the values of quantum observables without assuming a hidden variable theory, which are severely constrained by Kochen-Specker type no-go theorems [14, 29]. However, quantum set theory enables us to do so without assuming hidden variables but alternatively with the consistent use of quantum logic. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness. Section 2 provides preliminaries on complete orthomodular lattices, commutators of their subsets, quantum logic on Hilbert spaces, and the universe V (Q) of quantum Quantum Set Theory 3 set theory over a logic Q on a Hilbert space H . We give a characterization of the commutator of a subset of a complete orthomodular lattice, improving Takeuti's characterization, and give a factorization of the double commutant of a subset of a complete orthomodular lattice into the maximal Boolean factor and a complete orthomodular lattice without non-trivial Boolean factor. Section 3 introduces a one-to-one correspondence obtained in Refs. [32, 24] between the reals R(Q) in V (Q) and self-adjoint operators affiliated with the von Neumann algebra M = Q′′ generated by Q, determines commutators and equality in R(Q), and gives the embedding of intervals in R into V (Q). Section 4 formulates the standard probabilistic interpretation of quantum theory and also shows that the set of observational propositions for a quantum system can be embedded in a set of statements in quantum set theory without changing projection-valued truth value assignment. Section 5 extends the standard interpretation by introducing simultaneous determinateness, i.e., state-dependent commutativity of observables. We give several characterizations of simultaneous determinateness for finite number of quantum observables affiliated with an arbitrary von Neumann algebra in a given state, extending some previous results [23] on simultaneous determinateness for two observables. Section 6 extends the standard interpretation by introducing quantum equality, i.e., state-dependent equality for two arbitrary observables. We give several characterizations of quantum equality for two observables affiliated with an arbitrary von Neumann algebra in a given state, extending some previous results [23] on simultaneous determinateness for two observables. Sections 7 and 8 provide applications to quantum measurement theory. We discuss a state-dependent formulation of measurement of observables and simultaneous measurability, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness. The conclusion is given in Section 9. Whereas we will discuss the completely general case where M is an arbitrary von Neumann algebra, some results for the case where dim(H ) < ∞ and M = B(H ) have been previously reported in Ref. [25]. In this special case, we can avoid the use of quantum set theory to introduce simultaneous determinateness and quantum equality into the language of observational propositions, since simultaneous determinateness and quantum equality can be expressed, respectively, by observational propositions constructed by atomic formulas of the form X = x with an observable X and a real number x. However, to prove a transfer theorem ensuring that all the classical tautologies have the truth value 1, mentioned without proof in Ref. [25], Theorem 3, it is necessary, even in this special case, to develop quantum set theory and to define the embedding of the language of observational propositions into the language of quantum 4 Masanao Ozawa set theory. The required machinery will be, for the first time, fully constructed in this paper including the case with observables with continuous spectrum, though the full power of this machinery will be revealed when applied to mathematical theorems beyond tautologies after we have enriched the language of observational propositions, in the future research, with more sophisticated relations and functions than equality. 2 Quantum set theory 2.1 Quantum logic A complete orthomodular lattice is a complete lattice Q with an orthocomplementation, a unary operation ⊥ on Q satisfying (C1) if P ≤ Q then Q⊥ ≤ P⊥, (C2) P⊥⊥ = P, (C3) P∨P⊥ = 1 and P∧P⊥ = 0, where 0 = ∧Q and 1 = ∨Q, that satisfies the orthomodular law (OM) if P ≤ Q then P∨ (P⊥∧Q) = Q. In this paper, any complete orthomodular lattice is called a logic. A non-empty subset of a logic Q is called a subalgebra iff it is closed under ∧, ∨, and ⊥. A subalgebra A of Q is said to be complete iff it has the supremum and the infimum in Q of an arbitrary subset of A . For any subset A of Q, the subalgebra generated by A is denoted by Γ0A . We refer the reader to Kalmbach [13] for a standard text on orthomodular lattices. We say that P and Q in a logic Q commute, in symbols P |◦ Q, iff P = (P∧Q)∨(P∧ Q⊥). All the relations P |◦ Q, Q |◦ P, P⊥ |◦ Q, P |◦ Q⊥, and P⊥ |◦ Q⊥ are equivalent. The distributive law does not hold in general, but the following useful propositions hold (Ref. [13], pp. 24–25). Proposition 2.1. If P1,P2 |◦ Q, then the sublattice generated by P1,P2,Q is distributive. Proposition 2.2. If Pα |◦ Q for all α , then ∨ α Pα |◦ Q, ∧ α Pα |◦ Q, Q ∧ ( ∨ α Pα) =∨ α(Q∧Pα), and Q∨ ( ∧ α Pα) = ∧ α(Q∨Pα), From Proposition 2.1, a logic Q is a Boolean algebra if and only if P |◦ Q for all P,Q ∈ Q (Ref. [13, pp. 24–25]). Quantum Set Theory 5 For any subset A ⊆ Q, we denote by A ! the commutant of A in Q (Ref. [13], p. 23), i.e., A ! = {P ∈ Q | P |◦ Q for all Q ∈ A }. Then, A ! is a complete subalgebra of Q. A sublogic of Q is a subset A of Q satisfying A = A !!. For any subset A ⊆ Q, the smallest logic including A is A !! called the sublogic generated by A . Then, it is easy to see that a subset A is a Boolean sublogic, or equivalently a distributive sublogic, if and only if A = A !! ⊆ A !. 