Quantum measurement (Q739168)

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 book is organized in four parts, 22 chapters, a preface, and a subject index including a list of symbols. Each chapter is endowed with references, and suitable exercises. Chapter 1 is the introduction which contains an outline of the text including the general structure of statistical experiments. Part I, entitled ``Mathematics'', begins with Chapter 2 which states and explains fundamental concepts of unitary and Hilbert spaces. More advanced topics considered are the lattice of closed subspaces, the polar decomposition of operators, direct sums and tensor products of Hilbert spaces. Chapter 3 is devoted to classes of compact operators, trace class and Hilbert-Schmidt operators, operator ideals and dualities, tensor products and partial traces, the Schmidt decomposition in the bipartite case. In Chapter 4, operator integrals are introduced and the spectral representation of bounded symmetric operators is considered. Operator valued measures and integrals are discussed to some extend. Such considerations are continued for the unbounded case in Chapter 5. Now domains of definition of operators have to be respected and symmetry does not imply selfadjointness. Here also one-parameter unitary groups and Stone's theorem are considered. A final remark stresses the importance of this chapter for quantum mechanics. Chapter 6 points at $C^{\ast}$ and von Neumann algebras now introduced and used in the proceeding chapter as valuable tools in quantum theory. Chapter 7 begins with the definition of $n$-positive and completely positive linear maps. Then Stinespring and Naimark dilation theorems are derived considering Stinespring type representations of positive bilinear maps. These rigorous derivations form the soil for the the representation of normal complete positive operators, operations and instruments in the Kraus representation. After considering Maimark projections of vector valued measures and operator integrals operations and instruments are extensively considered. The final Chapter 8 of this part contains important examples of positive operator valued measures. Here, besides Fourier transforms, smearings of operator valued measures, also phase space operator measures, and related topics are considered. Part II, ``Elements'', begins with Chapter 9 where the quantum states being the convex set of positive trace one operators are considered, pure states being the extreme points. Coherent superpositions of pure states, and the indeterminacy of the pure components of one and the same mixed state is explained. Then effects in the dual space are introduced as the convex set of positive operators bounded by the unit operator. The extreme points of this set are projection operators, called sharp effects. The notion of compatibility is introduced. A general proof of Gleason's theorem is given. Observables are introduced as effect valued measures on a measurable space. In case the measure is projection valued the observable is called sharp. Borel spaces of the real line and selfadjoint operators are considered. Observables on one and the same measurable space form a convex set and the extrememe points are the sharp observales. Special properties are pointed out and discussed. The duality of state operations and effects, Wigner's theorem for state automorphisms, as well as Stone's theorem applied to quantum dynamics are described. Conditional probabilities and state operations are considered. The chapter finishes with the description of compound systems, state reductions, reduced dynamics, entanglement and separability. The meaning of complete positivity with all these respects is pointed out. Chapter 10, ``Measurements'', begins describing the requirements for a system with a pointer observable to react with the object to be measured. If these requirements are fulfilled, the system is called an instrument. The duality of state operations and effects can be used to define the observable associated to the instrument. Sequential, and, if possible. joint measurements are compared. A concept of mixed measurements is introduced. Several examples of measurement schemes are given. After describing repeatable and ideal measurements the role of entanglement after reaction of the object with the measuring system is analysed with the result that disturbances are inevitable. The appendix contains a proof of the statement that repeatable measurements are possible only for discrete spectra. Chapter 11, ``Joint measurability'', begins with definitions and propositions in special cases. Since the range of an effect valued measure need not be a Boolean algebra a concept of regular observables is introduced and enters the supposition of a theorem stating the compatibility of two observables. Then a theorem about joint measurability of two sharp observables is proved. The chapter closes with respective