Representation of quantum propositions in the quantale valued universe

Satoko Titani

                         titani@isc.chubu.ac.jp

Chubu University

1 Introduction

The lattice P (H) consisting of all closed subspaces of a Hilbert space H is a complete orthomodular lattice. P (H) is atomic and has the covering property. A complete orthomodular lattice which is atomic and has covering property is called a proposition system, whose elements are called propositions. A pure state is represented as a subspace of dimension 1, and

“Proposition p is certainly true in state x” means x ≤ p.

Quantum phenomenon is described as results of observations. For a propositional system P (H) the set of observations that we can perform on it has the struture of a unital quantale whose multiplication means “and then”, whose unit is the null observation and whose joins are disjunctions. A proposition system and quantale on it form a quantum system.

Let Q = Q(P (H)) be a quantale on P (H), consisting of sup-preserving mapping P (H) → P (H). P (H) is embedded in the quantale Q as the sub-quantale L(Q) of left-sided elements, which is a right Q-module. Then, propositions, pure states and symmetries are all represented as elements of Q. We consider the quantale Q as logic, and Q-valued set theory based on the logic. It is known that for any complete lattice L we can construct L-vaued universe V^{L} and develope a L-valued set theory on it (cf. [3]). Observables are represented as real numbers defined in V^{P (H)} (cf. [2]). It follows that propositions and states are represented as real numbers, and symmetries are represented as complex numbers in the quantale valued universe V^{Q}.

2 A quantum system (P (H), Q)

We assume through this paper that H is a separable complex Hilbert space, P (H) is the complete orthomodular lattice consisting of all closed subspaces of H, or equivalently, of all projections on H. A pure state of the physical system is a projection ∈ P (H) of rank 1. An observable is defined as a correspondence between the propositions associated with the measuring apparatus and those associated with the physical system. Thus, an observable is a mapping of a complete Boolean algebra consisting of Borel sets of real numbers, into the proposition sytem, which is considered as a self-adjoint operator on H.

Definition 2.1 A sup-lattice (complete lattice) Q with an associative binary operation & is called a quantale, if the operation & is distributive over arbitrary joins:

a& ⋁ S = ⋁{a&s | s∈S},  (⋁ S)&a = ⋁{s&a | s∈S},  for every a∈Q and S ⊂Q.

Let Q = Q(P (H)) be the set of all ⋁-preserving mapping P (H) → P (H). Q forms a quantale, where

φ ⩽ ψ \underleftrightarrow{def} φ(x) ⩽ ψ(x) for all x∈P (H)
(⋁_{α} φ_{α})(x) {def \atop =} ⋁_{α} φ_{α}(x) for all x∈P (H)

(φ&ψ)(x) {def \atop =} ψ(φ(x)) for all x∈P (H)

the pair (P (H), Q) is called a quantum system.

An element a of a quantale Q is said to be left-sided if 1&a ≤ a. The set L(Q) of left-sided elements of Q, L(Q) = {a∈Q | 1&a ≤ a}, is a subquantale of Q.

Theorem 2.1 ([1]) For each x ∈ P (H) we define κ_{x} by κ_{x}(y) = \begin{cases}0, & if y = 0, \\ x, & otherwise. \end{cases}. κ_{x} is in L(Q),

and P (H) is embedded in the quantale Q as L(Q), by x ↦ κ_{x}. We consider P (H) as L(Q). That is, P (H) = L(Q)⊂Q. P (H) is a Q-module.

Lemma 2.1 For x∈P (H) and a∈Q, κ_{a(x)} = κ_{x}&a.

3 Q-valued universe V^{Q}

Let (P (H), Q) be a quantum system. Q-valued universe V^{Q} is constructed by induction.

V_{α}^{Q} = {u | ∃β <α ∃Du⊂V_{β}^{Q}(u : Du → Q)} ; V^{Q} = ⋃_{α∈On} V_{α}^{Q}

Similarly, V^{P (H)} ⊂ V^{Q} is constructed. The formulas on the universe V^{P (H)} are constructed from atomic formulas of the form u = v, or u ∈ v, by logical operators ∧, ∨, ¬, &, ^{∗}, ^{⊥}, ∀, ∃, and by implications

→ interpreted as : (a → b) = \begin{cases}1 & if a ≤ b \\ 0 & otherwise \end{cases}. The formulas on the universe V^{Q} are constructed from

atomic formulas of the form u = v, or u ∈ v, and formulas on V^{P (H)} by ∧, ∨, ¬, &, ^{∗}, ∀, ∃, →. We call the set theory on the Q-valued universe V^{Q} quantale valued set theory. We also define an implication →_{q}

called quantum implication on P (H). If φ and ψ are formulas on V^{P (H)}, then φ →_{q} ψ is defined as

φ →_{q} ψ \underleftrightarrow{def} φ^{⊥} ∨ (φ ∧ ψ).

Truth value of a sentence φ is denoted by [[φ]]. Atomic formulas u = v and u ∈ v where u, v ∈ V^{Q} are interpreted by induction on the rank of u, v as follows :

[[u=v]] = ⋀_{x∈Du}(u(x) → [[x∈v]]) ∧ ⋀_{x∈Dv}(v(x) → [[x∈u]]) ; [[u∈v]] = ⋁_{x∈Dv} v(x) ∧ [[u=x]]

V^{Q} is a model of lattice valued set theory (cf. [3]). The equality =_{q} and the membership relation ∈_{q}

corresponding to the quantum implication →_{q} is defined by

u =_{q} v \underleftrightarrow{def} ∀x(x∈_{q}u →_{q} x∈_{q}v) ∧ ∀x(x∈_{q}v →_{q} x∈_{q}u) ; u∈_{q}v \underleftrightarrow{def} ∃x(x =_{q} u ∧ x∈v).