2.2 Commutators Let Q be a logic. Marsden [15] has introduced the commutator com(P,Q) of two elements P and Q of Q by com(P,Q) = (P∧Q)∨ (P∧Q⊥)∨ (P⊥∧Q)∨ (P⊥∧Q⊥). (1) Bruns and Kalmbach [4] have generalized this notion to finite subsets of Q by com(F ) = ∨ α :F→{id,⊥} ∧ P∈F Pα(P) (2) for all F ∈ Pω(Q), where Pω(Q) stands for the set of finite subsets of Q, and {id,⊥} stands for the set consisting of the identity operation id and the orthocomplementation ⊥. Generalizing this notion to arbitrary subsets A of Q, Takeuti [32] defined com(A ) by com(A ) = ∨ T (A ), (3) T (A ) = {E ∈ A ! | P1 ∧E |◦ P2 ∧E for all P1,P2 ∈ A }, (4) of any A ∈ P(Q), where P(Q) stands for the power set of Q, and showed that com(A ) ∈ T (A ). Subsequently, Pulmannová [28] showed: Theorem 2.3. For any subset A of a logic Q, we have (i) com(A ) = ∧{com(F ) | F ∈ Pω(A )}, (ii) com(A ) = ∧{com(P,Q) | P,Q ∈ Γ0(A )}. Here, we reformulate Takeuti's definition in a more convenient form. Let A ⊆ Q. Note that A !! is the sublogic generated by A , and A ! ∩A !! is the center of A !!, i.e., the set of elements of A !! commuting with all elements of A !!. Denote by L(A ) the sublogic generated by A , i.e., L(A ) = A !!, and by Z(A ) the center of L(A ), i.e., 6 Masanao Ozawa Z(A ) = A ! ∩A !!. A subcommutator of A is any E ∈ Z(A ) such that P1 ∧E |◦ P2 ∧E for all P1,P2 ∈ A . Denote by S(A ) the set of subcommutators of A , i.e., S(A ) = {E ∈ Z(A ) | P1 ∧E |◦ P2 ∧E for all P1,P2 ∈ A }. (5) By the relation Z(A ) ⊆ A !, we immediately obtain the relation ∨S(A ) ≤ com(A ). We shall show that the equality actually holds. Lemma 2.4. Let A be any subset of a logic Q. For any P1,P2 ∈ A and E ∈ A !, we have P1 ∧E |◦ P2 ∧E if and only if P1 ∧E |◦ P2. Proof. Let E ∈A ! and P1,P2 ∈A . We have (P1∧E)∧(P2 ∧E)⊥ = (P1∧E)∧P⊥2 , and hence [(P1 ∧E)∧ (P2 ∧E)]∨ [(P1 ∧E)∧ (P2 ∧E)⊥] = [(P1 ∧E)∧P2]∨ [(P1 ∧E)∧P⊥2 ]. It follows that P1 ∧E |◦ P2 ∧E if and only if P1 ∧E |◦ P2. For any P,Q ∈ Q, the interval [P,Q] is the set of all X ∈ Q such that P ≤ X ≤ Q. For any A ⊆ Q and P,Q ∈ A , we write [P,Q]A = [P,Q]∩A . Theorem 2.5. For any subset A of a logic Q, the following relations hold. (i) S(A ) = {E ∈ Z(A ) | [0,E]A ⊆ Z(A )}. (ii) ∨ S(A ) is the maximum subcommutator of A , i.e., ∨ S(A ) ∈ S(A ). (iii) S(A ) = [0, ∨ S(A )]L(A ). (iv) com(A ) = ∨ S(A ). Proof. (i) It is easy to see that P1∧E |◦ P2 for every P1,P2 ∈A if and only if [0,E]∩A ⊆ A !, and hence the assertion follows from Lemma 2.4. (ii) Let P1,P2 ∈ A . We have P1∧E |◦ P2 for every E ∈ S(A ) from Lemma 2.4, and P1∧ ∨ S(A ) |◦ P2 from Proposition 2.2. Since S(A ) ⊆ Z(A ), we have ∨S(A ) ∈ Z(A ). Thus, ∨S(A ) ∈ S(A ), and the assertion follows. (iii) If P ∈ [0,∨S(A )]L(A ) then P = P∧ ∨ S(A ) commutes with every element of L(A ). Thus, we have [0, ∨ S(A )]L(A ) = [0, ∨ S(A )]Z(A ). Now, let P ∈ [0,∨S(A )]Z(A ). Then, P1 |◦ P and P1 |◦ P2 ∧ ∨ S(A ), and hence P1 |◦ P∧ P2 ∧ ∨ S(A ) and P1 |◦ P2∧P. Thus, we have P ∈ S(A ), and the assertion follows. (iv) Since com(F )∈ Z(F ) for every finite subset F of A , we have com(A )∈ Z(A ), and hence we have com(A ) ∈ Z(A ). Thus, relation (iv) follows. The following proposition will be useful in later discussions. Quantum Set Theory 7 Theorem 2.6. Let B be a maximal Boolean sublogic of a logic Q and A a subset of Q including B, i.e., B ⊆A ⊆Q. Then, we have com(A )∈B and [0,com(A )]A ⊆B. Proof. Since com(A ) ∈ Z(A ) ⊆ B! = B, we have com(A ) ∈ B. Let P ∈ A . Then, P ∧ com(A ) |◦ Q for all Q ∈ B, so that P∧ com(A ) ∈ B! = B, and hence [0,com(A )]A ⊆ B. The following theorem clarifies the significance of commutators. Theorem 2.7. Let A be a subset of a logic Q. Then, L(A ) is isomorphic to the direct product of the complete Boolean algebra [0,com(A )]L(A ) and the complete orthomodular lattice [0,com(A )⊥]L(A ) without non-trivial Boolean factor. Proof. It follows from ∨ S(A ) ∈ Z(A ) that L(A ) ∼= [0, ∨ S(A )]L(A ) × [0, ∨ S(A )⊥]L(A ). Then, [0, ∨ S(A )]L(A ) is a complete Boolean algebra, since [0, ∨ S(A )]L(A ) ⊆ Z(A ). It follows easily from the maximality of ∨ S(A ) that [0, ∨ S(A )⊥]L(A ) has no non-trivial Boolean factor. Thus, the assertion follows from the relation ∨ S(A ) = com(A ). We refer the reader to Pulmannová [28] and Chevalier [5] for further results about commutators in orthomodular lattices. 2.3 Logic on Hilbert spaces Let H be a Hilbert space. For any subset S ⊆ H , we denote by S⊥ the orthogonal complement of S. Then, S⊥⊥ is the closed linear span of S. Let C (H ) be the set of all closed linear subspaces in H . With the set inclusion ordering, the set C (H ) is a complete lattice. The operation M 7→ M⊥ is an orthocomplementation on the lattice C (H ), with which C (H ) is a logic. Denote by B(H ) the algebra of bounded linear operators on H and Q(H ) the set of projections on H . We define the operator ordering on B(H ) by A ≤ B iff (ψ,Aψ) ≤ (ψ,Bψ) for all ψ ∈ H . For any A ∈ B(H ), denote by R(A) ∈ C (H ) the closure of the range of A, i.e., R(A) = (AH )⊥⊥. For any M ∈ C (H ), denote by P(M) ∈ Q(H ) the projection operator of H onto M. Then, RP(M) = M for all M ∈ C (H ) and PR(P) = P for all P ∈ Q(H ), and we have P ≤ Q if and only if R(P)⊆ R(Q) for all P,Q ∈ Q(H ), so that Q(H ) with the operator ordering is also a logic isomorphic to C (H ). Any sublogic of Q(H ) will be called a logic on H . The lattice operations are characterized by P∧Q = weak-limn→∞(PQ)n, P⊥ = 1−P for all P,Q ∈ Q(H ). 8 Masanao Ozawa Let A ⊆B(H ). We denote by A ′ the commutant of A in B(H ). A self-adjoint subalgebra M of B(H ) is called a von Neumann algebra on H iff M ′′ = M . For any self-adjoint subset A ⊆ B(H ), A ′′ is the von Neumann algebra generated by A . We denote by P(M ) the set of projections in a von Neumann algebra M . For any P,Q ∈ Q(H ), we have P |◦ Q iff [P,Q] = 0, where [P,Q] = PQ−QP. For any subset A ⊆ Q(H ), we denote by A ! the commutant of A in Q(H ). For any subset A ⊆Q(H ), the smallest logic including A is the logic A !! called the logic generated by A . Then, a subset Q ⊆Q(H ) is a logic on H if and only if Q =P(M ) for some von Neumann algebra M on H (Ref. [24], Proposition 2.1). We define the implication and the logical equivalence on Q by P → Q = P⊥ ∨ (P∧Q) and P ↔ Q = (P → Q)∧ (Q → P). We have the following characterization of commutators in logics on Hilbert spaces (Ref. [24], Theorems 2.5, 2.6). Theorem 2.8. Let Q be a logic on H and let A ⊆ Q. Then, we have the following relations. (i) com(A ) = P{ψ ∈ H | [A,B]ψ = 0 for all A,B ∈ A ′′}. (ii) com(A ) = P{ψ ∈ H | [P1,P2]P3ψ = 0 for all P1,P2,P3 ∈ A }. 