Quantum Shannon theory

Quantum Shannon theory is the branch of quantum information theory that generalizes Claude Shannon's classical information theory to the quantum domain. It replaces Shannon entropy with von Neumann entropy, classical channels with quantum channels, and probability distributions with density matrices. The central theorems — Schumacher compression, the Holevo bound, and the quantum channel capacity theorem — prove that quantum information has its own compression limits, communication bounds, and error-correcting codes, all governed by non-commutative mathematics rather than classical probability.

The field's foundational result is that quantum information cannot be cloned, cannot be compressed to the same limits as classical information, and cannot be transmitted through channels with the same capacity as classical bits. These are not limitations; they are structural features that reveal what information is in a quantum universe.