We present a mathematically precise derivation of some generalized orthogonality relations for the discrete series representations of SU(1,1). These orthogonality relations are applied to derive tomographical reconstruction formulas. Their physical interpretation is also discussed.

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component (...) systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category. (shrink)

The dynamical equations of quantum mechanics are rewritten in the form of dynamical equations for the measurable, positive marginal distribution of the shifted, rotated, and squeezed quadrature introduced in the so-called “symplectic tomography”. Then the possibility of a purely classical description of a quantum system as well as a reinterpretation of the quantum measurement theory is discussed and a comparison with the well-known quasi-probabilities approach is given. Furthermore, an analysis of the properties of this marginal distribution, (...) which contains all the quantum information, is performed in the framework of classical probability theory. Finally, examples of the harmonic oscillator's states dynamics are treated. (shrink)

Quantum theory has the property of “local tomography”: the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by “bilocal tomography”: the state of any composite system is determined by the statistics of measurements on pairs of components. (...) Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that real-vector-space quantum theory, while not locally tomographic, is bilocally tomographic. (shrink)

In spite of the interference manifested in the double-slit experiment, quantum theory predicts that a measure of interference defined by Sorkin and involving various outcome probabilities from an experiment with three slits, is identically zero. We adapt Sorkin’s measure into a general operational probabilistic framework for physical theories, and then study its relationship to the structure of quantum theory. In particular, we characterize the class of probabilistic theories for which the interference measure is zero as ones in which (...) it is possible to fully determine the state of a system via specific sets of ‘two-slit’ experiments. (shrink)

Initially Einstein, Podolsky, and Rosen (EPR) and later Bell shed light on the non-local properties exhibited by subsystems in quantum mechanics. Separately, Kochen and Specker analyzed sets of measurements of compatible observables and found that a consistent coexistence of these results is impossible, i.e., quantum indefiniteness of measurement results. As a consequence, quantum contextuality, a more general concept compared to non-locality, leads to striking phenomena predicted by quantum theory. Here, we report neutron interferometric experiments which investigate (...) entangled states in a single-particle system: entanglement is, in this case, achieved not between particles, but between degrees of freedom i.e., between spin, path, and energy degrees of freedom. Appropriate combinations of the spin analysis and the position of the phase shifter in the interferometer allow an experimental verification of the violation of a Bell-like inequality. In addition, state tomography, tomographic analysis of the density matrix of a quantum system, and Kochen-Specker-like phenomena are presented to characterize neutrons’ entangled states and their peculiarity. Furthermore, a coherent energy manipulation scheme is accomplished with a radio-frequency (RF) spin-flipper. This scheme allows the (total) energy degree of freedom to be entangled: the remarkable behavior of a triply entangled GHZ-like state is demonstrated. (shrink)

We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PS I-complete) POVM. We show that a measurement with 2D−1 outcomes cannot be PS I-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PS I-complete POVM has 2D outcomes. We also consider PS I-complete POVMs that have only rank-one POVM elements (...) and construct an example with 3D−2 outcomes, which is a generalization of the tetrahedral measurement for a qubit. The question of the minimal number of elements in a rank-one PS I-complete POVM is left open. (shrink)

We present strategies how to reconstruct (estimate) properties of a quantum channel described by the map E based on incomplete measurements. In a particular case of a qubit channel a complete reconstruction of the map E can be performed via complete tomography of four output states E[ρj] that originate from a set of four linearly independent “test” states ρj (j = 1,2,3,4) at the input of the channel. We study the situation when less than four linearly independent states (...) are transmitted via the channel and measured at the output. We present strategies how to reconstruct the channel when just one, two or three states are transmitted via the channel. In particular, we show that if just one state is transmitted via the channel then the best reconstruction can be achieved when this state is a total mixture described by the density operator ρ = I/2. To improve the reconstruction procedure one has to send via the channel more states. The best strategy is to complement the total mixture with pure states that are mutually orthogonal in the sense of the Bloch-sphere representation. We show that unitary transformations (channels) can be uniquely reconstructed (determined) based on the information of how three properly chosen input states are transformed under the action of the channel. (shrink)

Presents a guide to the basics of quantum mechanics and measurement.