2.4 Quantum set theory over logic on Hilbert spaces We denote by V the universe of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Let L (∈) be the first-order language with equality without constant symbols augmented by a binary relation symbol ∈, bounded quantifier symbols ∀x ∈ y, ∃x ∈ y (in addition to unbounded quantifier symbols ∀x, ∃x. For any class U , the language L (∈,U) is the one obtained by adding a name for each element of U . Let Q be a logic on H . For each ordinal α , let V (Q)α = {u| u : dom(u)→ Q and (∃β < α)dom(u)⊆V (Q)β }. (6) The Q-valued universe V (Q) is defined by V (Q) = ⋃ α∈On V (Q)α , (7) where On is the class of all ordinals. For every u ∈ V (Q), the rank of u, denoted by rank(u), is defined as the least α such that u ∈V (Q)α+1. It is easy to see that if u ∈ dom(v) then rank(u)< rank(v). For any u,v ∈V (Q), the Q-valued truth values of atomic formulas u = v and u ∈ v are assigned by the following rules recursive in rank. Quantum Set Theory 9 (i) [[u = v]]Q = ∧ u′∈dom(u)(u(u ′)→ [[u′ ∈ v]]Q)∧ ∧ v′∈dom(v)(v(v ′)→ [[v′ ∈ u]]Q). (ii) [[u ∈ v]]Q = ∨ v′∈dom(v)(v(v ′)∧ [[u = v′]]Q). To each statement φ of L (∈,V (Q)) we assign the Q-valued truth value [[φ ]]Q by the following rules. (iii) [[¬φ ]]Q = [[φ ]]⊥Q. (iv) [[φ1 ∧φ2]]Q = [[φ1]]Q ∧ [[φ2]]Q. (v) [[φ1 ∨φ2]]Q = [[φ1]]Q ∨ [[φ2]]Q. (vi) [[φ1 → φ2]]Q = [[φ1]]Q → [[φ2]]Q. (vii) [[φ1 ↔ φ2]]Q = [[φ1]]Q ↔ [[φ2]]Q. (viii) [[(∀x ∈ u)φ(x)]]Q = ∧ u′∈dom(u)(u(u ′)→ [[φ(u′)]]Q). (ix) [[(∃x ∈ u)φ(x)]]Q = ∨ u′∈dom(u)(u(u ′)∧ [[φ(u′)]]Q). (x) [[(∀x)φ(x)]]Q = ∧ u∈V (Q) [[φ(u)]]Q. (xi) [[(∃x)φ(x)]]Q = ∨ u∈V (Q) [[φ(u)]]Q. We say that a statement φ of L (∈,V (Q)) holds in V (Q) iff [[φ ]]Q = 1. A formula in L (∈) is called a ∆0-formula iff it has no unbounded quantifiers ∀x or ∃x. The following theorem holds [24]. Theorem 2.9 (∆0-Absoluteness Principle). For any ∆0-formula φ(x1, . . .,xn) of L (∈) and u1, . . .,un ∈V (Q), we have [[φ(u1, . . . ,un)]]Q = [[φ(u1, . . . ,un)]]Q(H ). Henceforth, for any ∆0-formula φ(x1, . . .,xn) and u1, . . . ,un ∈ V (Q), we abbreviate [[φ(u1, . . . ,un)]] = [[φ(u1, . . . ,un)]]Q, which is the common Q-valued truth value in all V (Q) such that u1, . . . ,un ∈V (Q). The universe V can be embedded in V (Q) by the following operation ∨ : v 7→ v defined by the ∈-recursion: for each v ∈ V , v = {ǔ| u ∈ v}×{1}. Then we have the following [24]. Theorem 2.10 (∆0-Elementary Equivalence Principle). For any ∆0formula φ(x1, . . .,xn) of L (∈) and u1, . . .,un ∈ V, we have 〈V,∈〉 |= φ(u1, . . .,un) if and only if [[φ(ǔ1, . . . , ǔn)]] = 1. For u ∈V (Q), we define the support of u, denoted by Ł(u), by transfinite recursion on the rank of u by the relation Ł(u) = ⋃ x∈dom(u) Ł(x)∪{u(x) | x ∈ dom(u)}. (8) 10 Masanao Ozawa For A ⊆ V (Q) we write Ł(A ) = ⋃u∈A Ł(u) and for u1, . . . ,un ∈ V (Q) we write Ł(u1, . . . ,un) = Ł({u1, . . . ,un}). Let A ⊆ V (Q). The commutator of A , denoted by com(A ), is defined by com(A ) = com(Ł(A )). (9) For any u1, . . . ,un ∈V (Q), we write com(u1, . . . ,un) = com({u1, . . . ,un}). For bounded theorems, the following transfer principle holds [24]. Theorem 2.11 (ZFC Transfer Principle). For any ∆0-formula φ(x1, . . .,xn) of L (∈) and u1, . . .,un ∈V (Q), if φ(x1, . . .,xn) is provable in ZFC, then we have com(u1, . . . ,un)≤ [[φ(u1, . . . ,un)]]. 3 Real numbers in quantum set theory Let Q be the set of rational numbers in V . We define the set of rational numbers in the model V (Q) to be Q. We define a real number in the model by a Dedekind cut of the rational numbers. More precisely, we identify a real number with the upper segment of a Dedekind cut assuming that the lower segment has no end point. Therefore, the formal definition of the predicate R(x), "x is a real number," is expressed by R(x) := ∀y ∈ x(y ∈ Q)∧∃y ∈ Q(y ∈ x)∧∃y ∈ Q(y 6∈ x) ∧∀y ∈ Q(y ∈ x ↔∀z ∈ Q(y < z → z ∈ x)). (10) The symbol ":=" is used to define a new formula, here and hereafter. We define R(Q) to be the interpretation of the set R of real numbers in V (Q) as follows. R(Q) = {u ∈V (Q)| dom(u) = dom(Q) and [[R(u)]] = 1}. (11) The set RQ of real numbers in V (Q) is defined by RQ = R (Q)×{1}. (12) Then, for any u,v ∈ R(Q), the following relations hold in V (Q) [24]. (i) [[(∀u ∈ RQ)u = u]] = 1. (ii) [[(∀u,v ∈ RQ)u = v → v = u]] = 1. (iii) [[(∀u,v,w ∈ RQ)u = v∧ v = w → u = w]] = 1. (iv) [[(∀v ∈ RQ)(∀x,y ∈ v)x = y∧ x ∈ v → y ∈ v]]. Quantum Set Theory 11 (v) [[(∀u,v ∈ RQ)(∀x ∈ u)x ∈ u∧u = v → x ∈ v]]. From the above, the equality is an equivalence relation between real numbers in V (Q). For any u1, . . . ,un ∈ R(Q), we have [[u1 = u2 ∧* * *∧un−1 = un]]≤ com(u1, . . . ,un), (13) and hence commutativity follows from equality in R(Q) [24]. Let M be a von Neumann algebra on a Hilbert space H and let Q = P(M ). A closed operator A (densely defined) on H is said to be affiliated with M , in symbols Aη M , iff U∗AU =A for any unitary operator U ∈M ′. Let A be a self-adjoint operator (densely defined) on H and let A = ∫ R λ dEA(λ ) be its spectral decomposition, where {EA(λ )}λ∈R is the resolution of identity belonging to A (Ref. [36], p. 119). It is wellknown that Aη M if and only if EA(λ ) ∈ Q for every λ ∈ R. Denote by M SA the set of self-adjoint operators affiliated with M . Two self-adjoint operators A and B are said to commute, in symbols A |◦ B, iff EA(λ ) |◦ EB(λ ′) for every pair λ ,λ ′ of reals. For any u ∈ R(Q) and λ ∈ R, we define Eu(λ ) by Eu(λ ) = ∧ λ<r∈Q u(ř). (14) Then, it can be shown that {Eu(λ )}λ∈R is a resolution of identity in Q and hence by the spectral theorem there is a self-adjoint operator ûη M uniquely satisfying û = ∫ R λ dEu(λ ). On the other hand, let Aη M be a self-adjoint operator. We define Ã ∈ V (Q) by Ã = {(ř,EA(r)) | r ∈ Q}. (15) Then, dom(Ã) = dom(Q) and Ã(ř) = EA(r) for all r ∈Q. It is easy to see that Ã ∈R(Q) and we have (û) = u for all u ∈ R(Q) and (Ã) = A for all A ∈ M SA. Therefore, the correspondence between R(Q) and M SA is a one-to-one correspondence. We call the above correspondence the Takeuti correspondence. Now, we have the following [24]. Theorem 3.1. Let Q be a logic on H . The relations (i) EA(λ ) = ∧ λ<r∈Q u(ř) for all λ ∈ Q, (ii) u(ř) = EA(r) for all r ∈ Q, for all u = Ã ∈ R(Q) and A = û ∈ M SA sets up a one-to-one correspondence between R(Q) and M SA. 12 Masanao Ozawa For any r ∈ R, we shall write r = (r1) , where r1 is the scalar operator on H . Then, we have dom(r) = dom(Q) and r(ť) = [[ř ≤ ť]], so that we have Ł(r) = {0,1}. Denote by B(Rn) the σ -filed of Borel subsets of Rn and B(Rn) the space of bounded Borel functions on Rn. A spectral measure [10] on Rn in M is a mapping E of B(Rn) into P(M ) satisfying ∑ j E(∆i) = 1 for any disjoint sequence {∆ j} in B(Rn) such that⋃ j ∆ j = Rn. Let X be a self-adjoint operator affiliated with M . For any f ∈ B(R), the bounded self-adjoint operator f (X) ∈ M is defined by f (X) = ∫ R f (λ )dEX(λ ). The spectral measure of X is a spectral measure EA on R in M defined by EX (∆) = χ∆(X) for any ∆ ∈ B(R). Then, we have EX (λ ) = EX ((−∞,λ ]). Proposition 3.2. Let r ∈ R, s, t ∈ R, and X η MSA. We have the following relations. (i) [[ř ∈ s]] = [[š ≤ ř]] = Es1(t). (ii) [[s ≤ t]] = [[š ≤ ť]] = Es1(t). (iii) [[X ≤ t]] = EX (t) = EX ((−∞, t]). (iv) [[t < X ]] = 1−EX (t) = EX ((t,∞)). (v) [[s < X ≤ t]] = EX (t)−EX(s) = EX ((s, t]). (vi) [[X = t]] = EX (t)−∨r<t,r∈Q EX (r) = EX ({t}). Proof. Relations (i), (ii), and (iii) follows from [24, Proposition 5.11]. We have com(t, X) = 1, so that (iv) follows from the ZFC Transfer Principle (Theorem 