3.1 Real and complex numbers in V^{Q}

We denote the set of all rational numbers, real numbers, and complex numbers by Q, R, and C, respectively. It is known that the set of rational numbers defined in V^{Q} coincides with the check set Q̌ corresponding to Q. We denote the check set corresponding to a set x by x̌.

A real number in V^{Q} is defined as an element of V^{P (H)} which is an upper part u of Dedekind cut of rational numbers. That is, u∈V^{Q} is called a real number if (0) ∧ (1) ∧ (2) ∧ (3), where

(0) ∀x∈Q(1&(x∈u) → (x∈u))

(1) u ⊂ Q̌, that is, ∀x(x∈u → x∈Q̌),

(2) ∃r ∈Q̌(r ∈u) ∧ ∃r ∈Q̌(r ∈u)^{⊥},

(3) ∀r ∈Q̌ ((r ∈u) ⟷ ∀s∈Q̌(r <s →_{q} s∈u)) .

Let u, v ∈V^{Q} are real number in V^{Q}, that is, [[u, v are reals ]] = 1. u, v are said to be compatible, if there exists a Boolean sub-algebra B of P (H) such that u, v ∈V^{B}. If u, v are compatible real numbers in V^{Q}, then u · v and u + v are defined ; and u + iv is called a complex , where i^{2} = −ˇ1.

Lemma 3.1 For each p∈P (H), we define p̂∈V^{Q} such that p̂⊂Q̌ by p̂(ř) = \begin{cases}0, & if r <0 \\ p^{⊥}, & if 0 ≤ r <1 \\ 1, & if 1 ≤ r \end{cases}.

Then [[p̂ =_{q} ˇ1]] = [[p̂ =_{q} ˇ0]]^{⊥} = p. Conversely, if u ∈ V^{Q} satisfies [[u ⊂ Q̌]] = 1 and [[u =_{q} ˇ1]] = [[u =_{q} ˇ0]]^{⊥} = p,

then [[u = p̂]] = 1.

Thus, we define proposition in V^{Q} as a real number u satisfying (u =_{q} ˇ1) ↔ (u =_{q} ˇ0)^{⊥}. A proposition p

is called a pure state if ∀r ∈ Prop(0 < r ≤ p → r = p). Propositions u, v are said to be orthogonal if (u =_{q} ˇ1) ↔ (v =_{q} ˇ1)^{⊥}. A positive real ρ is said to be a state, if there exist mutually orthogonal pure states

p̂_{1}, p̂_{2}, p̂_{3}, ⋯, and positive check reals μ_{1}, μ_{2}, μ_{3}, ⋯ with sum equal to ˇ1 such that ρ = ∑_{n=1}^{∞} μ_{n}p̂_{n}. The set of all states is denoted by St.

It is known that if we carry out a measurement corresponding to the proposition b in initial state ρ, then the probability of obtaining the answer “yes” is the trace of linear operator b ρ on H. It follows that there exists a function w_{ρ} : Prop → [ˇ0, ˇ1] ⊂ Ř, which associates the probability to b̂ such that

1. w_{ρ}(ˆ1) = ˇ1,

2. If {b̂_{n}} (n = 1, 2, ⋯) is a set of mutually orthogonal projections, then w_{ρ}(⋁_{n=1}^{∞} b̂_{n}) = ∑_{n=1}^{∞} w_{ρ}(b̂_{n})

3. If w_{ρ}(b̂) = w_{ρ}(ĉ) = 0 then w_{ρ}(b̂ ∨ ĉ) = 0.

If we obtain the answer “yes”, then the state of the system immediately after the experiment is (ρ∨b^{⊥})∧b, which is a pure state. We denote ((ρ ∨ b^{⊥}) ∧ b)ˆ by φ_{b}(ρ̂). Now we can define the product ρ̂ · b̂ of reals ρ̂ and b̂ as w_{ρ}(b̂) · φ_{b}(ρ̂). w_{ρ}(b̂)φ_{b}(ρ̂) + w_{ρ}(b̂^{⊥})φ_{b^{⊥}}(ρ̂) is a state, and we consider that the measurement correspondng to the proposition b is the mapping on St given by

ρ̂ ↦ w_{ρ}(b)φ_{b}(ρ̂) + w_{ρ}(b̂^{⊥})φ_{b^{⊥}}(ρ̂).

A symmetry is an automorphism of the orthomodular lattice of propositions, which is induced by a unitery operator on H. Let u be a unitary operator on H. u is a normal operator, and is represented as a complex in V^{Q}. Let ũ be mapping on P (H) defined by ũ : p ↦ upu^{∗}. Since ũ is an automorphism on P (H), ũ is an element of Q, and induces an automorphism σ_{u} on St. We call σ_{u} the symmetry associated with complex number u in V^{Q}.

References

[1] C.J.Mulvey and J.Wick Pelletier, Quantization of the Calculus of Relations, Canadian Mathematical Society, Conference Proceeding, Vol. 13, 1992.

[2] G.Takeuti, Two Applications of Logic to Mathematics, Iwanami and Princeton University Press, Tokyo and Princeton (1978).

[3] S.Titani, Lattice Valued Set Theory, Arch. Math. Logic 38-6(1999) pp.395-421