Quantum mechanics is humanity's finest scientific achievement. It explains why the sun shines and how your eyes can see. It's the theory behind the LEDs in your phone and the nuclear hearts of space probes. Every physicist agrees quantum physics is spectacularly successful. But ask them what quantum physics means, and the result will be a brawl. At stake is the nature of the Universe itself. What does it mean for something to be real? What is the (...) role of consciousness in the Universe? And do quantum rules apply to very small objects like electrons and protons, but not us? In What is Real?, Adam Becker brings to vivid life the brave researchers whose quest for the truth led them to challenge Bohr: David Bohm, who picked up Einstein's mantle and sought to make quantum mechanics deterministic, all while being hounded by the forces of McCarthyism; Hugh Everett, who argued that everything, big and small, must be governed by the same rules; and John Bell, who went to great lengths to eradicate the power of the god-like observer from the core of quantum physics. And they paid dearly, their reputations, careers, and sometimes lives ruined completely. But history has been kinder to them than their contemporaries were. As Becker shows, the brave intellectual giants have inspired a growing army of physicists and philosophers intent both on making a philosophically more satisfying theory of the universe and a more useful one as well. A gripping story of some of humanity's greatest ideas and the high cost with which many have pursued them, What is Real? is intellectual history at its passionate best. (shrink)

One of the most interesting programs in the foundations of quantum mechanics is the realist quantum logic approach associated with Putnam, Bub, Demopoulos and Friedman (and which is the focus of my own research.) I believe that realist quantum logic is our best hope for making sense of quantum mechanics, but I have come to suspect that the usual version may not be the correct one. In this paper, I would like to say why and to (...) propose an alternative. (shrink)

It is argued that the symmetry and anti-symmetry of the wave functions of systems consisting of identical particles have nothing to do with the observational indistinguishability of these particles. Rather, a much stronger conceptual indistinguishability is at the bottom of the symmetry requirements. This can be used to argue further, in analogy to old arguments of De Broglie and Schrödinger, that the reality described by quantum mechanics has a wave-like rather than particle-like structure. The question of whether quantum (...) statistics alone can give rise to empirically observable correlations between results of distant measurements is also discussed. (shrink)

Modern particle physics suggests an intriguing vision of physical reality: we are to imagine the symmetries of the world as fundamental, whereas the material constituents of the world are ontologically derivative of them. This paper develops a novel ontology for non-relativistic quantum mechanics which gives precise metaphysical content to this vision.

It is argued that certain recent advances in the construction of a theory of the collapses of Quantum Mechanical wave functions suggest the possibility of new and improved foundations for statistical mechanics, foundations in which epistemic considerations play no role.

It is almost universally believed that in quantum theory the two following statements hold: all transformations are achieved by a unitary interaction followed by a von-Neumann measurement; all mixed states are marginals of pure entangled states. I name this doctrine the dogma of purification ontology. The source of the dogma is the original von Neumann axiomatisation of the theory, which largely relies on the Schrődinger equation as a postulate, which holds in a nonrelativistic context, and whose operator version holds (...) only in free quantum field theory, but no longer in the interacting theory. In the present paper I prove that both ontologies of unitarity and state-purity are unfalsifiable, even in principle, and therefore axiomatically spurious. I propose instead a minimal four-postulate axiomatisation: associate a Hilbert space \ to each system \; compose two systems by the tensor product rule \; associate a transformation from system \ to \ to a quantum operation, i.e. to a completely positive trace-non-increasing map between the trace-class operators of \ and \; evaluate all joint probabilities through that of a special type of quantum operation: the state preparation. I then conclude that quantum paradoxes—such as the Schroedinger-cat’s, and, most relevantly, the information paradox—are originated only by the dogma of purification ontology, and they are no longer paradoxes of the theory in the minimal formulation. For the same reason, most interpretations of the theory interpret the same dogma, not the strict theory stripped of the spurious postulates. (shrink)

Contrary to Bell’s theorem it is demonstrated that with the use of classical probability theory the quantum correlation can be approximated. Hence, one may not conclude from experiment that all local hidden variable theories are ruled out by a violation of inequality result.

Standard canonical quantum mechanics makes much use of operators whose spectra cover the set of real numbers, such as the coordinates of space, or the values of the momenta. Discrete quantum mechanics uses only strictly discrete operators. We show how one can transform systems with pairs of integer-valued, commuting operators $P_i$ and $Q_i$ , to systems with real-valued canonical coordinates $q_i$ and their associated momentum operators $p_i$ . The discrete system could be entirely deterministic while the corresponding (p, (...) q) system could still be typically quantum mechanical. (shrink)

The territory of time travel has, from the days of H. G. Wells to the mid-1980's, been the exclusive province of writers of science fiction and fantasy. SF critics have even argued that time travel stories are so scientifically unlikely that they should be considered fantasy, not science fiction.

