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Springer eBooks

Real Single Quanta Measurements

James K. Freericks

Published Jan 1, 2026• 0 citations
Publisher

Abstract

The theoretical description of measurement in quantum mechanics is one of the most controversial topics in physics. It is deeply entrenched in the interpretations of quantum mechanics and combines both physics and philosophy into the discussion. Often it is described as the measurement problem because of some issues associated with the theory of measurement introduced by John von Neumann in 1932, related to the fact that the Born projection postulate is not unitary and so is not described by the time-dependent Schrödinger equation. We discuss that theory later in this chapter, but we start from a simpler approach focused on what is actually being done in most single-quantum experiments. Along the way, we encounter a number of counterintuitive and subtle issues, before we explain why we refuse to tackle the measurement problem itself. Specific experiments that we discuss in detail include how we can measure individual quanta and how we can manipulate entanglement and perform delayed choice experiments, quantum beats, atom interferometry, and quantum non-demolition experiments in cavity QED.

DOI
10.1007/978-3-031-87845-9_23

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The theoretical description of measurement in quantum mechanics is one of the most controversial topics in physics. It is deeply entrenched in the interpretations of quantum mechanics and combines both physics and philosophy into the discussion. Often it is described as the measurement problem because of some issues associated with the theory of measurement introduced by John von Neumann in 1932, related to the fact that the Born projection postulate is not unitary and so is not described by the time-dependent Schrödinger equation. We discuss that theory later in this chapter, but we start from a simpler approach focused on what is actually being done in most single-quantum experiments. Along the way, we encounter a number of counterintuitive and subtle issues, before we explain why we refuse to tackle the measurement problem itself. Specific experiments that we discuss in detail include how we can measure individual quanta and how we can manipulate entanglement and perform delayed choice experiments, quantum beats, atom interferometry, and quantum non-demolition experiments in cavity QED.
We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.
Abstract In this chapter, we present the theoretical foundations of quantum mechanics. Our presentation follows the von Neumann postulates. We discuss them stepping through the parts of a typical quantum experiment: the preparation of the system, the subsequent time evolution, and the final measurement. Specifically, we present normalized vectors in a Hilbert space, that describe preparations; projection-valued measures and Hermitian operators, that describe measurable observables; and the Schrodinger equation governing time evolution. In the last part of the chapter, we present two examples of quantum systems: the qubit and the quantum harmonic oscillator. We study the interaction between them in two different regimes
We provide an introduction to the theory of quantum measurements that is centered on the pivotalrole played by John von Neumann’s model. This introduction is accessible to students andresearchers from outside the field of foundations of quantum mechanics and presented withina historical context. We first explain the origins and the meaning of the measurement problemin quantum theory, and why it is not present in classical physics. We perform a chronologicalreview of the quantization of action and explain how this led to successive restrictions onwhat could be measured in atomic phenomena, until the consolidation of the orthodox interpretationof quantum mechanics. The clear separation between quantum system and classicalapparatus that causes these restrictions is subverted in von Neumann’s paradigmatic modelof quantum measurements, a subject whose concepts we explain, while also providing themathematical tools necessary to apply it to new problems. We show how this model was importantin discussing the interpretations of quantum mechanics and how it is still relevant inmodern applications. In particular, we explain in detail how it can be used to describe weakmeasurements and the surprising results they entail. We also discuss the limitations of vonNeumann’s model of measurements, and explain how they can be overcome with POVMs andKraus operators. We provide the mathematical tools necessary to work with these generalizedmeasurements and to derive master equations from them. Finally, we demonstrate how thesecan be applied in research problems by calculating the Quantum Zeno Effect.
The measurement problem in quantum mechanics (QM) is related to the inability to include learning about the properties of a quantum system by an agent in the formalism of quantum theory. It includes questions about the physical processes behind the measurement, uniqueness, and randomness of obtained outcomes and an ontic or epistemic role of the state. These issues have triggered various interpretations of quantum theory. They vary from refusing any connection between physical reality and a measurement process to insisting that a collapse of the wave-function is real and possibly involves consciousness. On the other hand, the actual mechanism of a measurement is not extensively discussed in these interpretations. This essay attempts to investigate the quantum measurement problem from the position of the scientific consensus. We begin with a short overview of the development of sensing in living organisms. This is performed for the purpose of stressing the relation between reality and our experience. We then briefly present different approaches to the measurement problem in chosen interpretations. We then state three philosophical assumptions for further consideration and present a decomposition of the measurement act into four stages: transformation, conversion, amplification and broadcasting, and, finally, perception. Each of these stages provides an intuition about the physical processes contributing to it. These conclusions are then used in a discussion about, e.g., objectivity, the implausibility of reversing a measurement, or the epistemic status of the wave-function. Finally, we argue that those in favor of some of the most popular interpretations can find an overlap between their beliefs and the consequences of considerations presented here.
Abstract The book starts with a description of classical mechanics then discusses the quantum phenomena that require us to give up our commonsense classical intuitions. We consider the physical and conceptual arguments that led to the standard von Neumann-Dirac formulation of quantum mechanics and how the standard theory explains quantum phenomena. This includes a discussion of how the theory’s two dynamical laws work with the standard interpretation of states to explain determinate measurement records, quantum statistics, interference effects, entanglement, decoherence, and quantum nonlocality. A careful understanding of how the standard theory works ultimately leads to the quantum measurement problem. We consider how the measurement problem threatens the logical consistency of the standard theory then turn to a discussion of the main proposals for resolving it. This includes collapse formulations of quantum mechanics like Wigner’s extension of the standard theory and the GRW approach and no-collapse formulations like pure wave mechanics, the various many-worlds theories, and Bohmian mechanics. In discussing alternative formulations of quantum mechanics we pay particular attention to the explanatory role played by each theory’s empirical ontology and associated metaphysical commitments and the conceptual trade-offs between theoretical options.
Foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity which persists throughout an experiment and therefore it does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and it does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions like a Bose-Einstein condensate system for which a single measurement yields a real ensemble. We discuss the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from quantum field theory.