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arXiv:2507.12310

Uncertainty and entropies of classical channels

Takla Nateeboon

Published Jul 16, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2507.12310

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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.
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.
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.
Chapter 7 discusses quantum conditional entropy, extending the concept of conditional majorization and introducing the notion of negative quantum conditional entropy. The chapter starts with the basic definition of conditional entropy, exploring its key properties like monotonicity and additivity. It further delves into the concepts of conditional min- and max-entropies, emphasizing their roles in quantifying uncertainty in quantum states and their operational significance in quantum information theory. The text presents conditional entropy as a measure sensitive to the effects of entanglement, showing that negative conditional entropy is a distinctive feature of quantum systems, contrasting with the classical domain where entropy values are nonnegative. This negativity is particularly pronounced in the context of maximally entangled states and is connected to the fundamental differences between classical and quantum information processing. Moreover, the chapter includes theorems and exercises to solidify understanding, like the invariance of conditional entropy under local isometric channels and its reduction to entropy for product states. It concludes by underscoring the inevitability of negative conditional entropy in quantum systems, a topic of both theoretical and practical importance in the quantum domain.
We develop a general operational framework that formalizes the concept of conditional uncertainty in a measure-independent fashion. Our formalism is built upon a mathematical relation which we call conditional majorization. We define conditional majorization and, for the case of classical memory, we provide its thorough characterization in terms of monotones, i.e., functions that preserve the partial order under conditional majorization. We demonstrate the application of this framework by deriving two types of memory-assisted uncertainty relations, (1) a monotone-based conditional uncertainty relation and (2) a universal measure-independent conditional uncertainty relation, both of which set a lower bound on the minimal uncertainty that Bob has about Alice's pair of incompatible measurements, conditioned on arbitrary measurement that Bob makes on his own system. We next compare the obtained relations with their existing entropic counterparts and find that they are at least independent.
Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring noncommuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabeling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in M. H. Partovi, Phys. Rev. A 84, 052117 (2011). Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.
Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of Rényi entropies, parameterized by a real number ααα. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterisation of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterisation of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We show that the most general conditional entropy is a generalized convex combination of conditional entropies defined as exponential averages of Rényi entropies of the conditioned distribution, parameterized by a real number and a probability measure on the positive reals.