By exploring principles found in psychology, math, physics, and shamanism, it becomes possible to link a cosmic perspective with ordinary life. This comprehensive work ventures into that challenging junction, journeying through the universe on paths of reason and magic, math and myth, bringing together humanity's traditional wisdom and shamanism with contemporary science.

We present a theory of discontinuous motion of particles in continuous space-time. We show that the simplest nonrelativistic evolution equation of such motion is just the Schroedinger equation in quantum mechanics. This strongly implies what quantum mechanics describes is discontinuous motion of particles. Considering the fact that space-time may be essentially discrete when considering gravity, we further present a theory of discontinuous motion of particles in discrete space-time. We show that its evolution will naturally result in the dynamical (...) collapse process of the wave function, and this collapse will bring about the appearance of continuous motion of objects in the macroscopic world. (shrink)

A quantum mechanical model of a counter monitoring the decay of an unstable microsystem is constructed. Detailed investigation of the time evolution of this model shows that the counter behaves essentially classically; i.e., its discharges may be considered as objective, observer-independent events. The possible relevance of this result for the physical interpretation of quantum mechanics is discussed.

The Bohr formulation—the original one—is now largely a matter of history. The usual career of a theory was its fate. It explained much, but it failed at the first signs of complication, even for the simplest molecules as hydrogen and helium. Some results were either only approximate in a loose or incomplete fashion, e.g. in the application of the correspondence principle to intensities, the only reliable predictions being only for an absence of certain lines, or they quite disagreed with experiment, (...) e.g. in the case of the hydrogen molecule and the hydrogen molecule ion, the normal and excited states of helium, the anomalous Zeeman effect, the Ramsauer effect, and in the treatment of problems where more than one electron was involved. In the last mentioned cases the perturbation method, so successful in solving “many-body” problems in celestial mechanics, failed completely because systems of several electrons, possessing frequencies of the order of light waves, cannot interact classically. A bit of much forced pragmatism was practiced in treating the many electron atom as a variation of the hydrogen atom. Besides, there were technical oddities, like ½ quantum numbers, and impenetrated anologies between visible and X-ray spectra. (shrink)

The “observer” in physics has always referred to someone who stands on the outside of a system looking in. In this paper an “inside observer” is defined, and an experiment is proposed that tests a given formulation of the problem of measurement in quantum mechanics.

Several interesting and important quantum field theories must contain the coupling constant in the boundary conditions. The theories considered include quantum electrodynamics of spin-1/2 fermions and gauge field theories.

This paper shows how different axiomatic and constructive approaches within quantum field theory can be understood in terms of the so-called ,picture theory' of Heinrich Hertz. Each approach will count as a different picture due to the different status of the various concepts (symbols) they are employing, like observables, gauge invariance, confinement or the space-time continuum. An important difference with the original Hertzian approach is the fact that the different approaches in quantum field theory have partially overlapping, partially (...) supplementing domains of application. This also marks some of the parallels and differences with contemporary debates on structural realism and model-theoretic approaches in the philosophy of physics. The objection that the talk about different pictures just relies on the fact that quantum field theory is unfinished will be countered. Finally, the Hertzian approach will be briefly elaborated and embedded into its philosophical successor projects of Cassirer and Goodman. German Dieser Aufsatz zeigt, wie verschiedene axiomatische und konstruktive Ansätze in der Quantenfeldtheorie mit Hilfe der sogenannten ,,Bildtheorie“ von Heinrich Hertz verstanden werden können. Aufgrund der unterschiedlichen Stellungen der verschiedenen Konzepte (Symbole), die diese Ansätze benutzen - wie etwa Observable, Eichinvarianz, Confinement oder Raum-Zeit-Kontinuum - wird jeder als ein eigenes Bild zu betrachten sein. Ein wichtiger Unterschied zu Hertz' ursprünglicher Fassung der Bildtheorie sind dabei die unterschiedlichen Anwendungsbereiche der Ansätze in der Quantenfeldtheorie. Diese überschneiden sich zum Teil, zum Teil ergänzen sie einander. Dadurch ergeben sich für die bildtheoretische Interpretation wichtige Parallelen und Unterschiede zu anderen Positionen in der Philosophie der Physik, insbesondere zum Strukturrealismus und zur Modell-Theorie. Neben diesen diskutiere ich den Einwand, dass die Redeweise von verschiedenen Bildern in der Quantenfeldtheorie nur deshalb möglich sei, weil die Theorie noch nicht abgeschlossen ist. Schließlich erweitere ich die Hertzsche Bildtheorie und setze sie dabei in Beziehung zu ihren philosophischen Nachfolgeprojekten bei Cassirer und Goodman. (shrink)