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.
Let M be a von Neumann algebra and let (Nt)t∈[0,T] be an increasing family of abelian von Neumann subalgebras encoding a (classical) information flow. Fix a faithful normal state φρ and a filtration of normal φρ-preserving conditional expectations Et:M→Nt satisfying the tower property. Using bounded functional-calculus cutoffs fn, we introduce a truncation-stable notion of localized (Nt,Et)-martingales for affiliated self-adjoint observables, and formulate a \emph{Local Informational Efficiency Principle} requiring symmetrically discounted traded prices to be martingales in this sense. Assuming a pricing state φ⋆ and a compatible family of normal φ⋆-preserving conditional expectations (Et⋆), we define for bounded terminal payoffs X∈MT the dynamic pricing operator Πt(X):=Bt1/2Et⋆(BT−1/2XBT−1/2)Bt1/2, where (Bt) is a strictly positive numéraire adapted to (Nt). We prove that (Πt) is normal, completely positive, Nt-bimodular, and time-consistent, and satisfies Πt(1)=Bt (equivalently, Πt(X):=Bt−1/2Πt(X)Bt−1/2 is unital). In the commutative reduction it agrees with risk-neutral valuation by conditional expectation. Finally, we develop an L2(M,φρ) prediction theory and introduce an operator-valued Fisher information relative to (Nt), obtaining a noncommutative Cramér--Rao lower bound for conditional mean-square prediction error; we compute the bound for compound Poisson lattice-jump models under ∑αγα(eαΔx−1)=r.