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Pedagogical Review of Quantum Measurement Theory with an Emphasis on Weak Measurements
Get PDFThe quantum theory of measurement has been with us since quantum mechanics was invented. It has recently been invigorated, partly due to the increasing interest in quantum information science. In this partly pedagogical review I attempt to give a self-contained overview of non-relativistic quantum theory of measurement expressed in density matrix formalism. I will not dwell on the applications in quantum information theory; it is well covered by several books in that field. The focus is instead on applications to the theory of weak measurement, as developed by Aharonov and collaborators. Their development of weak measurement combined with what they call post-selection - judiciously choosing not only the initial state of a system (pre-selection) but also its final state - has received much attention recently. Not the least has it opened up new, fruitful experimental vistas, like novel approaches to amplification. But the approach has also attached to it some air of mystery. I will attempt to demystify it by showing that (almost) all results can be derived in a straight-forward way from conventional quantum mechanics. Among other things, I develop the formalism not only to first order but also to second order in the weak interaction responsible for the measurement. I apply it to the so called Leggett-Garg inequalities, also known as Bell inequalities in time. I also give an outline, even if rough, of some of the ingenious experiments that the work by Aharonov and collaborators has inspired. As an application of weak measurement, not related to the approach by Aharonov and collaborators, the formalism also allows me to derive the master equation for the density matrix of an open system in interaction with an environment. An issue that remains in the weak measurement plus post-selection approach is the interpretation of the so called weak value of an observable. Is it a bona fide property of the system considered? I have no definite answer to this question; I shall only exhibit the consequences of the proposed interpretation.Quanta 2013; 2: 18–49
Are the Weak Measurements Really Measurements?
Get PDFWeak measurements can be seen as an attempt at answering the Which way? question without destroying interference between the pathways involved. Unusual mean values obtained in such measurements represent the response of a quantum system to this forbidden question, in which the true composition of virtual pathways is hidden from the observer. Such values indicate a failure of a measurement where the uncertainty principle says it must fail, rather than provide an additional insight into physical reality.Quanta 2013; 2: 50–57
A Career of Unyielding Exploration: In Memory of Ion C. Baianu (1947-2013)
Get PDFWe were deeply saddened to learn of the sudden death of our colleague, friend, and member of the Editorial Board of Quanta, Professor Ion C. Baianu, who unexpectedly passed away in Urbana, Illinois, USA, on February 10, 2013. Ion left behind his wife, Kimiko, his son, Stephen, and daughters, Antonia and Christina. He also left behind the achievements of a profound and illustrious career in his chosen fields of biophysics, spectroscopy, food science, and bioengineering.Quanta 2013; 2: 1–6
Introduction to Weak Measurements and Weak Values
Get PDFWe present a short review of the theory of weak measurement. This should serve as a map for the theory and an easy way to get familiar with the main results, problems and paradoxes raised by the theory.Quanta 2013; 2: 7–17
Einstein's Recoiling Slit Experiment, Complementarity and Uncertainty
Get PDFWe analyze Einstein's recoiling slit experiment and point out that the inevitable entanglement between the particle and the recoiling slit was not part of Bohr's reply. We show that if this entanglement is taken into account, one can provide a simpler answer to Einstein. We also derive the Englert-Greenberger-Yasin duality relation from the entanglement between the particle and the recoiling slit. In addition, we show that the Englert-Greenberger-Yasin duality relation can also be thought of as a consequence of the sum uncertainty relation for certain observables of the recoiling slit. Thus, the uncertainty relations and entanglement are both an integral part of the which-way detection process.Quanta 2013; 2: 58–65
Morlet Wavelets in Quantum Mechanics
Get PDFWavelets offer significant advantages for the analysis of problems in quantum mechanics. Because wavelets are localized in both time and frequency they avoid certain subtle but potentially fatal conceptual errors that can result from the use of plane wave or δ function decomposition. Morlet wavelets in particular are well-suited for this work: as Gaussians, they have a simple analytic form and they work well with Feynman path integrals. But to take full advantage of Morlet wavelets we need to supply an explicit form for the inverse Morlet transform and a manifestly covariant form for the four-dimensional Morlet wavelet. We construct both here.Quanta 2012; 1: 58–70
A Historical Survey of Sir Karl Popper's Contribution to Quantum Mechanics
Get PDFSir Karl Popper (1902-1994), though not trained as a physicist and embarrassed early in his career by a physics error pointed out by Einstein and Bohr, ultimately made substantial contributions to the interpretation of quantum mechanics. As was often the case, Popper initially formulated his position by criticizing the views of others - in this case Niels Bohr and Werner Heisenberg. Underlying Popper's criticism was his belief that, first, the Copenhagen interpretation of quantum mechanics abandoned scientific realism and second, the assertion that quantum theory was complete (an assertion rejected by Einstein among others) amounted to an unfalsifiable claim. Popper insisted that the most basic predictions of quantum mechanics should continue to be tested, with an eye towards falsification rather than mere adding of decimal places to confirmatory experiments. His persistent attacks on the Copenhagen interpretation were aimed not at the uncertainty principle itself and the formalism from which it was derived, but at the acceptance by physicists of an unclear epistemology and ontology that left critical questions unanswered.Quanta 2012; 1: 1–12
Popper's Experiment: A Modern Perspective
Get PDFKarl Popper had proposed an experiment to test the standard interpretation of quantum mechanics. The proposal survived for many year in the midst of no clear consensus on what results it would yield. The experiment was realized by Kim and Shih in 1999, and the apparently surprising result led to lot of debate. We review Popper's proposal and its realization in the light of current era when entanglement has been well studied, both theoretically and experimentally. We show that the "ghost-diffraction" experiment, carried out in a different context, conclusively resolves the controversy surrounding Popper's experiment.Quanta 2012; 1: 19–32
Problems in the Science and Mathematics of 'The Logic of Scientific Discovery'
Get PDFProfessor Sir Karl Popper (1902-1994) was one of the most influential philosophers of science of the twentieth century. However, in his most famous work 'The Logic of Scientific Discovery' he displays troubling misunderstandings of science and mathematics at a basic level. These call into question his conclusions concerning the philosophy of science.Quanta 2012; 1: 13–18
Popper and Bohr on Realism in Quantum Mechanics
Get PDFoai:ojs.quanta.ws:article/9Popper's program in the foundations of quantum mechanics defending objectivity and realism developed out of a profound dissatisfaction with the point of view associated with Bohr, which is usually designated the Copenhagen interpretation. Here I will argue that while Popper's aim is a noble one, his program does not succeed on two counts: he does not succeed in showing that Bohr's philosophy must be rejected as a variety of subjectivism, and his alternative interpretation of indeterminacy rests on a highly questionable assumption according to which simultaneously precise conjugate parameters are possible. Nevertheless I like Popper's propensity interpretation of probability and think that the propensity idea deserves further research.Quanta 2012; 1: 33–57