2.11). Relation (v) follows from (iii) and (iv). We have [[X = t]] = ∧ r∈Q X(ř)→ [[ř ∈ t]]∧ ∧ r∈Q t(ř)→ [[ř ∈ X ]] = ∧ r∈Q EX (r)⊥∨Et1(r)∧ ∧ r∈Q Et1(r)⊥∨EX (r) = ∧ r<t,r∈Q EX (r)⊥∧ ∧ t≤r∈Q EX (r) = [1− ∨ r<t,r∈Q EX (r)]∧EX(t) = EX (t)− ∨ r<t,r∈Q EX (r) = EX ({t}). Thus, relation (vi) follows. Quantum Set Theory 13 4 Standard probabilistic interpretation of quantum theory Let S be a quantum system described by a von Neumann algebra M on a Hilbert space H . According to the standard formulation of quantum theory, the observables of S are defined as self-adjoint operators affiliated with M , the states of S are represented by density operators on H , and a vector state ψ is identified with the state |ψ〉〈ψ|. We denote by O(M ) the set of observables, by S (H ) the space of density operators. Observables X1, . . . ,Xn ∈O(M ) are said to be mutually commuting iff X j |◦ Xk for all j,k = 1, . . . ,n. If X1, . . . ,Xn ∈ O(M ) are bounded, this condition is equivalent to [X j,Xk] = 0 for all j,k = 1, . . . ,n. The standard probabilistic interpretation of quantum theory defines the joint probability distribution function FX1,...,Xnρ (x1, . . . ,xn) for mutually commuting observables X1, . . . ,Xn ∈O(M ) in ρ ∈ S (H ) by the Born statistical formula: FX1,...,Xnρ (x1, . . . ,xn) = Tr[E X1(x1) * * *EXn(xn)ρ ]. (16) To clarify the logical structure presupposed in the standard probabilistic interpretation, we define observational propositions for S by the following rules. (R1) For any X ∈ O(M ) and x ∈ R, the expression X ≤o x is an observational proposition. (R2) If φ1 and φ2 are observational propositions, ¬φ1 and φ1∧φ2 are also observational propositions. Thus, every observational proposition is built up from "atomic" observational propositions X ≤o x by adding finite number of connectives ¬ and ∧. We denote by Lo(M ) the set of observational propositions. We introduce the connective ∨ by definition. (D1) φ1 ∨φ2 := ¬(¬φ1 ∧¬φ2). For each observational proposition φ , we assign its projection-valued truth value [[φ ]]o ∈ Q(H ) by the following rules [2]. (T1) [[X ≤o x]]o = EX (x). (T2) [[¬φ ]]o = [[φ ]]⊥o . (T3) [[φ1 ∧φ2]]o = [[φ1]]o ∧ [[φ2]]o. From (D1), (T2) and (T3), we have (D2) [[φ1 ∨φ2]]o = [[φ1]]o ∨ [[φ2]]o. 14 Masanao Ozawa We define the probability Pr{φ‖ρ} of an observational proposition φ in a state ρ by (P1) Pr{φ‖ρ}= Tr[[[φ ]]oρ ]. We say that an observational proposition φ holds in a state ρ iff Pr{φ‖ρ}= 1. The standard interpretation of quantum theory restricts observational propositions to be standard defined as follows. (W1) An observational proposition including atomic formulas X1 ≤o x1, . . . ,Xn ≤o xn is called standard iff X1, . . . ,Xn are mutually commuting. All the standard observational propositions including only given mutually commuting observables X1, . . . ,Xn comprise a complete Boolean algebra under the logical order ≤ defined by φ ≤ φ ′ iff [[φ ]]o ≤ [[φ ′]]o and obey inference rules in classical logic. Suppose that X1, . . . ,Xn ∈ O(M ) are mutually commuting. Let x1, . . . ,xn ∈ R. Then, X1 ≤o x1 ∧* * *∧Xn ≤o xn is a standard observational proposition. We have [[X1 ≤o x1 ∧* * *∧Xn ≤o xn]]o = EX1(x1)∧* * *∧EXn(xn) = EX1(x1) * * *EXn(xn). (17) Hence, we reproduce the Born statistical formula as Pr{X1 ≤o x1 ∧* * *∧Xn ≤o xn‖ρ}= Tr[EX1(x1) * * *EXn(xn)ρ ]. (18) From the above, our definition of the truth values of observational propositions are consistent with the standard probabilistic interpretation of quantum theory. From Proposition 3.2 and (T1), we conclude [[X ≤ x]] = [[X ≤o x]]o (19) for all X ∈ O(M ) and x ∈ R. To every observational proposition φ the corresponding statement φ in L (∈,R(Q)) is given by the following rules for any X ∈ O(M ) and x ∈ R, and observational propositions φ ,φ1,φ2. (Q1) X ≤o x := X ≤ x. (Q2) ¬φ := ¬φ . (Q3) φ1 ∧φ2 := φ1 ∧ φ2. Then, it is easy to see that the relation [[φ ]] = [[φ ]]o (20) Quantum Set Theory 15 holds for any observational proposition φ . Thus, all the observational propositions are embedded in the set of statements in L (∈,R(Q)) with the same projection-valued truth value. We denote by Sp(X) the spectrum of an observable X ∈ O(M ), i.e., the set of all λ ∈ R such that X − λ1 has a bounded inverse operator on H . An observable X ∈ O(M ) is called finite iff Sp(X) is a finite set, and infinite otherwise. Denote by Oω(M ) is the set of finite observables in O(M ). Let X ∈ Oω(M ). Then, Sp(X) coincides with the set of eigenvalues of X . Let δ (X) = min x,y∈Sp(X),x 6=y {|x− y|/2,1}. (21) For any x ∈ R, we define the observational proposition X =o x by X =o x := x−δ (X)< X ≤o x+δ (X). (22) Then, it is easy to see that we have [[X =o x]]o = E X ({x}) (23) for all x ∈ R. In Ref. [25] we have introduced observational propositions for the case where dim(H ) < ∞ and M = B(H ) by rules (R'1), (R'2) of well-formed formulas and rules (T'1)–(T'3) for projection-valued truth value assignment as follows. (R'1) For any X ∈ O(B(H )) and x ∈ R, the expression X =o′ x is an observational proposition. (R'2) If φ1 and φ2 are observational propositions, ¬φ1 and φ1∧φ2 are also observational propositions. (T'1) [[X =o′ x]]o′ = E X (x). (T'2) [[¬φ ]]o′ = [[φ ]]⊥o′ . (T'3) [[φ1 ∧φ2]]o′ = [[φ1]]o′ ∧ [[φ2]]o′ . Denote by Lo′(B(H )) the set of observational propositions constructed by rules (R'1) and (R'2). In this language, for any observables X ∈ O(B(H )) and any real number x ∈ R, we can introduce the observational proposition X ≤o′ x in Lo′(B(H )) by X ≤o′ x := ∨ x j∈Sp(X)∩(−∞,x] X =o′ x j, (24) 16 Masanao Ozawa where the observational proposition ∨ j φ j is defined by ∨ j φ j = φ1 ∨ * * * ∨ φn for any finite sequence of observational propositions φ1, . . . ,φn. Then, we have [[X ≤o′ x]]o′ = EX (x). (25) Now, we can conclude that if dim(H ) < ∞, the language Lo(B(H )) and Lo′(B(H )) are equivalent in the sense that there is a one-to-one correspondence Φ of Lo′(B(H )) onto Lo(B(H )) such that [[Φ(φ)]]o = [[φ ]]o′ , Φ(X =o′ x) = (X =o x), and Φ(X ≤o′ x) = (X ≤o x) for all φ ∈ Lo′(B(H )), X ∈ O(B(H )), and x ∈ R. Thus, in what follows for the case where dim(H ) < ∞ we shall identify the language Lo′(B(H )) introduce in Ref. [25] with the language Lo(B(H )); in this case we have O(B(H )) = Oω(B(H )). 