A reinterpretation of von Neumann's theory of measurement

Jeffrey Bub

Kenjiro Yanagi

Fedor Herbut

Teruaki Nagasawa

One of the major unsolved problems in physics is the contradiction betwee n the reversibility of the microscopic world, as described by quantum mech anics, and the irreversibility of the macroscopic world, as described by c lassical thermodynamics and statistical mechanics.Specifically, time evol ution governed by the Schrödinger equation is unitary and possesses time-r eversal symmetry, and the von Neumann entropy of a microscopic system is c onserved.However, the entropy increase observed in macroscopic, isolated systems according to the second law of thermodynamics cannot be explained by unitary time evolution and von Neumann entropy alone.This doctoral dissertation presents a unified framework that uses observe rs to explain the contradiction.Based on von Neumann's concept of macrosc opic entropy, the framework redefines these concepts through the lens of q uantum information theory and specifically, Petz's theory of quantum suffi ciency and resource theory.This research proposes that macroscopic irreversible phenomena do not vio late physical laws.Instead, these phenomena result from the loss of stati stical information caused by limitations in an observer's measurement capa bilities or by coarsening.This doctoral thesis specifically focuses on ob servational entropy, which is a generalization of von Neumann's macroscopi c entropy.Chapter 4 of this study reinterprets von Neumann's H-theorem within a con temporary framework.The chapter provides a mathematical proof of the irre versible increase of observational entropy in isolated quantum systems.It was demonstrated that, when the system's initial state is macroscopic, or lacking microscopic details with respect to a specific measurement, obser Macroscopic States, Operations, and Entropy: An Obser ver-Centric Approach (巨視的な状態、操作、エントロピー ：観測者中心のアプローチ) 長澤 輝明

Pier A. Mello

Rathindra Nath Sen

Von Neumann's theory of measurement in Quantum Mechanics was spelled out in the last chapter of his book, which was published in 1932. This book was highly mathematical for its time, and in 1939 London and Bauer provided a simplified account of the measurement theory part of it (London and Bauer, 1939). Von Neumann died in 1957. Thirty years after the publication of von Neumann's book, Wigner published a review containing his views on the shortcomings of von Neumann's theory, but omitting any discussion of its mathematical core, namely von Neumann's analysis of composite systems (Wigner, 1963). He also published a set of lecture notes entitled Interpretation of Quantum Mechanics (Wigner, 1983) in which some of his concerns were spelled out in greater detail, and an article addressed ‘to an audience of non-physicists’ (Wigner, 1964). Wigner's own contributions to measurement theory were discussed by Shimony in a talk at the Wigner centennial conference (Shimony, 2002). The English translation of von Neumann's book was published in 1955.1 The account that follows is based on these sources. We shall assume that the reader is acquainted with notions such as wave function collapse and the Heisenberg cut, but we shall not assume familiarity with the technicalities of von Neumann's theory. This chapter is organized accordingly. Section 8.1 explains what we mean by the term von Neumann's measurement theory and gives an overview of the subject. It is followed by Sections 8.2 and 8.3, in which the theory is spelled out in detail. Section 8.4 recounts Wigner's reservations.