Quantum Measurement (Theoretical and Mathematical Physics) by Paul Busch | (PDF) Free Download

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This is a book about the Hilbert space formulation of quantum mechanics and its measurement theory. It contains a synopsis of what became of the Mathematical Foundations of Quantum Mechanics since von Neumann’s classic treatise with this title. Fundamental non-classical features of quantum mechanics―indeterminacy and incompatibility of observables, unavoidable measurement disturbance, entanglement, nonlocality―are explicated and analysed using the tools of operational quantum theory.The book is divided into four parts: 1. Mathematics provides a systematic exposition of the Hilbert space and operator theoretic tools and relevant measure and integration theory leading to the Naimark and Stinespring dilation theorems; 2. Elements develops the basic concepts of quantum mechanics and measurement theory with a focus on the notion of approximate joint measurability; 3. Realisations offers in-depth studies of the fundamental observables of quantum mechanics and some of their measurement implementations; and 4. Foundations discusses a selection of foundational topics (quantum-classical contrast, Bell nonlocality, measurement limitations, measurement problem, operational axioms) from a measurement theoretic perspective.The book is addressed to physicists, mathematicians and philosophers of physics with an interest in the mathematical and conceptual foundations of quantum physics, specifically from the perspective of measurement theory.

User’s Reviews

Editorial Reviews: Review “This book is addressed to students of physics or mathematics who like mathematically rigorous formulations and deductions. It may as well serve lecturers or researchers as a useful reference text. The reader gets well introduced into the physical problems and the mathematical tools used for their solution. … The content of this book is highly rich. … This book is best recommended.” (K.-E. Hellwig, zbMath 1416.81001, 2019)“The book is constructed as a series of theorems, proofs, propositions, lemmas and corollaries … . Each chapter throughout the book is supplemented by a detailed set of references for further reading. Consequently, this monograph will be very useful as a reference work. … this is a volume that is really intended for mathematicians.” (Stephen J. Blundell, Contemporary Physics, Vol. 58 (4), September, 2017) From the Back Cover This is a book about the Hilbert space formulation of quantum mechanics and its measurement theory. It contains a synopsis of what became of the Mathematical Foundations of Quantum Mechanics since von Neumann’s classic treatise with this title. Fundamental non-classical features of quantum mechanics―indeterminacy and incompatibility of observables, unavoidable measurement disturbance, entanglement, nonlocality―are explicated and analysed using the tools of operational quantum theory.The book is divided into four parts: 1. Mathematics provides a systematic exposition of the Hilbert space and operator theoretic tools and relevant measure and integration theory leading to the Naimark and Stinespring dilation theorems; 2. Elements develops the basic concepts of quantum mechanics and measurement theory with a focus on the notion of approximate joint measurability; 3. Realisations offers in-depth studies of the fundamental observables of quantum mechanics and some of their measurement implementations; and 4. Foundations discusses a selection of foundational topics (quantum-classical contrast, Bell nonlocality, measurement limitations, measurement problem, operational axioms) from a measurement theoretic perspective.The book is addressed to physicists, mathematicians and philosophers of physics with an interest in the mathematical and conceptual foundations of quantum physics, specifically from the perspective of measurement theory.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This textbook is a detailed introduction to the topic of quantum measurement that follows some of the mainstream thought on the field but also adds some highly original and interesting insights. The following should be considered a high-level summary of some of the main points that the reviewer gained from studying the book. The discussion is not exhaustive due to time and space constraints.The book contains the important mathematical constructions used in the theory of quantum measurement, such as positive operator valued measures (POVM) and the spectral representations of unbounded self-adjoint operators. In obtaining this spectral representation it is important to remember that there may not be a compact set whose complement would be mapped to the null operator, and that use is made of the Cayley transform of a self-adjoint operator, which gives a unitary operator which has a spectral representation. Interestingly, this spectral representation is shown using a particular representation of a real polynomial defined on a compact subset of the real line, and not using the Gelfand-Naimark theorem for commutative C*-algebras, as is usually the case.The authors also give a detailed discussion of dilation theory, which is an important topic not only for quantum measurement theory but also for quantum information theory. Loosely speaking it says that any semispectral measure