In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B X constitute a convex set whose extreme points are degenerate probability measures.
We restrict ourselves to finite level systems. Elementary classical probability theory is woven around the notions of a finite sample space or an alphabet A, events which are subsets of A, random variables which are real or complex-valued functions on A and probability distributions which are, once again, functions on A with values in the unit interval adding to unity. An event E ⊂ A can be equivalently replaced by the two-valued indicator function 1 E which is one on the set E and zero on the complement E′ of E in A. Thus all the basic objects, namely, events, random variables and distributions are functions on A. Complex-valued functions on A constitute a commutative algebra A A with complex conjugation as an involution. Thus an event can be identified as an element f ∈ A A satisfying f = f2 = f̄ where the bar indicates complex conjugation. A real-valued random variable is an element f ∈ A A satisfying f = f̄ whereas a probability distribution is an element p ∈ A A satisfying p(x) ≥ 0 ∀ x ∈ A and ∑ x∈A p( x )=1 . x∈A∑p(x)=1. . Note that every nonnegative function on A has the form f f̄ for some f ∈ A A .
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=j=1∑n⟨βj∣A∣βj⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=j=1∑nPj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,j=1∑npj=1