Classical information theory, built on Shannon entropy, and quantum information theory, built on von Neumann entropy, quantify uncertainty and compression properties of probability distributions and density operators. Both are powerful, but both treat information as a property of states of individual systems or ensembles. In this manuscript we argue that entanglement—the pattern of non-classical correlations between systems—is more fundamental than either Shannon or von Neumann information. We motivate this claim with a condensate thought experiment in which many bosons occupy the same single-particle state: Shannon and von Neumann entropies assign them identical information content, yet they can be distinguished by their distinct entanglement histories with external systems. Formally, states with identical marginal density operators but different purifications represent different global information, accessible only through their entanglement structure. Entropic quantities are then seen as coarse-grained summaries of this underlying relational information. We connect this perspective to an entanglement-geometric view of physics, including Entanglement Manifold Cosmology (EMC), in which spacetime and physical “truth” emerge from the geometry of correlations. The goal is an information-theoretic narrative that is rigorous enough for physics and information theory while readable to a non-specialist who is comfortable with basic linear algebra and quantum mechanics.
Classical information theory, built on Shannon entropy, and quantum information theory, built on von Neumann entropy, quantify uncertainty and compression proper ties of probability distributions and density operators. Both are powerful, but both treat information as a property of states of individual systems or ensembles. In this manuscript we argue that entanglement the pattern of non-classical correlations be tween systems is more fundamental than either Shannon or von Neumann information. We motivate this claim with a condensate thought experiment in which many bosons occupy the same single-particle state: Shannon and von Neumann entropies as sign them identical information content, yet they can be distinguished by their distinct entanglement histories with external systems. Formally, states with identical marginal density operators but different purifications represent different global information, accessible only through their entanglement structure. Entropic quantities are then seen as coarse-grained summaries of this underlying relational information. We connect this perspective to an entanglement-geometric view of physics, including Entanglement Manifold Cosmology (EMC), in which spacetime and physical truth emerge from the geometry of correlations. The goal is an information-theoretic narrative that is rigorous enough for physics and information theory while readable to a non-specialist who is comfortable with basic linear algebra and quantum mechanics.
The notion of entropy was introduced by Clausius in order to discuss the thermal behavior of physical systems. After the work of Clausius, Boltzmann, Gibbs and others, the beginning of the exact treatment of entropy was due to von Neumann and Shannon. Their starting point was very di erent, von Neumann was motivated by quantum mechanics and Shannon founded communication theory. The entropy of a state describing a physical system is a quantity expressing the uncertainty or randomness of the system. Shannon regarded this uncertainty attached to a physical system as the amount of information carried by the system, so that the entropy of a state can be read as the information carried by the state. His idea comes from the following consideration: If a physical system has a large uncertainty and one receives information on the system by some procedure, then the so obtained information is more valuable than that received from a system having less uncertainty. It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behavior of certain probabilities in the course of independent trials. First we will discuss this phenomena for nite quantum systems. By a nite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a nite quantum system is a nite dimensional C*algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is "tracial " in the sense that Tr ab = Tr ba (a; b 2 A) Every functional! on A is determined by a density operator D! 2 A in the form!(a) = Tr D! a (a 2 A): The entropy S(!) of a functional! is de ned by means of its density operator as S(!) = Tr (D!): This notion was introduced by von Neumann in 1927 and we term it von Neumann's entropy or shortly entropy (cf. [1]). In fact, von Neumann de ned the entropy such