Abstract This chapter introduces the theory and practice of various approaches to measuring quantum systems, focusing on quantum-optical settings, including monitored spontaneous emission, the quantum eraser, and unambiguous state discrimination. Beginning with a review of classical probability and update rules, it explains the motivation for the consideration of density matrices and generalized quantum measurements, treats the connection with decoherence, and goes on to introduce and discuss retrodiction, including ‘interaction-free measurement’, Hardy’s paradox, and ‘weak measurement’.
Abstract This chapter introduces the theory and practice of various approaches to measuring quantum systems, focusing on quantum-optical settings, including monitored spontaneous emission, the quantum eraser, and unambiguous state discrimination. Beginning with a review of classical probability and update rules, it explains the motivation for the consideration of density matrices and generalized quantum measurements, treats the connection with decoherence, and goes on to introduce and discuss retrodiction, including ‘interaction-free measurement’, Hardy’s paradox, and ‘weak measurement’.
We present a comprehensive technical review of quantum measurement theory, emphasizing the mathematical structures underlying the measurement problem and their role in modern quantum information science. Beginning with spectral theory and the Born rule, we develop the theory of positive operator-valued measures (POVMs), Kraus operators, and completely positive channels as essential generalizations of projective measurements. We discuss how decoherence theory addresses the preferred-basis and interference problems while leaving the outcome problem formally unresolved, motivating interpretational responses ranging from collapse models to many-worlds scenarios. Throughout, we emphasize the applicability of POVM theory to quantum state discrimination, quantum cryptography, and quantum error correction. The review prioritizes physical motivation and mathematical hints over detailed proofs, suitable for researchers transitioning into quantum foundations or quantum information theory.
Quantum mechanics famously posits two distinct evolution rules for the state of a system: a deterministic, unitary time evolution under the Schr¨odinger equation, and a probabilistic “collapse” of the wavefunction upon measurement as dictated by the Born rule. The apparent inconsistency of these two dynamical laws – one continuous and one discontinuous – constitutes the quantum measurement problem. This foundational problem, which underlies the emergence of definite outcomes from quantum superpositions, has persisted since the theory’s inception. In this review, we present a comprehensive and rigorous examination of the measurement problem and survey the leading approaches that seek to resolve or circumvent it. We begin by formulating the measurement problem and its conceptual challenges. We then discuss in detail the major interpretations of quantum mechanics and theoretical frameworks addressing the problem: the Copenhagen interpretation, Everett’s many-worlds interpretation, de Broglie–Bohm pilot-wave theory, objective collapse models, the role of decoherence and environment-induced superselection, and epistemic approaches such as QBism. For each interpretation, we describe the core principles and mathematical formalism, assessing how (and whether) it attempts to solve the measurement problem. We also review modern developments and experiments relevant to quantum measurements, including tests of Bell inequalities, observations of decoherence and macroscopic superpositions, weak measurements and quantum eraser experiments, and thought experiments ‘a la Wigner’s friend. We highlight how these results inform the ongoing debate. Finally, we discuss open problems and challenges that remain in fully resolving the measurement problem.
This book offers a complete revision for its introduction to the quantum theory of light, including notable developments as well as improvements in presentation of basic theory and concepts, with continued emphasis on experimental aspects. The author provides a thorough overview on basic methods of classical and quantum mechanical measurements in quantum optics, enabling readers to analyze, summarize, and resolve quantum optical problems. The broad coverage of concepts and tools and its practical, experimental emphasis set it apart from other available resources. New discussions of timely topics such as the concept of the photon and distinguishability bring the entire contents up to date. Key Features: Provides a complete update of a classic textbook for the field. Features many new topics, including optical coherence, coherent and incoherent imaging, turbulence-free interferometry. Includes new chapters for intensity fluctuation correlation and thermal light ghost imaging, and biphoton imaging. Offers a complete overhaul of the introductory theory to give a more coherent and thorough treatment. Expands on discussions of optical tests of quantum theory, Popper’s experiment, Einstein’s locality questions, and the delayed choice quantum eraser.
This study offers a comprehensive technical analysis of the mechanisms through which classical behavior emerges from the dynamics of quantum measurement. Beginning with unitary evolution and the principles of measurement, we construct the framework of environment-induced decoherence, the selection of pointer states, and the reduction of interference. Our examination focuses on how effective classicality emerges through reduced density matrices, quantum Darwinism, and consistent histories, highlighting the importance of system-environment interactions in determining stable observables. The review investigates into mathematical constructs such as spectral theory, completely positive maps, and quantum trajectories, linking them to experimental observations in mesoscopic systems, optomechanics, and superconducting circuits. We explore the constraints of decoherence in addressing the outcome problem and review interpretational models that tackle the remaining issues. The discussion emphasizes applications in quantum information, metrology, and tests of macroscopic superpositions. The article favors physical insights and mathematical guidance over exhaustive proofs, making it suitable for researchers in quantum foundations and quantum technologies.
In these notes, based on lectures given as part of the Les Houches summer school on Quantum Optics and Nanophotonics in August, 2013, I have tried to give a brief survey of some important approaches and modern tendencies in quantum measurement. I wish it to be clear from the outset that I shy explicitly away from the "quantum measurement problem," and that the present treatment aims to elucidate the theory and practice of various ways in which measurements can, in light of quantum mechanics, be carried out; and various formalisms for describing them. While the treatment is by necessity largely theoretical, the emphasis is meant to be on an experimental "perspective" on measurement -- that is, to place the priority on the possibility of gaining information through some process, and then attempting to model that process mathematically and consider its ramifications, rather than stressing a particular mathematical definition as the {\it sine qua non} of measurement. The textbook definition of measurement as being a particular set of mathematical operations carried out on particular sorts of operators has been so well drilled into us that many have the unfortunate tendency of saying "that experiment can't be described by projections onto the eigenstates of a Hermitian operator, so it is not really a measurement," when of course any practitioner of an experimental science such as physics should instead say "that experiment allowed us to measure something, and if the standard theory of measurement does not describe it, the standard theory of measurement is incomplete." Idealisations are important, but when the real world breaks the approximations made in the theory, it is the theory which must be fixed, and not the real world.
Recently we proposed the linguistic interpretation of quantum mechanics (called quantum and classical measurement theory), which was characterized as a kind of metaphysical and linguistic turn of the Copenhagen interpretation. This turn from physics to language does not only extend quantum theory to classical systems but also yield the quantum mechanical world view (i.e., quantum philosophy or quantum language). The purpose of this paper is to formulate the double-slit experiment, the quantum eraser experiment, Wheeler's delayed choice experiment, Hardy's paradox and the three boxes paradox (the weak value associated with a weak measurement due to Aharonov, et al.) in the linguistic interpretation of quantum mechanics. Through these arguments, we assert that the linguistic interpretation is just the final version of so called Copenhagen interpretation. And therefore, we conclude that the Copenhagen interpretation does not belong to physics (i.e., the realistic world view) but the linguistic world view.