Denoted by M(A),QM(A)and SQM(A)the sets of all measures,quantum measures and subadditive quantum measures on a σ-algebra A,respectively.We observe that these sets are all positive cones in the real vector space F(A)of all real-valued functions on A and prove that M(A)is a face of SQM(A).It is proved that the product of m grade-1 measures is a grade-m measure.By combining a matrix Mμto a quantum measureμon the power set An of an n-element set X,it is proved thatμν(resp. μ⊥ν)if and only if μν M M(resp.MμMv=0).Also,it is shown that two nontrivial measuresμandνare mutually absolutely continuous if and only ifμ·ν∈QM(An).Moreover,the matrices corresponding to quantum measures are characterized. Finally,convergence of a sequence of quantum measures on An is introduced and discussed;especially,the Vitali-Hahn-Saks theorem for quantum measures is proved.
Denoted by M(A),QM(A)and SQM(A)the sets of all measures,quantum measures and subadditive quantum measures on a σ-algebra A,respectively.We observe that these sets are all positive cones in the real vector space F(A)of all real-valued functions on A and prove that M(A)is a face of SQM(A).It is proved that the product of m grade-1 measures is a grade-m measure.By combining a matrix Mμto a quantum measureμon the power set An of an n-element set X,it is proved thatμν(resp. μ⊥ν)if and only if μν M M(resp.MμMv=0).Also,it is shown that two nontrivial measuresμandνare mutually absolutely continuous if and only ifμ·ν∈QM(An).Moreover,the matrices corresponding to quantum measures are characterized. Finally,convergence of a sequence of quantum measures on An is introduced and discussed;especially,the Vitali-Hahn-Saks theorem for quantum measures is proved.
Denoted by M(A), QM(A) and SQM(A) the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F(A) of all real-valued functions on A and prove that M(A) is a face of SQM(A). It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix M µ to a quantum measure µ on the power set A n of an n-element set X, it is proved that µ ≪ ν (resp. µ ⊥ ν) if and only if M µ ≪ M ν (resp. M µ M ν =0). Also, it is shown that two nontrivial measures µ and ν are mutually absolutely continuous if and only if µ·ν∈QM(A n ). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on A n is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved.
Grade-d measures on a σ-algebra A⊆2X over a set X are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum mechanics. Every signed polymeasure λ on (X,A)d produces a grade-d measure as its diagonal λ˜(A):=λ(A,…,A), and we prove that as soon as d≥2, measures (as opposed to polymeasures) do not suffice: the separate σ-additivity of a λ producing μ=λ˜ cannot, generally, be amplified to global σ-additivity. This amends a result in the literature, asserting the contrary in case d=2.
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.
Let F : ℬ(ℝ)→ℱ(ℋ) be a commutative positive operator-valued measure from the Borel σ-algebra of the reals to the space of bounded positive operators on the Hilbert space ℋ. It is well known that there exist a self-adjoint operator A and a Markov kernel μ(·)(λ) such that F(Δ) = μΔ(A), where Δ ∈ ℬ(ℝ). We prove that, for any Δ ∈ ℬ(ℝ), if F(Δ) is a discrete operator, then A is a linear combination of the moments of F. This result allows us to characterize the quantum observable represented by A by means of the Naimark dilation of F. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals {Ti}i ∈ ℕ, such that Tif = ∫ f(t) dμt(i), corresponding to an infinite sequence of probability measures {μ(·)(i)}i ∈ ℕ on the Borel σ-algebra ℬ([0, 1]) such that μ(·)(i) ≠ μ(·)(j), where i, j ∈ ℕ and i≠j. There exists a real bounded one-to-one continuous function f such that T i f = ∫ f ( t ) d μ t ( i ) ≠ ∫ f ( t ) d μ t ( j ) = T j f , i , j ∊ ℕ , i ≠ j .
We show that quantum measures and integrals appear naturally in any L2-Hilbert space H. We begin by defining a decoherence operator D(A, B) and it’s associated q-measure operator µ(A) = D(A, A) on H. We show that these operators have certain positivity, additivity and continuity properties. If ρ is a state on H, then Dρ(A, B) = tr [ρD(A, B)] and µρ(A) = Dρ(A, A) have the usual properties of a decoherence functional and q-measure, respectively. The quantization of a random variable f is defined to be a certain self-adjoint operator fb on H. Continuity and additivity properties of the map f 7→ fb are discussed. It is shown that if f is nonnegative, then fb is a positive operator. A quantum integral is de- fined by R f dµρ = tr(ρfb). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.
I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.