5 Simultaneous determinateness In this section, we shall examine basic properties of the commutator com(X1, . . . , Xn) for observables X1, . . . ,Xn ∈ O(M ). Let X1, . . . ,Xn ∈ O(M ). We denoted by {X1, . . . ,Xn}′′ the von Neumann algebra generated by projections EX j(λ ) for all j = 1, . . . ,n and λ ∈ R, and denote by Z (X1, . . . ,Xn) the center of {X1, . . . ,Xn}′′, i.e., Z (X1, . . . ,Xn) = {X1, . . . ,Xn}′′ ∩{X1, . . . ,Xn}′. The cyclic subspace C (X1, . . . ,Xn;ρ) of H generated by X1, . . . ,Xn, and ρ is defined by C (X1, . . . ,Xn;ρ) = {X1, . . . ,Xn}′′ran(ρ), where ran stands for the closure of the range. Then, C (X1, . . . ,Xn;ρ) is the least invariant subspace under {X1, . . . ,Xn}′′ containing ρ . Denote by C(X1, . . . ,Xn;ρ) the projection of H onto C (X1, . . . ,Xn;ρ). Then, C(X1, . . . ,Xn;ρ) is the smallest projection P in {X1, . . . ,Xn}′ such that Pρ = ρ . Under the Takeuti correspondence, the commutator of observables are characterized as follows. Theorem 5.1. For any X1, . . . ,Xn ∈ O(M ), the following relations hold. (i) com(X1, . . . , Xn) = P{ψ ∈ H | [A,B]ψ = 0 for all A,B ∈ {X1, . . . ,Xn}′′}. (ii) com(X1, . . . , Xn) = P{ψ ∈ H | [EX j(r1),EXk(r2)]EXl (r3)ψ = 0 for all r1,r2,r3 ∈ Q and j,k, l = 1, . . . ,n}. (iii) com(X1, . . . , Xn) = max{E ∈ P(Z (X1, . . . ,Xn)) | X jE |◦ XkE for all j,k = 1, . . . ,n}. Quantum Set Theory 17 Proof. Let A = L(X1, . . . , Xn). Then, com(X1, . . . , Xn) = com(A ). We have L(X1, . . . , Xn) = {EX j(r j) | r j ∈ Q and j = 1, . . . ,n}∪{0,1}, and hence L(X1, . . . , Xn)′′ = {X1, . . . ,Xn}′′. Thus, relations (i) and (ii) follow from Theorem 2.8 (i) and (ii), respectively. From Theorem 2.5 we have com(X1, . . . , Xn) = max{E ∈ Z(A ) | P1 ∧E |◦ P2 ∧E for all P1,P2 ∈ A } = max{E ∈ P(Z (X1, . . . ,Xn)) | X jE |◦ XkE for all j,k = 1, . . . ,n}, from which relation (iii) follows. We say that observables X1, . . . ,Xn ∈ O(M ) are simultaneously determinate in a state ρ iff Tr[com(X1, . . . , Xn)ρ ] = 1. A probability measure μ on B(Rn) is called a joint probability distribution of X1, . . . ,Xn ∈ O(M ) in ρ ∈ S (H ) iff for any polynomial p( f1(X1), . . . , fn(Xn)) of observables f1(X1), . . . , fn(Xn), where f1, . . . , fn ∈ B(R), we have Tr[p( f1(X1), . . . , fn(Xn))ρ ] = ∫ * * * ∫ Rn p( f1(x1), . . . , fn(xn))dμ(x1, . . . ,xn). (26) A joint probability distribution of X1, . . . ,Xn in ρ is unique, if any. Since simultaneous determinateness is considered to be a state-dependent notion of commutativity, it is expected that simultaneous determinateness is equivalent to the state-dependent existence of the joint probability distribution. This is indeed shown below together with other useful characterizations of this notion. Theorem 5.2. For any observables X1, . . . ,Xn ∈ O(M ) and a state ρ ∈ S (H ), the following conditions are all equivalent. (i) X1, . . . ,Xn are simultaneously determinate in ρ , i.e., Tr[com(X1, . . . , Xn)ρ ] = 1 (ii) com(X1, . . . , Xn)ρ = ρ . (iii) C(X1, . . . ,Xn;ρ)≤ com(X1, . . . , Xn). (iv) [A,B]ρ = 0 for all A,B ∈ {X1, . . . ,Xn}′′. (v) There exists a joint probability distribution of X1, . . . ,Xn in ρ . (vi) X jC(X1, . . . ,Xn;ρ) |◦ XkC(X1, . . . ,Xn;ρ) for all j,k = 1, . . . .n. (vii) There exists a spectral measure E in M on Rn satisfying E(∆1 ×** *×∆n)ρ = EX1(∆1)∧* * *∧EXn(∆n)ρ (27) for all ∆1, . . . ,∆n ∈ B(R). 18 Masanao Ozawa (viii) There exists a probability measure μ on Rn satisfying μ(∆1 ×** *×∆n) = Tr[EX1(∆1)∧* * *∧EXn(∆n)ρ ] (28) for any ∆1, . . . ,∆n ∈ B(R). Proof. Let B = {X1, . . . ,Xn}′′ and C =C(X1, . . . ,Xn;ρ), (i)⇒(ii): The assertion follows from the relation ‖P√ρ −√ρ‖2HS = 1−Tr[Pρ ] for any projection P, where ‖* * *‖HS is the Hilbert-Schmidt norm. (ii)⇒(iii): Since com(X1, . . . , Xn) ∈ B′, (iii) follows from (ii) by minimality of C(X1, . . . ,Xn;ρ). (iii)⇒(iv): It follows from (iii) that ran(ρ)⊆ ran(com(X1, . . . , Xn)) so that (iv) follows from Theorem 5.1 (i). (iv)⇒(v): It follows from assumption (iv) and Proposition 2.2 in Ref. [11] that the GNS representation (H ,π ,Ω) of B induced by ρ is abelian (i.e., π(B) is abelian) and normal. Let j = 1, . . . ,n. Let f j be a bounded Borel function on R. By normality of π , there is a self-adjoint operator π(X j) affiliated with π(B) such that Eπ(X j)(∆) = π(EX j(∆)) for all ∆ ∈ B(R), and hence we have π( f j(X j)) = f j(π(X j)). Thus, the relation μ(∆1 ×** *×∆n) = (Ω,Eπ(X1)(∆1) * * *Eπ(Xn)(∆n)Ω), where ∆1, . . . ,∆n ∈ B(R), defines a probability measure μ on B(Rn) satisfying ∫ * * * ∫ Rn p( f1(x1), . . . , fn(xn))dμ(x1, . . . ,xn) = (Ω,π (p( f1(X1), . . . , fn(Xn)))Ω) for any polynomial p( f1(X1), . . . , fn(Xn)) of f1(X1), . . . , fn(Xn). Thus, assertion (iv) follows from the relation Tr[Aρ ] = (Ω,π(A)Ω) for any A ∈ B satisfied by the GNS representation (H ,π ,Ω). (v)⇒(i): Suppose that there exists a joint probability distribution μ of X1, * * *Xn in ρ . Then, for any j,k, l = 1, . . . ,n and r1,r2,r3 ∈ Q, we have Tr[|[EX j(r1),EX2(r2)]EXl (r3)|2ρ ] = 0 and we have [EX j(r1),EX2(r2)]EXl (r3)ρ = 0. From Theorem 5.1 (ii), it follows that com(X1, . . . , Xn)ρψ = ρψ for all ψ ∈ H . Thus, we have com(X1, . . . , Xn)ρ = ρ and hence X1, . . . ,Xn are simultaneously determinate in a state ρ . Quantum Set Theory 19 (iii)⇒(vi): Let G = com(X1, . . . ,Xn) and C =C(X1, . . . ,Xn;ρ). From Theorem 5.1 (iii), we have X jG |◦ XkG for all j,k = 1, . . . ,n. Since X jG |◦ C for all j, assertion (vi) follows from (iii). (vi)⇒(vii): Obvious. (vii)⇒(ii): Let μ be a probability measure on Rn satisfying (28). Let j,k, l ∈ {1, . . . ,n}. By taking an appropriate marginal measure of μ there exists a probability measure μ ′ on R3 such that μ ′(∆1 ×∆2 ×∆3) = Tr[EX j(∆1)∧EXk(∆2)∧EXl (∆3)ρ ] for all ∆1,∆2,∆3 ∈ B(R). Let ∆1,∆2,∆3 ∈ B(R) and P = EXl (∆3)−EXk(∆c2)∧EXl (∆3)−EX j(∆c1)∧EXk(∆2)∧EXl (∆3) −EX j(∆1)∧EXk(∆2)∧EXl (∆3), where ∆c stands for the complement of ∆ ∈ B(R). Then, by the additivity of μ ′ we have Tr[Pρ ] = μ ′(R×R×∆3)−μ ′(R×∆c2 ×∆3)−μ(∆c1 ×∆2 ×∆3)−μ(∆1 ×∆2 ×∆3) = 0. Since Tr[(P √ρ)†(P√ρ)] = Tr[Pρ ], we have P√ρ = 0, so that EX j(∆1)EXk(∆2)Pρ = 0, and hence EX j (∆1)EXk(∆2)EXl (∆3)ρ = EX j (∆1)∧EXk(∆2)∧EXl (∆3)ρ . By symmetry we also have EXk(∆2)EX j(∆1)EXl (∆3)ρ = EX j(∆1)∧EXk(∆2)∧EXl (∆3)ρ . Thus, we have [EX j(∆1),EXk(∆2)]EXl (∆3)ρψ = 0 for all ψ ∈H . Since ∆1,∆2,∆3 were arbitrary, it follows from Theorem 5.1 that ran(ρ)⊆ ran(com(X1, . . . , Xn)), and (ii) follows. The equivalence between (i) and (v) in the above theorem was previously reported in Theorem 2 of Ref. [25] for the case where M = B(H ) with H < ∞. The equivalence of (ii), (vii), and (viii) was given in Theorem 5.1 of Ref. [23] for the case n = 2. Note that for any X1, . . . ,Xn ∈ O(M ) there exists a proposition φ in Lo(M ) such that [[φ ]]o = com(X1, . . . , Xn), since com(X1, . . . , Xn) ∈ Oω(M ) and [[com(X1, . . . , Xn) =o 1]]o = com(X1, . . . , Xn). However, it is not in general possible to construct such φ from atomic propositions of the form X j ≤o λ for j = 1, . . . ,n with λ ∈ R. In what follows, we shall show that this is possible for finite observables. For any finite observables X1, . . . ,Xn ∈ Oω(M ) we define the observational proposition como(X1, . . . ,Xn) by como(X1, . . . ,Xn) := ∨ x1∈Sp(X1),...,xn∈Sp(Xn) X1 =o x1 ∧* * *∧Xn =o xn. (29) Then, we have the following theorem. 