Minimal-disturbance measurement as a specification in von Neumann's quantal theory of measurement

In von Neumann's theory an incomplete observableA is measured by measuring any complete observableB whose functionA is. This procedure is narrowed down in this paper by the additional requirement of preservation of the sharp value of any observable compatible withA. The requirement is shown to be equivalent to the unique change of state:ρ → (trρP _n)^?1 P _nρP_n (P _n is the eigenprojector ofA corresponding to the obtained eigenvaluea _n, ρ is the statistical operator of the initial state, and by assumption trρP _n > 0). This characterises the minimal-disturbance measurement. A necessary and sufficient condition is derived for the selection of the above observableB so that its measurement implies the minimal-disturbance measurement ofA. For arbitraryρ andA, there exists aB satisfying the condition. Hence, this constitutes a reasonable specification within von Neumann's theory, reducing the latter to the physically preferable minimal-disturbance measurement theory.     

A reinterpretation of dense gas kinetic theory

In dense gas kinetic theory it is standard to express all reduced distribution functions as functionals of the singlet distribution function. Since the singlet distribution function includes aspects of correlated particles as well as describing the properties of freely moving particles, it is here argued that these aspects should more clearly be distinguished and that it is the distribution function for free particles that is the prime object in terms of which dense gas kinetic theory should be expressed. The standard equations of dense gas kinetic theory are rewritten from this point of view and the advantages of doing so are discussed.     

The Wheeler-Feynman absorber theory: A reinterpretation?

The “reinterpretation” of the Wheeler-Feynman absorber theory of radiation, as presented by H. Price in Refs. 1 and 2, is shown not to be a reinterpretation of the mathematical framework of the theory, but an alteration of the theory, which renders it asymmetric. It is shown that Price is mistaken in accusing Wheeler and Feynman of presenting flawed arguments as to whether the advanced and retarded components are distinct; and about the reasons for excluding the time-reversed version of the absorber theory.     

An accurate von Neumann's law for three-dimensional foams

The diffusive coarsening of 2D soap froths is governed by von Neumann's law. A statistical version of this law for dry 3D foams has long been conjectured. A new derivation, based on a theorem by Minkowski, yields an explicit analytical von Neumann's law in 3D which is in very good agreement with detailed simulations and experiments. The average growth rate of a bubble with F faces is shown to be proportional to F1/2 for large F, in contrast to the conjectured linear dependence. Accounting for foam disorder in the model further improves the agreement with data.     

Roots for quantum ergodic theory invon Neumann's work

In ergodic theory von Neumann emphasized the spectral analysis of the unitary implementor and the possibility to express point translations as automorphisms over abelian algebras. Replacing the abelian algebras by noncommutative algebras good ergodic behaviour asks for type II and III algebras. The possibility for existing K-systems and Anosov systems in this framework is discussed. Von Neumnns example of a type III algebra is examined from this viewpoint.     

Generalization of von Neumann's theorem to irreproducible measurements within the framework of quantum field theory. V

Various papers in the series are summarized and the results are extended to the general possibility of mutual self-measurement in a closed macrosystem whose state is specified not by a vector but by some statistical operator in Hilbert space. All the nonconservative effects still apply. Further, these effects occur in a microsystem that remains in a mixed state.Report to a meeting of the Division of Nuclear Physics, Academy of Sciences of the USSR, Moscow, February, 1978.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 85–92, September, 1979.I am indebted to I. Krivskii for a valuable discussion.     

Generalization of von Neumann's theorem to nonreproducible measurements in the framework of quantum field theory. I

One of von Neumann's three theorems - the one on the reduction of the wave packet – is extended to the most general case of nonreproducible measurements in the framework of quantum field theory. The proof of the generalization uses only the most general principles of theory: unitarity of the operator and the standard probability interpretation of the state vector of a physical system.Lecture at the Fall School 1976 on Axiomatic Quantum Field Theory (Uzhgorod, September 15–21, 1976) and at the Nuclear-Physics Section Session of the Academy of Sciences of the USSR (Moscow, February 1–4, 1978).Uzhgorod Branch of Hadron Theory. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 108–113, April, 1978.I am very grateful to Academician N. N. Bogolyubov for approval of the resulta of this paper (preliminary publications of some results were given in [16–22]), to Professors M. I. Podgoretskii (JINR, Dubna), V. Ya. Fainberg (P. N. Lebedev Physics Institute, Moscow) for numerous fruitful and stimulating discussions, to Professor Yu. M. Shirokov (V. A. Steklov Mathematics Institute, Moscow) for valuable comments, and to L Yu. Krivskii and E. P. Sabada for helpful discussions.     

