A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.
A quantum effect is an operator
A on a complex Hilbert space
H that satisfies
0≤A≤I,
E(H) is the set of all quantum effects on
H. In 2001, Professor Gudder and Nagy studied the sequential product
A∘B=A1/2BA1/2 of
A,B∈E(H). In 2005, Professor Gudder asked: Is
A∘B=A1/2BA1/2 the only sequential product on
E(H)? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on
E(H); secondly, we present a general method for constructing sequential products on
E(H); finally, we study some properties of the sequential products constructed by the method
The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. For A, B∊E(H), the operation of sequential product A∘B=A1∕2BA1∕2 was proposed as a model for sequential quantum measurements. A nice investigation of properties of the sequential product has been carried over [Gudder, S. and Nagy, G., “Sequential quantum measurements,” J. Math. Phys. 42, 5212 (2001)]. In this note, we extend some results of this reference. In particular, a gap in the proof of Theorem 3.2 in this reference is overcome. In addition, some properties of generalized infimum A⨅B are studied.
Our basic concept is the set
E(H) of effects on a finite dimensional complex Hilbert space
H. If
a,b∈E(H), we define the sequential product
a[I]b of
a then
b. The sequential product depends on the operation
I used to measure
a. We begin by studying the properties of this sequential product. It is observed that
b↦a[I]b is an additive, convex morphism and we show by examples that
a↦a[I]b enjoys very few conditions. This is because a measurement of
a can interfere with a later measurement of
b. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on
a[I]b that imply commutativity. We introduce the concept of an effect
b given an effect
a and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.
Unsharp quantum measurements can be modelled by means of the class E ( H ) \mathcal {E}(\mathcal {H}) of positive contractions on a Hilbert space H \mathcal {H} , in brief, quantum effects. For A , B ∈ E ( H ) A,B\in \mathcal {E}(\mathcal {H}) the operation of sequential product A ∘ B = A 1 / 2 B A 1 / 2 A\circ B=A^{1/2}BA^{1/2} was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption A ∘ B ≥ B A\circ B\geq B implies A B = B A = B AB=BA=B . Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.
A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator E E on a Hilbert space that satisfies 0 ≤ E ≤ I 0\le E\le I . We define effects E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n to be sequentially independent if the result of any sequential measurement of E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.