20 Masanao Ozawa Theorem 5.3. For any finite observables X1, . . . ,Xn ∈ Oω(M ), we have [[como(X1, . . . ,Xn)]]o = com(X1, . . . , Xn). (30) Proof. Let X1, . . . ,Xn ∈ Oω(H ). Let x(1)j < * * *< x (n j) j ∈ R be the ascending sequence of eigenvalues of X j. Then, we have L(X1, . . . , Xn) = {EX j(x) | x = x(1)j , . . . ,x (n j) j ; j = 1, . . . ,n}∪{0}. Since L(X1, . . . ,Xn) is a finite set, it is easy to see that the relations [[como(X1, . . . ,Xn)]]o = com({EX j({x}) | x = x(1)j , . . . ,x (n j) j ; j = 1, . . . ,n}) = com(L(X1, . . . , Xn)) = com(X1, . . . , Xn) hold. The observational proposition como(X1, . . . ,Xn) was previously introduced in Ref. [25] for the case where M = B(H ) and dim(H ) < ∞. The following theorem is a straightforward generalization of Theorem 1 in Ref. [25]. Theorem 5.4. Finite observables X1, . . . ,Xn ∈Oω(M ) are simultaneously determinate in a vector state ψ if and only if the state ψ is a superposition of common eigenvectors of X1, . . . ,Xn. 6 Quantum equality In this section, we shall examine basic properties of the Q-valued equality relation [[X = Ỹ ]] defined through V (Q) for any two observables X ,Y ∈ O(M ), where Q = P(M ). From Theorem 6.3 of Ref. [24], we have the following characterizations. Theorem 6.1. For any X ,Y ∈ O(M ), the following relations hold. (i) [[X = Ỹ ]] = P{ψ ∈ H | EX (r)ψ = EY (r)ψ for all r ∈ Q}. (ii) [[X = Ỹ ]] = P{ψ ∈ H | f (X)ψ = f (Y )ψ for all f ∈ B(R)}. (iii) [[X = Ỹ ]] = P{ψ ∈ H | (EX (∆)ψ,EY (Γ)ψ) = 0 for any ∆,Γ ∈ B(R) with ∆∩Γ = /0}. We introduce a new atomic observational proposition X =o Y in Lo(M ) for all X ,Y ∈ O(M ) by the following additional rules for formation of observational propositions and for projection-valued truth values: Quantum Set Theory 21 (R3) For any X ,Y ∈ O(M ), the expression X =o Y is an observational proposition. (T4) [[X =o Y ]]o = [[X = Ỹ ]]. We extend the correspondence between observational propositions and formulas in L (∈,V (Q)) by the following rule for any X ,Y ∈ O(M ). (Q4) X =o Y := X = Ỹ . Then, from (T4) it is easy to see that the relation [[φ ]] = [[φ ]]o (31) holds for any observational proposition φ . We denote by Lo(M ,=) the set of observational propositions constructed by rules (R1), (R2), (R3). Then, the language Lo(M ,=) is embedded in the set of statements in L (∈,R(Q)) by rules (Q1), (Q2), (Q3), (Q4) with the same projection-valued truth value by rules (T1), (T2), (T3), (T4). In general, the equality relation in V (Q) is not an equivalence relation in V (Q) [32]. From Theorem 6.3 of Ref. [24], however, we conclude that that Q-valued equality between two observables is indeed a Q-valued equivalence relation as follows. Theorem 6.2. For any observables X ,Y,Z ∈ O(M ), the following relations hold. (i) [[X =o X ]]o = 1. (ii) [[X =o Y ]]o = [[Y =o X ]]o. (iii) [[X =o Y ]]o ∧ [[Y =o Z]]o ≤ [[X =o Z]]o. We say that observables X and Y are equal in a state ρ , in symbols X =ρ Y , iff Pr{X =o Y‖ρ} = 1, or equivalently iff [[X =o Y ]]oρ = ρ . In general, we say that observables X and Y are equal in a state ρ with probability Pr{X =o Y‖ρ}. On the other hand, we have explored another relation called quantum perfect correlation in Ref. [23] as follows. Two observables X and Y are called perfectly correlated in a state ρ iff Tr[EX(∆)EY (Γ)ρ ] = 0 for any disjoint Borel sets ∆,Γ ∈B(R). It is noted that the quantity Tr[EX (∆)EY (Γ)ρ ] = 0 for ∆,Γ ∈ B(R) is called the weak joint distribution of X and Y in ρ , and known to be experimentally accessible by weak measurement and post-selection [26]. We shall show that the above two relations are equivalent together with other equivalent conditions to conclude that the relation X =ρ Y and the probability Pr{X =o Y‖ρ} are experimentally accessible. Theorem 6.3. For any observables X ,Y ∈O(M ) and ρ ∈S (H ), the following conditions are all equivalent. 22 Masanao Ozawa (i) X =ρ Y , i.e., [[X =o Y ]]oρ = ρ . (ii) X and Y are perfectly correlated in ρ , i.e., Tr[EX (∆)EY (Γ)ρ ] = 0 for all ∆,Γ ∈ B(R) with ∆∩Γ = /0. (iii) Xψ = Y ψ for all ψ ∈ C (X ,ρ). (iv) 〈 ψ,EX (∆)ψ 〉 = 〈 ψ,EY (∆)ψ 〉 for all ψ ∈ C (X ,ρ). (v) EX (∆)ρ = EY (∆)ρ for all ∆ ∈ B(R). (vi) f (X)C(X ;ρ) = f (Y )C(X ;ρ) for all f ∈ B(R). (vii) C(X ;ρ) =C(Y ;ρ) and XC(X ;ρ) = YC(X ;ρ). (viii) There exists a joint probability distribution μX ,Yρ (x,y) of X ,Y in ρ that satisfies μX ,Yρ ({(x,y) ∈ R2 | x = y}) = 1. (32) Proof. The assertions follow from Theorem 6.1 above and Theorems 3.2, Theorem 3.4, Theorem 4.3, and Theorem 5.3 in Ref. [23]. The equivalence between (i) and (viii) was previously reported in Theorem 4 in Ref. [25] for the case where H < ∞ and M = B(H ). Let φ(X1, . . . ,Xn) be an observational proposition that is constructed by rules (R1), (R2), (R3) and includes symbols for observables only from the list X1, . . . ,Xn, i.e., φ(X1, . . . ,Xn) includes only atomic observational propositions of the form X j ≤o x j or X j = Xk, where j,k = 1, . . . ,n and x j is the symbol for an arbitrary real number. In this case, φ(X1, . . . ,Xn) is said to be an observational proposition in Lo(X1, . . . ,Xn). Then, φ(X1, . . . ,Xn) is said to be contextually well-formed in a state ρ iff X1, . . . ,Xn are simultaneously determinate in ρ . The following theorem answers the question as to in what state ρ the probability assignment satisfies rules for calculus of classical probability, and shows that for well-formed observational propositions φ(X1, . . . ,Xn) the projection-valued truth value assignment satisfies Boolean inference rules. Theorem 6.4. Let φ(X1, . . . ,Xn) be an observational proposition in Lo(X1, . . . ,Xn). If φ(X1, . . . ,Xn) is a tautology in classical logic, then we have com(X1, . . . , Xn)≤ [[φ(X1, . . . ,Xn)]]o. Moreover, if φ(X1, . . . ,Xn) is contextually well-formed in a state ρ , then φ(X1, . . . ,Xn) holds in ρ . Proof. Suppose that an observational proposition φ = φ(X1, . . . ,Xn) is a tautology in classical logic. Let φ be the corresponding formula in L (∈,V (Q)). Then, it is easy Quantum Set Theory 23 to see that there is a formula φ0(u1, . . . ,un,v1, . . . ,vm) in L (V (Q)) provable in ZFC satisfying φ0(X1, . . . , Xn, r1, . . . , rm) = φ with some real numbers r1, . . . ,rm. Then, by the ZFC Transfer Principle (Theorem 2.11), we