EPR-relations,von Neumann's standard forms and a proof concerning a conjecture of E. Scheibe

In the first part of this paper it is shown how EPR-situations are correlated with von Neumann's standard form of quantum mechanical states describing a system consisting of two dynamical independent subsystems. These standard forms are the mathematical tools for a proof of a conjecture of E. Scheibe: If the 4 selfadjoint operators in Bell's — inequality are pairwise EPR — related, then this inequality is valid in the strong form (with the same upper bound as in statistical mechanics). In the last section the question is discussed whether observations made on the two subsystems together with EPR-relations between them, determine the state of the composed system.Retired, however for a time back at     

Quantum decision theory as quantum theory of measurement

We present a general theory of quantum information processing devices, that can be applied to human decision makers, to atomic multimode registers, or to molecular high-spin registers. Our quantum decision theory is a generalization of the quantum theory of measurement, endowed with an action ring, a prospect lattice and a probability operator measure. The algebra of probability operators plays the role of the algebra of local observables. Because of the composite nature of prospects and of the entangling properties of the probability operators, quantum interference terms appear, which make actions noncommutative and the prospect probabilities nonadditive. The theory provides the basis for explaining a variety of paradoxes typical of the application of classical utility theory to real human decision making. The principal advantage of our approach is that it is formulated as a self-consistent mathematical theory, which allows us to explain not just one effect but actually all known paradoxes in human decision making. Being general, the approach can serve as a tool for characterizing quantum information processing by means of atomic, molecular, and condensed-matter systems.     

The reinterpretation of wave mechanics

The author begins by recalling how he was led in 1923–24 to the ideas of wave mechanics in generalizing the ideas of Einstein's theory of light quanta. He made himself at that time a concrete physical picture of the coexistence of waves and particles and, in 1927, attempted to give them precise form in his theory of the double solution. As other ideas prevailed at the time, he abandoned the development of his conception. But for the past twenty years, once again convinced, like Einstein, that present-day quantum mechanics is only a statistical theory and does not give a true picture of physical reality, he has again taken up his old ideas and developed them considerably. He has in particular introduced an element of randomness into the theory and has thus attained to a hidden thermodynamics of particles, the results of which appear to be very interesting.     

Space-time structure and measurement theory

The present paper is a naïve operational approach to measurement theory in a truly relativistic framework. Both experiments and states exist in finite regions of space-time. The causality structure of the underlying Minkowski space is described in terms of these.A Mellon Postdoctoral Fellow partially supported by the U.S. Atomic Energy Commission under contract number AT-30-1-3829.An N.S.F. Postdoctoral Fellow supported by N.S.F. development grant GU 2056.     

Symmetry breaking and measurement theory

For studying the problem of measurement we accept the statistical interpretation of quantum mechanics, but we take into account, that in a sequence of measurements, offdiagonal terms arise which cannot obviously be interpreted as the probability of a sequence of macroscopic events. We demonstrate, however, their vanishing for ordinary measurement processes by observing that the measurement apparatus undergoes a symmetry breaking transition during the measurement.1. On leave from the Central Research Institute for Physics, Budapest, Hungary.     

Nondemolition principle of quantum measurement theory

We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous-spectrum and continuous-in-time observations. The collapsed state-vector after the objectification is simply treated as a random vector of the a posterioristate given by the quantum filtering, i.e., the conditioning of the a prioriinduced state on the corresponding reduced algebra. The nonlinear phenomenological equation of continuous spontaneous localization has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments.     

Perturbation theory of von Neumann entropy

In quantum information theory, von Neumann entropy plays an important role; it is related to quantum channel capacities. Only for a few states can one obtain their entropies. In a continuous variable system, numeric evaluation of entropy is not easy due to infinite dimensions. We develop the perturbation theory for systematically calculating von Neumann entropy of a non-degenerate system as well as a degenerate system.     

Direct measurement theory, Ito-Stratonovich theory, and weakly dissipative Kolmogorov-Arnold-Moser theory

The direct measurement theory studies linear functionals as applied to the problems of quantum mechanics in addition to considering quadratic functionals on the space of wave functions, well established since the beginning of the 20th century. The theory is based on the time invariance principle of an appropriate space for linear functionals. In this case, it turns out that the second-order Schr?dinger equation is factorized: factors “respect” the effect of one of two groups, i.e., the group of inertial gas motion or the nonlinear group. In the weakly dissipative Kolmogorov-Arnold-Moser (KAM) theory, the former group is of extraordinary interest in connection with the formation of caustic curves which, in turn, cause the appearance of advanced and delayed potentials, which makes it possible to estimate anew the ideas of Ito-Stratonovich in the theory of stochastic processes.     

Problems of the quantum theory of measurement

After a short review of the present situation of the quantum theory of measurement, a formulation of the measuring process is given, which allows an “objective” interpretation. Starting from the unitary time transformation of the states of the measured systemS and the measuring deviceM, it is shown that after appropriate specification of the interaction HamiltonianH _int betweenS andM and the macroscopic structure ofM, the statistical operatorW ofS+M approximately develops into the mixture, which is desired as result of the measuring process. During the process the interference terms practically vanish in the sense of weak operator convergence, while on the other hand their Hilbert-Schmidt norm remains constant.     

On a reinterpretation of decay experiments

Discussion of the basic formula conventionally used to interpret decay measurements shows that it is incompatible with the accepted physical interpretation of quantum mechanics. Two alternative theoretical correlates of the actual experiments are discussed. The first assumes a sequence of observations performed on the same object, each causing a collapse of the wave function. The second theory constructs a time-of-sojourn operator on the assumption that the protracted interaction between object and measurement apparatus is “gentle.” The first theoretical formula is developed with the result that the decay probability shows an asymptotic decrease which is at least exponential, contrary to the conventional theory.     

Quantum theory of state reduction and measurement

The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrödinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrödinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.