Research On Tensor Product And Sequential Product Of Effect Algebras

Effect algebras have been introduced as an algebraic structure for investigat-ing the foundations of quantum mechanics. This framework gives a unification of the operational and quantum logic approaches to quantum mechanics and yields a natural definition of a tensor product. However, the definition of tensor product of effect algebras has been introduced in category terms without constructing the con-crete form of it. Hence, it is difficult to know what the tensor product of two given effect algebras is. In this paper, we construct the forms of{0,1}(?)E, Cm(a)(?)Cn(b), C2(x)(?)C4(y,z), C2(x)(?)C’4(y,z), C2(x)(?)[0,1], C’4(x,y)(?)C’4(u,v) and ε(H)(?)ε(κ). Abstract effect algebras are not necessarily formed as ε(H), and the effect algebra which can be represented as ε(H) is called representable effect algebra, otherwise, it is called unrepresentable effect algebra. We also discuss the representability of the above tensor products of effect algebras.Unsharp quantum measurements experiments can be modelled by means of ε(H). For A,B∈ε(H) the operation of sequential product AοB=A1/2BA1/2was proposed as a model for sequential measurements. Two measurements a and b can not be performed simultaneously in general, so they are frequently executed sequentially. We denote by a o b a sequential measurement in which a is performed first and b second. We call it the sequential product of a and b. A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The commutator of a subset A of a SEA E is defined naturally, denoted by C(A). Some properties of sequential product are discussed, especially when E is the Hilbert space effect algebra, the commutativity of A implies the commutativity of C(C(A)).