Classical and Quantum Information Measures Characterizing Convertibility: An Axiomatic Approach

In the world of information science, researchers often ask a simple but profound question: how much can we change a system without losing its essential identity? Imagine you have a collection of distinct signals, like a set of different colored lights or a series of unique sound patterns. If you pass these signals through a machine that processes information—perhaps a filter that blurs them or a converter that translates them—you expect them to become harder to tell apart. This is a fundamental rule of nature: information cannot be created out of thin air by a physical process; it can only be preserved or lost. Scientists measure this loss of distinctiveness using mathematical tools called divergences. These are not physical distances, but rather values that tell us how different two states are from one another. When we have a whole group of states, the question becomes even more complex: can we transform an entire group of signals into a completely different group using a single machine?

This is the central puzzle tackled in a recent doctoral thesis by Frits Verhagen, a researcher at the Centre for Quantum Technologies in Singapore. The work sits at the intersection of classical information theory, which deals with standard data like bits, and quantum information theory, which handles the strange, probabilistic rules of the subatomic world. The thesis investigates the precise conditions under which one set of states can be converted into another, either by using a vast number of copies of the original states or by using a special "catalyst" that helps the conversion happen without being consumed. The goal was to find a definitive list of rules that determine when such a transformation is possible. By treating these sets of states as mathematical objects with specific algebraic properties, Verhagen was able to derive a complete set of conditions that act as a universal test for convertibility.

The research focuses on two main scenarios. The first is a classical setting involving multiple probability distributions, which are essentially lists of numbers representing the likelihood of different outcomes. In this context, the problem is known as matrix majorization. The second scenario moves into the quantum realm, examining pairs of states where one part is classical and the other is a pure quantum state. In both cases, the researcher sought to identify the specific "divergences" that must be ordered correctly for a conversion to succeed. Previous studies had solved this problem only under very strict conditions, such as requiring all the states to have the same range of possible outcomes. Verhagen's work breaks these limitations, providing a solution for cases where the states have different or partially overlapping ranges of possibilities.

To achieve this, the thesis employs a sophisticated mathematical framework developed by other researchers, which treats the problem as an algebraic structure known as a preordered semiring. In plain terms, this means the researcher organized the states into a system where they can be added together and multiplied, much like numbers, but with a built-in rule that says one state is "greater than" another if it can be transformed into it. Within this system, the researcher identified specific functions, called monotone homomorphisms and derivations, that act as the measuring sticks for information. These functions correspond to the divergences mentioned earlier. The core discovery is that if a set of states scores higher than another set on every single one of these measuring sticks, then the conversion is possible.

The results are remarkably precise. For the classical case involving multiple probability distributions, the thesis shows that the conversion is possible if and only if a specific family of generalized measures, which extend the well-known Rényi divergences, are strictly ordered in favor of the starting state. These measures account for the complex ways in which the supports of the distributions—their non-zero entries—overlap or differ. The work proves that this family of measures is minimal, meaning that if you were to remove even a small part of it, you would no longer have enough information to guarantee that a conversion is possible. This establishes that the identified set of rules is the most efficient and complete set of conditions required.

In the quantum setting, the findings are equally significant. The researcher analyzed pairs of classical-quantum states where the quantum components are pure, meaning they are in a single, well-defined state rather than a mixture. Here, the conversion is characterized by a different family of measures known as the $\alpha$-$z$ relative entropies. The thesis demonstrates that for these specific types of quantum states, the conversion is possible if and only if these relative entropies are ordered correctly. This result is particularly important because it provides the first clear operational meaning for these relative entropies in a setting where the parameters $\alpha$ and $z$ are truly independent of each other. Before this work, it was not entirely clear how these mathematical quantities related to the physical ability to transform one state into another.

The thesis also addresses the practical reality of imperfect conversions. In the real world, we often do not need a perfect transformation; a result that is very close to the target is usually sufficient. The research extends the main findings to this "asymptotic" regime, showing that if the ordering of the measures is non-strict (meaning the starting state is at least as good as the target state, not necessarily better), then a conversion is possible with an error that can be made arbitrarily small. This covers both the large-sample scenario, where many copies are used, and the catalytic scenario, where a helper state is used and returned unchanged. The work confirms that the same set of measures governs both exact and approximate conversions, unifying the theory.

One of the most compelling aspects of this research is its application to quantum thermodynamics, the study of how heat and work behave at the quantum level. The thesis shows how the results on matrix majorization can be used to determine when a quantum system can be transformed into another while preserving the thermal equilibrium state of the system. This resolves previous technical issues in the field and provides a rigorous foundation for understanding the limits of energy conversion in quantum machines. By removing restrictive assumptions about the states involved, the work offers a more general and robust framework for understanding the flow of information and energy.

The methodology used in the thesis relies on a deep connection between algebra and information theory. By mapping the physical problem of state conversion onto an algebraic structure, the researcher could use powerful theorems from real algebra to derive the necessary conditions. This approach allowed for a rigorous proof that the identified set of divergences is not just sufficient but also necessary. The work carefully rules out the possibility that a smaller or different set of measures could suffice, demonstrating that the identified family is the exact boundary of what is physically possible. The results are presented as mathematical proofs, ensuring that the conclusions are certain and not merely suggested by simulations or approximations.

Ultimately, this thesis provides a comprehensive map for navigating the landscape of state convertibility. It tells us exactly what we need to check to know if one set of information states can be turned into another. Whether dealing with classical data or complex quantum systems, the rules are now clear: the transformation is possible if and only if the starting state dominates the target state across a specific, minimal family of information measures. This clarity is a significant step forward for the field, offering a solid theoretical foundation for future developments in quantum information processing and thermodynamics. The work stands as a testament to the power of abstract mathematical reasoning to solve concrete physical problems, bridging the gap between pure theory and the practical limits of information manipulation.