propositions for convex combinations of observables on one and the same measurable space. In Chapter 12, ``Preparation uncertainty'', the usual point of view that for two given observables there is no state for which both have a sharp value is adopted. The discussion is restricted to sharp observables represented by selfadjoint operators with spectrum in $\mathbb{R}$. After considering the standard deviation the more general $\alpha$-spread and an overall width are introduced and discussed. In examples these conxepts are used aside of the standard deviation to formulate uncertainty relations in certain circumstance like joint measurability etc. Chapter 13, ``Measurement uncertainty'', develops tools to quantify approximation errors and degrees of unsharpness necessary to achieve compatible approximations. Problems to quantify measurement errors are pointed out. If a measurement of an observable $\mathbf{A}$ causes the state change $\rho \mapsto \rho^{\prime}$ the measurement of $\mathbf{B}$ on $\rho^{\prime}$ can be interpreted as a measurement of the observable $\mathbf{\tilde{B}}$ on $\rho$. The distortion $\mathbf{B} \mapsto \mathbf{\tilde{B}}$ results from the measurement disturbance measuring $\mathbf{A}$. The Wasserstein distance is generalized to a distance of observables which is related to uncertainty relations. For measurement instruments systematic and random errors are discriminated and investigated. Approximations to sharp observables are considered for to find compatible ones is exemplified by phase space observables. Value deviations of the approximation are considered. Solutions to such problems are investigated and discussed to some extend. Part III, ``Realizations'', begins with Chapter 14 where joint measurability is applied in qubit systems. Here the linear operators form the $\mathbb{C}^4$ given by the span of the Pauli matrices $\{\sigma_i\}_{i=0,1,2,3}$. Preparation uncertainty relations and compatibility of pairs of qubit effects are treated using the structure of the linear $\mathbb{C}^2$ operators. Also the compatibility of three qubit effects is included. Approximate joint measurements, error measures, optimal approximations, and uncertainty relations follow. Applications in the more involved case of position $Q$ and momentum $P$ are the topics of Chapter 15. After considering Weyl commutation relations and Weyl pairs it is shown that there are periodic functions $g$ and $h$ such that $g(Q)$ and $h(P)$ commute but these functions are no suitable approximations of $Q$ and $P$. Several further aspects related to the incompatibility are discussed. Chapter 16, entitled ``Number and phase'', begins introducing covariant operators with respect to an orthonormal base and proves a characterizing property. The number operator of a single-mode electromagnetic field has this property and is called the canonical phase observable which is a member of a whole convex set of phase observables, i.e., operator valued measures on the Borel sets of the unit circle. These are investigated in detail. Operations, phase space phase operators, and number-phase complementarity are considered. Chapter 17 concerns time and energy in non-relativistic classical and quantum mechanics. Covariant time observables are considered. In Chapter 18, informational complete sets of effects, i.e., for density operators $\rho_1,\; \rho_2,\; \rho_1 \neq \rho_2$ there is at least an effect $A$ in this set s. t. $\mathrm{tr}(\rho_1A) \neq \mathrm{tr}\rho_2A)$, are considered. The problem of state reconstruction, the Pauli problem, informational complete observables, and related questions are presented. Chapter 19 is devoted to modern schemes for to implement measurements. It begins with the Arthurs-Kelly model of sequential joint measurement for to realize a covariant phase space observable. Then phase shifters and beam splitters for photon detection are described. Homodyne photon detection schemes are intensivly described including tomography and generalized Markov kernels. The chapter ends representing the Mach-Zehnder Interferometer. Part IV is entitled ``Foundations''. Chapter 20 is concerned with the Bell inequalities, non-locality, incompatibility and other conclusions. Chapter 21 is devoted to the problem of measurement limitations by general conservation laws which exclude the existence of suitable interactions. The Wigner-Araki-Yanase theorem concerning completely additive conserved quantities is considered. A measurement-theoretic interpretation of superselection rules is given. The final Chapter 22 of the book describes the measurement problem in general. The result of pointer readings, and the problem of objectivation are considered. The content of this book is highly rich. After introducing into the mathematical tools applied in quantum mechanics all aspects of quantum measurement are worked out and explained analytically. This book is best recommended.