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.
A convex sequential effect algebra (COSEA) is an algebraic system with three physically motivated operations, an orthogonal sum, a scalar product and a sequential product. The elements of a COSEA correspond to yes-no measurements and are called effects. In this work we stress the importance of contexts in a COSEA. A context is a finest sharp measurement and an effect will act differently according to the underlying context with which it is measured. Under a change of context, the possible values of an effect do not change but the way these values are obtained may be different. In this paper we discuss direct sums and the center of a COSEA. We also consider conditional probabilities and the spectra of effects. Finally, we characterize COSEA's that are isomorphic to COSEA's of positive operators on a complex Hilbert space. These result in the traditional quantum formalism. All of this work depends heavily on the concept of a context.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabilities of observables. We introduce a definition for joint probabilities and discuss why we consider this to be superior to the standard definition.
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
We first show that the convex effect algebras (CEA) approach to quantum mechanics is more general than the general probabilistic theories approach. We then restrict our attention to finite-dimension CEA's. After an introductory Section~1, we present basic definitions in Section~2. Section~3 studies convex subeffect algebras and observables. In Section~4 we consider strong CEA's and strong observables. We show that a CEA is strong if and only if it is classical. Informationally complete observables on classical CEA's are studied in Section~5. Section~6 considers quantum CEA's in Hilbert spaces.