اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدQuantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces
114
0
0.0
(
0
)
تأليف
A. Vourdas
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d- dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities, is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2), to the subspaces H1,H2. As an application, it is shown that the proof of CHSH inequalities for a system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it is not valid for Dempster- Shafer probabilities. The violation of these inequalities in experiments, supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.
قيم البحث
اقرأ أيضاً
We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable
to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data.
The set of subsystems of a finite quantum system (with variables in Z(n)) together with logical connectives, is a distributive lattice. With regard to this lattice, the (where P(m) is the projector to) obeys a supermodularity inequality, and it is in
terpreted as a lower probability in the sense of the Dempster-Shafer theory, and not as a Kolmogorov probability. It is shown that the basic concepts of the Dempster-Shafer theory (lower and upper probabilities and the Dempster multivaluedness) are pertinent to the quantum formalism of finite systems.
We show that including both the system and the apparatus in the quantum description of the measurement process, and using the concept of conditional probabilities, it is possible to deduce the statistical operator of the system after a measurement wi
th a given result, which gives the probability distribution for all possible consecutive measurements on the system. This statistical operator, representing the state of the system after the first measurement, is in general not the same that would be obtained using the postulate of collapse.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the measurement problem of quantum mechanics. Efforts to justify the B
orn rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohms deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohms quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
We use a novel form of quantum conditional probability to define new measures of quantum information in a dynamical context. We explore relationships between our new quantities and standard measures of quantum information such as von Neumann entropy.
These quantities allow us to find new proofs of some standard results in quantum information theory, such as the concavity of von Neumann entropy and Holevos theorem. The existence of an underlying probability distribution helps to shed some light on the conceptual underpinnings of these results.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
