Quantum mechanics via quantum tomography

Arnold Neumaier

Published Oct 11, 2021• 0 citations

Abstract

Starting from first principles inspired by quantum tomography rather than Born's rule, this paper gives a new, elementary, and self-contained deductive approach to quantum mechanics. A suggestive notion for what constitutes a quantum detector and for the behavior of its responses leads to a logically impeccable definition of measurement. Applications to measurement schemes for optical states, position measurements and particle tracks demonstrate that this definition is applicable without any idealization to complex realistic experiments. The various forms of quantum tomography for quantum states, quantum detectors, quantum processes, and quantum instruments are discussed. The traditional dynamical and spectral properties of quantum mechanics are derived from a continuum limit of quantum processes. In particular, the Schr\odinger equation the state vector of a pure, nonmixing quantum system and the Lindblad equation for the density operator of a mixing quantum system are shown to be consequences of the new approach. A slight idealization of the measurement process leads to the notion of quantum fields, whose smeared quantum expectations emerge as reproducible properties of regions of space accessible to measurements.

Related papers

Quantum tomography explains quantum mechanics

Arnold Neumaier

Starting from a new principle inspired by quantum tomography rather than from Born's rule, this paper gives a self-contained deductive approach to quantum mechanics and quantum measurement. A suggestive notion for what constitutes a quantum detector and for the behavior of its responses leads to a logically impeccable definition of measurement. Applications to measurement schemes for optical states, position measurements and particle tracks demonstrate the applicability to complex realistic experiments without any idealization. The various forms of quantum tomography for quantum states, quantum detectors, quantum processes, and quantum instruments are discussed. The traditional dynamical and spectral properties of quantum mechanics are derived from a continuum limit of quantum processes, giving the Lindblad equation for the density operator of a mixing quantum system and the Schrödinger equation for the state vector of a pure, nonmixing quantum system. Normalized density operators are shown to play the role of quantum phase space variables, in complete analogy to the classical phase space variables position and momentum. A slight idealization of the measurement process leads to the notion of quantum fields, whose smeared quantum expectations emerge as reproducible properties of regions of space accessible to measurements. The new approach is closer to actual practice than the traditional foundations. It is more general, and therefore more powerful. It is simpler and less technical than the traditional approach, and the standard tools of quantum mechanics are not difficult to derive. This makes the new approach suitable for introductory courses on quantum mechanics. A variety of quotes from the literature illuminate the formal exposition with historical and philosophical aspects.

arXiv:2110.05294Oct 11, 20210 citations

Born's rule and measurement

Arnold Neumaier

Born's rule in its conventional textbook form applies to the small class of projective measurements only. It is well-known that a generalization of Born's rule to realistic experiments must be phrased in terms of positive operator valued measures (POVMs). This generalization accounts for things like losses, imperfect measurements, limited detection accuracy, dark detector counts, and the simultaneous measurement of position and momentum. Starting from first principles, this paper gives a self-contained, deductive introduction to quantum measurement and Born's rule, in its generalized form that applies to the results of measurements described by POVMs. It is based on a suggestive definition of what constitutes a detector, assuming an intuitive informal notion of response. The formal exposition is embedded into the context of a variaety of quotes from the literature illuminating historical aspects of the subject. The material presented suggests a new approach to introductory courses on quantum mechanics.

arXiv:1912.09906Dec 20, 20191 citation

Quantum mechanics

Mark E. Tuckerman

Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles

Aug 1, 20230 citations

Quantum Tomography twenty years later

M. Asorey, Alberto Ibort, G Marmo +1 more

We show a sample of some relevant developments in classical and quantum tomography that have taken place over the last twenty years. We will present a general conceptual framework that provides a simple unifying mathematical picture for them and, as an effective use of it, three subjects have been chosen that offer a wide panorama of the scope of classical and quantum tomography: tomography along lines and submanifolds, coherent state tomography and tomography in the abstract algebraic setting of quantum systems.

arXiv:1510.08140Physica ScriptaJun 1, 201556 citations

A classical interpretation of quantum mechanics and the measurement problem

Christian Jansson

In this paper a didactic approach is described which immediately leads to an understanding of those postulates of quantum mechanics used most frequently in quantum computation. Moreover, an interpretation of quantum mechanics is presented which is motivated by retaining the point of view of classical mechanics as much as possible, and which is consistent with relativity theory. Everything can be written down in terms of well-known mathematical formulations that can be found in every textbook about quantum mechanics. Therefore, in this version, almost no formulas are used.

arXiv:0801.3189Jan 21, 20081 citation

A refinement of quantum mechanics by algorithmic randomness

Kohtaro Tadaki

The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In this paper, based on the toolkit of algorithmic randomness, we present a refinement of the Born rule, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. Algorithmic randomness is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We then present an operational refinement of the Born rule for mixed states, as an alternative rule to it, based on algorithmic randomness. In particular, we give a precise definition for the notion of mixed state. We then show that all of the refined rules of the Born rule for both pure states and mixed states can be derived from a single postulate, called the principle of typicality, in a unified manner. We do this from the point of view of the many-worlds interpretation of quantum mechanics. Finally, we make an application of our framework to the BB84 quantum key distribution protocol in order to demonstrate how properly our framework works in practical problems in quantum mechanics, based on the principle of typicality.

arXiv:1804.10174Apr 26, 20182 citations