Localizing quantum information

In classical information theory, Shannon entropy is localizable, meaning it quantifies information that can be meaningfully distributed between parts of a system. Standard von Neumann entropy, however, fails this property in quantum settings, as evidenced by the Bell state where the entropy of a subsystem can exceed the entropy of the whole. The authors seek a quantum entropy that generalizes Shannon entropy more faithfully by satisfying the requirements of localizability.

The authors investigate an adjusted von Neumann entropy, defined as the standard von Neumann entropy plus a logarithmic correction term based on the dimension of the system. They work within the framework of finite-dimensional C*-algebras, which allows for the modeling of finite quantum systems with superselection sectors. They employ an axiomatic approach to define a quantum entropy and evaluate how this adjusted version behaves under marginalization and in the presence of Bell states.

The study establishes that the adjusted von Neumann entropy is the unique quantum entropy that characterizes Bell states by the property that the bipartite system contains exactly as much information as each of its individual parts. Furthermore, it is the smallest quantum entropy that is monotonic, ensuring that discarding a part of a system never increases the total information content. This adjusted measure satisfies the same linear inequalities as Shannon entropy, including strong subadditivity, and provides a more coherent framework for understanding information localization in quantum systems.

This work bridges the gap between classical information theory and quantum mechanics by providing a more intuitive, localizable measure of information. By aligning quantum entropy with the classical notion of localizable information, the authors offer a framework that may be particularly useful for quantum gravity, where entropy bounds are often assigned to spatial regions. It suggests that many of the "miracles" of quantum entropy, such as strong subadditivity, are actually remnants of properties shared with classical Shannon entropy.