have com(X1, . . . , Xn)≤ [[φ ]]. Thus, the assertion follows from relation (31). The above theorem was previously announced as Theorem 3 in Ref. [25] for the case where H < ∞ and M = B(H ), the proof of which needs quantum set theory and the embedding of the language of observational propositions into the language of quantum set theory developed in this paper. Note that for any X ,Y ∈ O(M ) there exists a proposition φ in Lo(M ) such that [[φ ]]o = [[X = Ỹ ]]. In fact, we have [[X = Ỹ ]]∈Oω(M ) and [[[[X = Ỹ ]] =o 1]] = [[X = Ỹ ]]. However, it is not in general possible to construct such φ from atomic propositions of the form X j ≤o λ for j = 1, . . . ,n with λ ∈ R. In what follows, we shall show that this is possible for finite observables. For any finite observables X ,Y , we define the observational proposition X = Y by X =o Y := ∨ x∈Sp(X) X =o x∧Y =o x. (33) Then, we have the following. Theorem 6.5. For any finite observables X ,Y ∈ Oω(H ), we have [[X =o Y ]]o = [[X = Ỹ ]]. (34) Proof. Let ψ ∈ R([[X = Y ]]o). Then, for any x ∈ Sp(X), we have EX ({x})ψ = EX ({x})∩EY ({x})ψ = EY ({x})ψ, and for any x 6∈ Sp(X), we have EX ({x})ψ = 0 = EY ({x})ψ. Thus, ψ ∈ R([[X = Ỹ ]]) follows from Theorem 6.1 (i). Conversely, suppose ψ ∈ R([[X = Ỹ ]]). Then, for all x ∈ R, we have EX ({x})ψ = EY ({x})ψ so that we have EX ({x})∧EY ({x})ψ = EX ({x})ψ . Thus, we have [[X = Y ]]oψ = ψ . Therefore, the assertion follows. As shown in Ref. [25] for the finite dimensional case, state-dependent equality between finite observables are generally characterized in terms of eigenvectors as follows. Theorem 6.6. Finite observables X and Y are equal in a vector state ψ if and only if the state ψ is a superposition of common eigenvectors of X and Y with common eigenvalues. 24 Masanao Ozawa 7 Measurements of observables In this and next sections, we shall discuss measurements for a quantum system described by a von Neumann algebra M on a Hilbert space H . A probability operator-valued measure (POVM) for a von Neumann algebra M on Rn is a mapping Π : B(Rn)→ M satisfying the following conditions. (M1) Π(∆)≥ 0 for all ∆ ∈ B(Rn). (M2) ∑ j Π(∆ j) = 1 for any disjoint sequence of Borel sets ∆1,∆2, . . . ∈ B(Rn) such that Rn = ⋃ j ∆ j . A measuring process for M is defined to be a quadruple (K ,σ ,U,M) consisting of a Hilbert space K , a state (density operator) σ on K , a unitary operator U on H ⊗K , and an observable M on K satisfying TrK [U †(X ⊗EM(∆))U(1⊗σ)]∈ M (35) for every X ∈M and ∆ ∈B(R), where TrK stands for the partial trace on K [20, 16]. A measuring process M(x)= (K ,σ ,U,M)with output variable x describes a measurement carried out by an interaction, called the measuring interaction, from time 0 to time ∆t between the measured system S described by M and the probe system P described by B(K ) that is prepared in the state σ at time 0. The outcome of this measurement is obtained by measuring the observable M, called the meter observable, in the probe at time ∆t. The unitary operator U describes the time evolution of S+P from time 0 to ∆t. We shall write M(0) = 1⊗M, M(∆t) =U†M(0)U , X(0) = X ⊗1, and X(∆t) = U†X(0)U for any observable X ∈ O(M ). We can use the probabilistic interpretation for the system S+P. The output distribution Pr{x ∈ ∆‖ρ}, the probability distribution of the output variable x of this measurement on input state ρ ∈S (H ), is naturally defined as Pr{x ∈ ∆‖ρ}= Pr{M(∆t) ∈ ∆‖ρ ⊗σ}= Tr[EM(∆t)(∆)ρ ⊗σ ]. The POVM of the measuring process M(x) is defined by Π(∆) = TrK [EM(∆t)(∆)(I⊗σ)]. Then, Π(∆) ∈ M for all ∆ ∈ B(R) by Eq. (35) and Π : B(R) → M is a POVM for M on R satisfying (M3) Pr{x ∈ ∆‖ρ}= Tr[Π(∆)ρ ]. Quantum Set Theory 25 Conversely, from a general result in Ref. [20] it can be easily seen that for every POVM Π for M on R, there is a measuring process M(x) = (K ,σ ,U,M) for M satisfying (P3). In fact, for any fixed ρ0 ∈ S (H ) the relation I (∆)∗X = Tr[Xρ0]Π(∆) for all X ∈ M and ∆ ∈ B(R) defines a completely positive instrument for B(H ) on R, and by Theorem 5.1 in Ref. [20] there exists a measuring process M(x) = (K ,σ ,U,M) for B(H ) such that Tr[Xρ0]Π(∆) = TrK [U†(X ⊗EM(∆))U(1⊗σ)] for all X ∈ M and ∆ ∈ B(R). Then, it is easy to see that M(x) is a measuring process for M and satisfies (P3). For further accounts of the universality of the class of measurement models described by measuring processes we refer the reader to Ref. [20] for quantum systems with finite degrees of freedom and to Ref. [16] for those with infinite degrees of freedom. Let A ∈ O(M ) and ρ ∈ S (H ). A measuring process M(x) = (K ,σ ,U,M) for M with the POVM Π is said to measure A in ρ if A(0) =ρ⊗σ M(∆t), and weakly measure A in ρ iff Tr[Π(∆)EA(Γ)ρ ] = Tr[EA(∆∩Γ)ρ ] for any ∆,Γ ∈ B(R). A measuring process M(x) is said to satisfy the Born statistical formula (BSF) for A in ρ iff it satisfies Pr{x ∈ ∆‖ρ} = Tr[EA(∆)ρ ] for all x ∈ R. The following theorem characterizes measurements of an observable in a given state [23]. Theorem 7.1. Let M(x) = (K ,σ ,U,M) be a measuring process for M with the POVM Π(∆). For any observable A ∈ O(M ) and any state ρ ∈S (H ), the following conditions are all equivalent. (i) M(x) measures A in ρ . (ii) M(x) weakly measures A in ρ . (iii) M(x) satisfies the BSF for A in any vector state ψ ∈ C (A,ρ). In the conventional approach, a measuring process M(x) = (K ,σ ,U,M) with the POVM Π is considered to be a measurement of an observable A iff Π = EA [20], since in this case the probability distribution of A predicted by the Born formula is reproduced by the probability distribution of Π in any state. However, in this approach it is not clear whether a measurement of an observable A actually reproduces the value of the observable A just before the measurement. The following theorem, which is an immediate consequence of Theorem 7.1, ensures that this is indeed the case (cf. the remark after Theorem 8.2 in Ref. [23]). Theorem 7.2. Let M(x) = (K ,σ ,U,M) be a measuring process for M with the POVM Π. Then, M(x) measures A ∈ O(M ) in any ρ ∈ S (H ) if and only if Π = EA for all x ∈ R. 26 Masanao Ozawa 8 Simultaneous measurability For any measuring process M(x) = (K ,σ ,U,M) for M and a real-valued Borel function f , the measuring process M( f (x)) with output variable f (x) is defined by M( f (x)) = (K ,σ ,U, f (M)). Observables A,B are said to be simultaneously measurable in a state ρ ∈S (H ) by M(x) iff there are Borel functions f ,g such that M( f (x)) and M(g(x)) measure A and B in ρ , respectively. Observables A,B are said to be simultaneously measurable in ρ iff there is a measuring process M(x) such that A and