Uncertainty relations represent one of the most fundamental constraints in quantum theory and have been extensively studied from the early days of quantum mechanics to the present.The importance of uncertainty relations has further increased with the emergence of innovative quantum information processing protocols and the recent advances in high-precision control of quantum-scale physical systems.In particular, the uncertainty relations derived by Xiao et al. for quantum testers, also known as process positiveoperator-valued measures (PPOVMs), provide quantitative trade-offs that characterize the (im)possibility of quantum processes, and thus offer formulas of great significance from both foundational and applied perspectives.On the other hand, it is possible to obtain more practical uncertainty relations for quantum testers by restricting classes of quantum processes or by considering a formulation that allows an arbitrary number of testers.The purpose of this thesis is to present theoretical results on uncertainty relations for quantum testers obtained from these viewpoints.Specifically, the thesis addresses the following three topics:1. derivation of an exact uncertainty relation for a simple physical system; derivation of a scalable uncertainty relation applicable to an arbitrary
Uncertainty relations represent one of the most fundamental constraints in quantum theory and have been extensively studied from the early days of quantum mechanics to the present.The importance of uncertainty relations has further increased with the emergence of innovative quantum information processing protocols and the recent advances in high-precision control of quantum-scale physical systems.In particular, the uncertainty relations derived by Xiao et al. for quantum testers, also known as process positiveoperator-valued measures (PPOVMs), provide quantitative trade-offs that characterize the (im)possibility of quantum processes, and thus offer formulas of great significance from both foundational and applied perspectives.On the other hand, it is possible to obtain more practical uncertainty relations for quantum testers by restricting classes of quantum processes or by considering a formulation that allows an arbitrary number of testers.The purpose of this thesis is to present theoretical results on uncertainty relations for quantum testers obtained from these viewpoints.Specifically, the thesis addresses the following three topics:1. derivation of an exact uncertainty relation for a simple physical system; derivation of a scalable uncertainty relation applicable to an arbitrary
The uncertainty principle is one of the features of quantum theory. Fine-grained uncertainty relations (FGURs) are a contemporary interpretation of this principle. Each FGUR is derived from a scenario where multiple measurements of a quantum state are stochastically performed. While state measurements are fundamental, measuring quantum processes, namely, completely positive and trace preserving maps, is also crucial both theoretically and practically. These measurements are mathematically characterized by quantum testers. In this study, we develop FGURs in terms of quantum testers. Because state preparation is a type of quantum process, our framework encompasses the conventional case as a special instance. The generalized FGURs' bounds are typically challenging to compute. Thus, we also provide estimates for these bounds. Specifically, we explore quantum testers involving maximally entangled states in detail. Consequently, some FGURs for quantum testers are derived as explicit forms for specific settings.
Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.
The foundational theory of quantum enhanced metrology for parameter estimation is of fundamental importance to the progression of science and technology as the scientific method is built upon empirical evidence, the acquisition of which is entirely reliant on measurement. Quantum mechanical properties can be exploited to yield measurement results to a greater precision (lesser uncertainty) than that which is permitted by classical methods. This has been mathematically demonstrated by the derivation of theoretical bounds which place a fundamental limit on the uncertainty of a measurement. Furthermore, quantum metrology is of immediate interest in the application of quantum technologies since measurement plays a central role. This thesis focuses on the role of quantum correlations and uncertainty relations which govern the precision bounds. We show how correlations can be distributed amongst limited resources in realistic scenarios, as permitted by current experimental capabilities, to achieve higher precision measurements than current approaches. This is extended to the setting of multiparameter estimation in which we demonstrate a more technologically feasible method of correlation distribution than those previously posited which perform as well as, or worse than, our scheme. Furthermore, a quantum metrology protocol is typically comprised of three stages: probe state preparation, sensing and then readout, where the time required for the first and last stages is usually neglected. We consider the more realistic sensing scenario of time being a limited resource which is divided amongst the three stages and demonstrate the most efficient use of this resource. Additionally, we take an information theoretic approach to quantum mechanical uncertainty relations and derive a one-parameter class of uncertainty relations which supplies more information about the quantum mechanical system of interest than conventional uncertainty relations. Finally, we demonstrate how we can use this class of uncertainty relations to reconstruct information of the state of the quantum mechanical system.
Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics.The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space.However, many physical systems are dynamic in nature and described with the formalism of a quantum channel.In this paper, we show that the uncertainty principle can be reformulated to include process measurements that are performed on quantum channels.Since both the preparation of quantum states and the implementation of quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality.More specifically, we obtain expressions that generalize the Maassen-Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.
Uncertainty relations are one of the most important foundations of quantum physics. In the textbook literatures, uncertainty relations usually refer to the preparation uncertainty. Its original formulation based on variances of two observables limits on the ability to prepare an ensemble of quantum systems for which non-commuting observables will have arbitrary uncertainty. The preparation uncertainty relation has been widely investigated. On the other hand, a unitary operator is a fundamental tenet of quantum theory. Every evolution of a closed quantum system is governed by acting unitary operators on the state of the system and the evolution of an open system can be represented by acting unitary operators on an enlarged system consisting of the quantum system as a subsystem. Therefore, naturally, to understand and quantitatively capture the essence of uncertainty relations for unitary operators is important and timely. Here we report an experimental investigation of a set of uncertainty relations for two unitary operators, which are theoretically derived by using a sequence of fine-grained inequalities. We test these uncertainty relations with single photons and interferometric networks. The unitary uncertainty relation is saturated by any pure qubit state. For higher-dimensional states, it is stronger than the best known bound introduced in the previous literatures. The lower bounds of the unitary uncertainty relations can be even further strengthened by the symmetry of permutation. The experimental findings agree with the predictions of quantum theory and respect the new uncertainty relations.