Information Theory Applied to Measurements

Claudia Zander, Angel Ricardo Plastino

We investigate the main features of a measure of fidelity between states in a general family of probabilistic theories admitting classical probability theory and standard quantum theory as particular instances. We apply the aforementioned measure to investigate informationtheoretical \nfeatures of these theories related to the conservation of information during the evolution \nof closed physical systems. In particular, we derive a generalization of a fundamental result in \nquantum theory relevant for the measurement problem: Zurek’s recent extension of the no-cloning theorem.

Léon Brillouin

Jennifer R. Glick

Thomas Guff, Nathan A. McMahon, Yuval R. Sanders +1 more

Abstract Resource theories are broad frameworks that capture how useful objects are in performing specific tasks. In this paper we devise a formal resource theory quantum measurements, focusing on the ability of a measurement to acquire information. The objects of the theory are equivalence classes of positive operator-valued measures, and the free transformations are changes to a measurement device that can only deteriorate its ability to report information about a physical system. We show that catalysis and purification, protocols that are possible in other resource theories, are impossible in our resource theory for quantum measurements. Standard measures of information gain are shown to be resource monotones, and the resource theory is applied to the task of quantum state discrimination.

K. -E. Hellwig

Mark M. Wilde

All physical systems register bits of information, whether it be an atom, an electrical current, the location of a billiard ball, or a switch. Information can be classical, quantum, or a hybrid of both, depending on the system. For example, an atom or an electron or a superconducting system can register quantum information because the quantum theory applies to each of these systems, but we can safely argue that the location of a billiard ball registers classical information only. These atoms or electrons or superconducting systems can also register classical bits because it is always possible for a quantum system to register classical bits. The term information , in the context of information theory, has a precise meaning that is somewhat different from our prior “everyday” experience with it. Recall that the notion of the physical bit refers to the physical representation of a bit, and the information bit is a measure of how much we learn from the outcome of a random experiment. Perhaps the word “surprise” better captures the notion of information as it applies in the context of information theory. This chapter begins our formal study of classical information. Recall that Chapter 2 reviewed some of the major operational tasks in classical information theory. Here, our approach is somewhat different because our aim is to provide an intuitive understanding of information measures, in terms of the parties who have access to the classical systems. We define precise mathematical formulas that measure the amount of information encoded in a single physical system or in multiple physical systems. The advantage of developing this theory is that we can study information in its own right without having to consider the details of the physical system that registers it. We first introduce the entropy in Section 10.1 as the expected surprise of a random variable. We extend this basic notion of entropy to develop other measures of information in Sections 10.2–10.6 that prove useful as intuitive informational measures, but also, and perhaps more importantly, these measures are the answers to operational tasks that one might wish to perform using noisy resources. While introducing these quantities, we discuss and prove several mathematical results concerning them that are important tools for the practicing information theorist.