This chapter presents the theoretical framework, based on Gleason’s theorem, allowing to describegeneralized measurements, in addition to von Neumann measurements, in terms of POVMs, probability operators and post-measurement operators. It mentions the Naimark theorem according to which a generalized measurement is a von Neumann measurement if one describes it in a Hilbert space of higher dimension. Examples of generalized measurements are given: imperfect measurements, simultaneous measurements of noncommuting operators. It presents the Zurek model that accounts for the decoherence process occurring in a measurement and shows that the quantum measurement process, including state collapse, is not a physical evolution. Finally, it studies the case of successive measurements using Bayes statistics in which the state collapse appears as an updating of the information about a system, and the fundamental property of repeatability of quantum mechanics.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.
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 Measurement in the quantum realm is not deterministic as in the classical one, and many of the quantum principles delineated in Chapter 1 are expanded. The results of individual quantum measurements cannot be predicted; quantum mechanics gives us only the probabilities of observing each of the possible measurement outcomes. Anchored in the concrete example of light polarization, Chapter 5 takes on the Born rule or the probability principle of quantum mechanics involving the square of state amplitudes. What is known before a measurement and what is known after (measurement principle) explains the no-cloning theorem. The standard terminology of eigenvectors and eigenstates is also introduced.
We begin with a study of operations and the effects they measure. We define the probability that an effect
a occurs when the system is in a state
ρ by
Pρ(a)=Tr(ρa). If
Pρ(a)=0 and
I is an operation that measures
a, we define the conditional probability of an effect
b given
a relative to
I by
Pρ(b∣a)=Tr[I(ρ)b]/Pρ(a). We characterize when Bayes' quantum second rule
Pρ(b∣a)=Pρ(a)Pρ(b)Pρ(a∣b) holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If
A and
B are observables and an instrument
I measures
A, we define the observable
B conditioned on
A relative to
I and denote it by
(B∣A). Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure
A. We then extend this to Bayes' quantum first rule for expectations. We show that two observables
B and
C are jointly commuting if and only if there exists an atomic observable
A such that
B=(B∣A) and
C=(C∣A). We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples. Quanta 2024; 13: 1–10.