can be obtained from a spectral measure by embedding the Hilbert space into a larger Hilbert space (whence the terminology “dilation”).Quantum measurement theory makes use of Schrodinger operations or S-operations, which are positive linear maps between trace class operators, and these maps have trace between 0 and 1. Heisenberg operations or H-operations are normal positive linear maps between spaces of bounded linear operators, and the images of these maps are positive unit bounded operators, and this image is called the effect operator. H-operations are automatically completely positive. As the terminology indicates, Heisenberg operations are dual to Schrodinger operations.Effect operators and operations lead to the notions of Heisenberg and Schrodinger instruments, which are loosely speaking operations parameterized by measurable sets. As expected, these notions are dual to each other, and there are completely positive instruments, the set of which is convex, and each instrument defines a unique semispectral measure. In fact, there are infinitely many instruments for a given semispectral measure, and each instrument corresponding to such a semispectral measure E are said to be E-compatible. Completely positive instruments have a minimal Stinespring type representation, and a unital, normal, completely positive, E-compatible map between spaces of bounded linear operators is called a channel.The extreme points of the convex set of completely positive instruments can, for the case where the Hilbert space is the complex numbers, be related to state purification in quantum information theory and be identified as the convex of probability measures, i.e., classical states. Describing a physical system S in quantum mechanics consists of fixing a complex separable Hilbert space H and on H defining an equivalence class of preparations called states (positive trace one operators), equivalence classes of measurements called observables (normalized positive operator measures), and giving the probability measures, or effects (positive unit bounded operators) for the possible measurement outcomes.Describing a physical system in this way is referred to as a Hilbert space realization, and is denoted by (S, E, O), where S(H) is the set of positive trace one operators r: H -> H, E(H) is the set of positive trace one operators E: H -> H, and O is the set of observables. It is important to be able to justify why a Hilbert space realization is considered to be an actual quantum system, i.e. has no classical properties. S(H) is a convex set whose elements (the mixed states) can be combined using given weights, with pure states being those cannot be expressed as such. One feature of the set of quantum states that is not exhibited by “classical” sates is that pure states can be added to form another pure state. Another “nonclassical” feature of S(H) is that mixed states cannot be decomposed uniquely.The set of effects E(H) is partially ordered with the zero operator as the lower bound and the identity operator as its upper bound. E(H) is a convex subset of L(H) whose extreme points are the set of projections P(H), and the weak closure of P(H) is E(H). There is a difference between P(H) and E(H) that arises from their structures of partial algebras and ortho-ordered sets: the partial algebra structure of P(H) determines its order structure, but this is not true for E(H). One can also define a generalized probability measure on P(H), but this cannot be done on E(H).Physical quantities are associated with a value space (Ω, A), where Ω is any set and A is a sigma algebra contained in the subsets of Ω.A value space is a prerequisite for doing measure theory, which is reflected by the requirement that A be a sigma algebra. If O is physical quantity with value space (Ω, A) which is measured on a quantum system S prepared in a state r, then if a result is registered, then a probability can be assigned to this result. In this way, an observable in quantum physics can be described in terms of measurement statistics and allow the usual quantities such as expectation values to be estimated. An observable of a quantum system S described by a Hilbert space H is therefore implemented by a semispectral measure E: A -> L(H), with (Ω, A) being the value space of the observable E. Two effects are compatible if they can be measured together by a single observable. Projections for example are compatible if and only if they commute.Measurements are performed on a physical system S in order to determine its properties. The system will be prepared in a state and brought into contact with a quantum mechanical system called the measuring apparatus to arrive at a result related to the measurement observable. This result is determined by reading the value of a pointer, or pointer observable. The measurement of an observable E of the system S consists of a probe P with its Hilbert space K, prepared in an initial state along with a pointer observable Zand a unitary map U which models the measurement coupling between the system and the probe. It is usually assumed that before the measurement the system and the probe are dynamically and probabilistically independent of each other, where “dynamically independent” and “probabilistically independent” are assumed to be meaningful notions.

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