Information Theoretic Interpretations of von Neumann Entropy

John F. Cyranski

Mario Berta, Fabian Furrer, Volkher B. Scholz

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.

Dėnes Petz

Zhu Han, Dusit Tao Niyato

Semantic communication aims to transmit the task-relevant uncertainty that remains after common knowledge has been taken into account. However, the semantic communication capacity and rate distortion analysis have not been well investigated theoretically. Inspired by quantum entanglement across a black-hole horizon, this paper develops a quantum-inspired framework, where we present a von Neumann entropy bridge between new information and existing common knowledge. Conditioned on the existing common knowledge and the new information, we show that the corresponding von Neumann entropy is continuous and nondecreasing. To give this von Neumann entropy a practical interpretation, we define an auxiliary classical spectral-mode source, whose conditional probabilities are the eigenvalues of the reduced state. Then the resulting semantic capacity and tradeoff are discussed between the fully new information (the Shannon capacity case) and fully common knowledge (no new information) limits. Furthermore, we derive a conditional rate-distortion function, a source-channel feasibility condition, and a task-specific semantic-sample throughput over a cost-constrained discrete memoryless channel. A binary example illustrates the effects of common knowledge, semantic distinguishability, distortion tolerance, and the physical channel capacity.

Wilhelm Ochs

D�nes Petz

This paper is an introduction to the von Neumann entropy in a historic approach. Von Neumann's gedanken experiment is repeated, which led him to the formula of thermodynamic entropy of a statistical operator. In the analysis of his ideas we stress the role of superselection sectors and summarize von Neumann's knowledge about quantum mechanical entropy. The final part of the paper is devoted to important developments discovered long after von Neumann's work. Subadditivity and the interpretation of the von Neumann entropy as channel capacity are among those.