The crucial role of channels in physics and information theory motivates the task of characterizing the entropy, or uncertainty, of a channel. Games of chance become a natural candidate for this task, as a system's performance in a gambling game depends solely on the uncertainty of its output. In this work, we construct families of games that induce preorders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all preorders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical channel entropy.
The crucial role of channels in physics and information theory motivates the task of characterizing the entropy, or uncertainty, of a channel. Games of chance become a natural candidate for this task, as a system's performance in a gambling game depends solely on the uncertainty of its output. In this work, we construct families of games that induce preorders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all preorders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical channel entropy.
Given entropy's central role in multiple areas of physics and science, one important task is to develop a systematic and unifying approach to defining entropy. Games of chance become a natural candidate for characterising the uncertainty of a physical system, as a system's performance in gambling games depends solely on the uncertainty of its output. In this work, we construct families of games which induce pre-orders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all pre-orders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical dynamic entropy.
In this thesis, I studied a mathematical development to define and quantify the uncertainty inherent in classical channels. This thesis starts with the introduction and background on how to formally think about uncertainty in the domain of classical states. The concept of probability vector majorization and its variants, relative majorization and conditional majorization, are reviewed. This thesis introduces three conceptually distinct approaches to formalize the notion of uncertainty inherent in classical channels. These three approaches define the same preordering on the domain of classical channels, leading to characterizations from many perspectives. With the solid foundation of uncertainty comparison, classical channel entropy is then defined to be an additive monotone with respect to the majorization relation. The well-known entropies in the domain of classical states are uniquely extended to the domain of channels via the optimal extensions, providing not only a solid foundation but also the quantifiers of uncertainty inherent in classical channels.
Quantum channels represent a broad spectrum of operations crucial to quantum information theory, encompassing everything from the transmission of quantum information to the manipulation of various resources. In the domain of states, the concept of majorization serves as a fundamental tool for comparing the uncertainty inherent in both classical and quantum systems. This paper establishes a rigorous framework for assessing the uncertainty in both classical and quantum channels. By employing a specific class of superchannels, we introduce and elucidate three distinct approaches to channel majorization: constructive, axiomatic, and operational. Intriguingly, these methodologies converge to a consistent ordering. This convergence not only provides a robust basis for defining entropy functions for channels but also clarifies the interpretation of entropy in this broader context. Most notably, our findings reveal that any viable entropy function for quantum channels must assume negative values, thereby challenging traditional notions of entropy.
Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features. An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty. We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity. Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels.
The von Neumann entropy is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the “relative entropy distance” between ρ and the maximally mixed state, here we define the entropy of a channel N as the difference of the number of physical qubits of the channel output with the “relative entropy distance” between N and the completely depolarizing channel. We establish that this definition satisfies all of the axioms, recently put forward in [Dour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. We define Rényi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.
The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the "relative entropy distance"between ρ and the maximally mixed state, here we define the entropy of a channel N as the difference of the number of physical qubits of the channel output with the "relative entropy distance"between N and the completely depolarizing channel. We prove that this definition satisfies all of the axioms, recently put forward by Gour [IEEE Trans. Inf. Theory 65, 5880 (2019)IETTAW0018-944810.1109/TIT.2019.2907989], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. The entropy of a channel can be negative for certain channels, but this negativity has an operational interpretation in terms of the channel merging protocol. We define Rényi and min-entropies of a channel and prove that they satisfy the axioms required for a channel entropy function. Among other results, we also prove that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.