B are simultaneously measurable in ρ by M(x). Simultaneous measurability and simultaneous determinateness are not equivalent notions under the state-dependent formulation, as the following theorem clarifies; the case where dim(H )< ∞ was previously reported in Ref. [25], Theorem10. Theorem 8.1. (i) Two observables A,B ∈O(M ) are simultaneously determinate in a state ρ ∈ S (H ) if and only if there exists a POVM Π for M on R2 satisfying Π(∆×R) = EA(∆) on C (A,B,ρ) for all ∆ ∈ R, (36) Π(R×Γ) = EB(Γ) on C (A,B,ρ) for all Γ ∈ R. (37) (ii) Two observables A,B ∈ O(M ) are simultaneously measurable in a state ρ ∈ S (H ) if and only if there exists a POVM Π for M on R2 satisfying Π(∆×R) = EA(∆) on C (A,ρ) for all ∆ ∈ R, (38) Π(R×Γ) = EB(Γ) on C (B,ρ) for all Γ ∈ R. (39) (iii) Two observables A,B ∈ O(M ) are simultaneously measurable in a state ρ ∈ S (H ) if they are simultaneously determinate in ρ . Proof. Let C = C (A,B,ρ) and C =C(A,B;ρ). (i) (only if part): Let G= com(Ã, B). Then, G∈M and AG |◦ BG. Let Π be the joint spectral measure of AG and BG, i.e., Π(∆×Γ) = EAG(∆)EBG(Γ) for all ∆,Γ ∈ B(R). Then, Π is a POVM for M on R2. Suppose that A and B are simultaneously determinate in a state ρ . Then, ran(ρ) ⊆ ran(com(Ã, B)). By the minimality of C(A,B,ρ) among (A,B)-invariant subspaces, we have C ≤ G and AG,BG |◦ C. Thus, we have Π(∆×R)C = EAG(∆)C = EAC(∆)C = EA(∆)C and similarly Π(R×Γ)C = EB(Γ)C for all ∆,Γ ∈ B(R). Thus, Π satisfies Eqs. (36) and (37). (i) (if part): Let Π be a POVM for M on R2 satisfying (36) and (37). Let Π′ be a positive operator valued measure for B(H ) on R2 defined by Π′(∆×Γ) = CΠ(∆× Quantum Set Theory 27 Γ)C for all ∆,Γ∈B(R). Let Π′′ be a POVM for B(C ) on R2 obtained by restricting Π′ to C . Let ∆,Γ∈B(R). By the definition of C, we have EA(∆)C =CEA(∆) =CEA(∆)C and EA(∆)C is a projection. Similarly, EB(Γ)C = CEB(Γ) = CEB(Γ)C and EB(Γ)C is a projection. Thus, we have Π′′(∆×R) = CΠ(∆×R)C = EA(∆)C, and similarly Π′′(R×Γ) =CΠ(R×Γ)C = EB(Γ)C. Since ∆ and Γ were arbitrary, the marginals of Π′′ are projection-valued. By a well-know theorem (e.g., Ref. [8], Theorem 3.2.1), the marginals commute and Π′′ is the product of their marginals. Thus, we have AC |◦ BC, and hence by Theorem 5.2, A and B are simultaneously determinate. (ii) (only if part): Suppose that A,B ∈O(M ) are simultaneously measurable in ρ ∈ S (H ). Then, we have a measuring process M(x) = (K ,σ ,U,M) for M and realvalued Borel functions f ,g such that M( f (x)) measures A in ρ and M(g(x)) measures B in ρ . Let Π0 be the POVM of M(x). Let Π be a POVM on R2 such that Π(∆×Γ) = Π0( f−1(∆)∩g−1(Γ)). Then, it is easy to see that Π satisfies Eqs. (38) and (39). (ii) (if part) Let Π be a POVM for H on R2 satisfying Eqs. (38) and (39). Then, by the remark after condition (M3) in Section 7 there exists a measuring process M(x) = (K ,σ ,U,M) for M and real-valued Borel functions f ,g such that Π(∆×Γ) = TrK [U†(I ⊗E f (M)(∆)Eg(M)(Γ))U(I⊗σ)]. (40) Then, we have Π(∆×R) = TrK [U†(I ⊗E f (M)(∆))U(I⊗σ)], (41) so that from Eq. (38) we have M( f (x)) measures A in ρ . Similarly, we can show that M(g(x)) measures B in ρ . Assertion (iii) follows from (i) and (ii). Discussions on physical significance of the state-dependent formulation of simultaneous measurability have been given in Ref. [25] for the finite dimensional case. Further discussions on the state-dependent formulation of quantum measurement theory will appear elsewhere. 9 Conclusion Quantum set theory originated from the method of forcing introduced by Cohen [6, 7] for the independence proof of the continuum hypothesis and from quantum logic introduced by Birkhoff and von Neumann [3] for logical axiomatization of quantum mechanics. After Cohen's work, Scott and Solovay [30] reformulated the forcing method by Boolean-valued models of set theory [1], which have become a central method in 28 Masanao Ozawa the field of axiomatic set theory. In 1978 Takeuti [31] started Boolean-valued analysis, which provides systematic applications of logical meta-theorems for Booleanvalued models to not meta-mathematical problems mainly in analysis. Among others, Boolean-valued analysis made a great successes in operator algebras [34, 33, 18] and especially in solving a long-standing open problem in the structure theory of type I algebras applying the forcing method for cardinal collapsing [17, 19, 21]. As a successor of those attempts, quantum set theory, a set theory based on the Birkhoff-von Neumann quantum logic, was introduced by Takeuti [32], who established the one-to-one correspondence between reals in the model (quantum reals) and quantum observables. Quantum set theory was recently developed by the present author [24, 27] to obtain the transfer principle to determine quantum truth values of theorems of the ZFC set theory, and to clarify the operational meaning of equality between quantum reals, which extends the probabilistic interpretation of quantum theory, To formulate the standard probabilistic interpretation of quantum theory, we have introduced the language of observational propositions with rules (R1) and (R2) for well-formed formulas constructed from atomic formulas of the form X ≤o x, rules (T1), (T2), and (T3) for projection-valued truth value assignment, and rule (P1) for probability assignment. Then, the standard probabilistic interpretation gives the statistical predictions for standard observational propositions formulated by (W1), which concern only a commuting family of observables. The Born statistical formula is naturally derived in this way. We have extended the standard interpretation by introducing the notion of simultaneous determinateness and atomic formulas of the form X =Y for equality. To extended observational propositions formed through rules (R1), . . ., (R4), the projection-valued truth values are assigned by rule (T1), . . ., (T4), and the probabilities are assigned by rule (P1). Then, we can naturally extend the standard interpretation to a general and state-dependent interpretation for observational propositions including the relations of simultaneous determinateness and equality. Quantum set theory ensures that any contextually well-formed formula provable in ZFC has the probability assigned to be 1. This extends the classical inference for quantum theoretical predictions from commuting observables to simultaneously determinate observables. We apply this new interpretation to construct a theory of measurement of observables in the state-dependent approach, to which the standard interpretation cannot apply. 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