Information Theory and Statistical Mechanics II (1957)

Paul A. Benioff

A. Katz

Qiuping Alexandre Wang

E. T. Jaynes

Treatment of the predictive aspect of statistical mechanics as a form of statistical inference is extended to the density-matrix formalism and applied to a discussion of the relation between irreversibility and information loss. A principle of "statistical complementarity" is pointed out, according to which the empirically verifiable probabilities of statistical mechanics necessarily correspond to incomplete predictions. A preliminary discussion is given of the second law of thermodynamics and of a certain class of irreversible processes, in an approximation equivalent to that of the semiclassical theory of radiation.It is shown that a density matrix does not in general contain all the information about a system that is relevant for predicting its behavior. In the case of a system perturbed by random fluctuating fields, the density matrix cannot satisfy any differential equation because $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)$ does not depend only on $\ensuremath{\rho}(t)$, but also on past conditions The rigorous theory involves stochastic equations in the type $\ensuremath{\rho}(t)=\mathcal{G}(t, 0)\ensuremath{\rho}(0)$, where the operator $\mathcal{G}$ is a functional of conditions during the entire interval ($0\ensuremath{\rightarrow}t$). Therefore a general theory of irreversible processes cannot be based on differential rate equations corresponding to time-proportional transition probabilities. However, such equations often represent useful approximations.

Roger Balian

Shiro Ishikawa

I have proposed OST (Observable System Theory) as a foundational theory of system theory.3 This theory adopts two principles, namely, the Born principle and the Causal principle, and furthermore explicitly uses the one-shot measurement principle as a rule of use. OST is one of the most fundamental scientific theories, alongside mathematics and physics, and it can provide a unified framework for discussing quantum mechanics (information), statistics (system theory), practical logic, epistemology (philosophy), statistical mechanics, and so on.In this paper, I clarify that Fisherian statistics, Bayesian statistics, and quantum mechanics share a common mathematical foundation, namely OST. In statistics, an OST-based understanding of the Kalman filter is presented. Moreover, the so-called “interpretation problem” of quantum mechanics after measurement is unnecessary, and the “Wigner’s friend paradox” is resolved within OST. In addition, the FR paradox (= Frauchiger–Renner paradox) is understood as a paradox arising from the violation of the one-shot measurement principle, and thus it suggests the validity of OST. Through these discussions, I claim the following two points:(i) OST-based statistics will, in the future, become the standard understanding of statistics.(ii) OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics