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Proceedings of 2005 Symposium on Applied Functional Analysis : information sciences and related fields / editors, Toshiaki Murohushi, Wataru Takahashi, Makoto Tsukada. Yokohama Publishers,c2007.

The Einstein-Rupp experiments have been unduly neglected in the history of quantum mechanics. While this is to be explained by the fact that Emil Rupp was later exposed as a fraud and had fabricated the results, it is not justified, due to the importance attached to the experiments at the time. This paper discusses Rupp's fraud, the relation between Albert Einstein and Rupp, and the Einstein-Rupp experiments, and argues that these experiments were an influence on Niels Bohr's development of complementarity and Werner Heisenberg's formulation of the uncertainty relations., One Hundred Years of the Bohr Atom, 1913-2013. Conference at the Royal Danish Academy of Sciences and Letters, Copenhagen, 11-14 June 2013. Published as Scientia Danica. Series M, Mathematica et physica, 1: One Hundred Years of the Bohr Atom, Proceedings, 2015, pp. 310-343

In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the "uncertainty regions'' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r(s), where r(s)/s goes to 1 as s tends to infinity., 30 pages, 22 figures, 1 cut-out paper model, video abstract available on https://youtu.be/h01pHekcwFA

The standard uncertainty relations (UR) in quantum mechanics are typically used for unbounded operators (like the canonical pair). This implies the need for the control of the domain problems. On the other hand, the use of (possibly bounded) functions of basic observables usually leads to more complex and less readily interpretable relations. Also, UR may turn trivial for certain states if the commutator of observables is not proportional to a positive operator. In this letter we consider a generalization of standard UR resulting from the use of two, instead of one, vector states. The possibility to link these states to each other in various ways adds additional flexibility to UR, which may compensate some of the above mentioned drawbacks. We discuss applications of the general scheme, leading not only to technical improvements, but also to interesting new insight., Comment: 13 pages

Considering that the movements of complex system entities take place on continuous, but non-differentiable, curves, concepts, like non-differentiable entropy, informational non-differentiable entropy and informational non-differentiable energy, are introduced. First of all, the dynamics equations of the complex system entities (Schrödinger-type or fractal hydrodynamic-type) are obtained. The last one gives a specific fractal potential, which generates uncertainty relations through non-differentiable entropy. Next, the correlation between informational non-differentiable entropy and informational non-differentiable energy implies specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. Finally, for a harmonic oscillator, the constant value of the informational non-differentiable energy is equivalent to a quantification condition.

A discrete-event simulation approach which provides a cause-and-effect description of manyexperiments with photons and neutrons exhibiting interference and entanglement is applied to a recentsingle-neutron experiment that tests (generalizations of) Heisenberg's uncertainty relation.The event-based simulation algorithm reproduces the results of thequantum theoretical description of the experimentbut does not require the knowledge of the solution of a wave equation nor does itrely on concepts of quantum theory.In particular, the data satisfies uncertainty relations derived in the context of quantum theory.

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order $\alpha$ rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one., Comment: This version 2 contains minor corrections and reformulations

We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region -- this is the Zeno effect. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which kills interference between histories. We show how this works explicitly by coupling to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally-induced fluctuations in momentum are sufficiently large and we establish the associated timescale. We link our results to earlier work on the hbar -->0 limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper demonstrating the suppression of quantum-mechanical reflection by decoherence, 45 pages, 8 figures

While the slogan "no measurement without disturbance" has established itself under the name Heisenberg effect in the consciousness of the scientifically interested public, a precise statement of this fundamental feature of the quantum world has remained elusive, and serious attempts at rigorous formulations of it as a consequence of quantum theory have led to seemingly conflicting preliminary results. Here we show that despite recent claims to the contrary [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)], Heisenberg-type inequalities can be proven that describe a trade-off between the precision of a position measurement and the necessary resulting disturbance of momentum (and vice versa). More generally, these inequalities are instances of an uncertainty relation for the imprecisions of any joint measurement of position and momentum. Measures of error and disturbance are here defined as figures of merit characteristic of measuring devices. As such they are state independent, each giving worst-case estimates across all states, in contrast to previous work that is concerned with the relationship between error and disturbance in an individual state., Comment: Version 2 contains a more explicit description of the significance of the error-disturbance relation, formulated here for figures of merit of measuring devices, and its contrast with approaches that use state-dependent measures of error and disturbance

The thesis starts with a high-level introduction into cryptography and quantum mechanics. Chapter 2 gives a theoretical foundation by introducing probability theory, information theory, functional analysis, quantum mechanics and quantum information theory. Chapter 3, 4 and 5 are editions of work published earlier. In Chapter 3, we present a quantum-information-theoretic tool to analyze random sampling in a quantum setting. In particular, we present two new rigorous security proofs that make use of our new sampling tool: one for BB84 quantum key distribution, and one for a quantum reduction from oblivious transfer (OT) to bit commitment. Chapter 4 studies the problem of message authentication from a weak key (which is a key that is not uniformly random, e.g., a password) in a new scenario. In this scenario, the weak key is a one-time session key that is derived from a public source of randomness with the help of a long-term key (e.g., a password). We propose a new four-round protocol for message authentication from a weak (session) key. In Chapter 5 we present a new entropic uncertainty relation and furthermore we consider the task of password-based identification. We devise a new quantum identification protocol that is secure in two security models simultaneously.

This thesis is concerned with different characterizations of multi-particle quantum correlations and with entropic uncertainty relations. The effect of statistical errors on the detection of entanglement is investigated. First, general results on the statistical significance of entanglement witnesses are obtained. Then, using an error model for experiments with polarization-entangled photons, it is demonstrated that Bell inequalities with lower violation can have higher significance. The question for the best observables to discriminate between a state and the equivalence class of another state is addressed. Two measures for the discrimination strength of an observable are defined, and optimal families of observables are constructed for several examples. A property of stabilizer bases is shown which is a natural generalization of mutual unbiasedness. For sets of several dichotomic, pairwise anticommuting observables, uncertainty relations using different entropies are constructed in a systematic way. Exponential families provide a classification of states according to their correlations. In this classification scheme, a state is considered as k-correlated if it can be written as thermal state of a k-body Hamiltonian. Witness operators for the detection of higher-order interactions are constructed, and an algorithm for the computation of the nearest k-correlated state is developed. Diese Arbeit befasst sich mit Charakterisierungen von Mehrteilchen-Quantenkorrelationen und mit entropischen Unschärferelationen. Der Einfluss statistischer Fehler auf die Detektion von Verschränkung wird untersucht. Zuerst werden allgemeine Resultate zur statistischen Signifikanz von Verschränkungszeugen erzielt, dann wird unter Verwendung eines Fehlermodells für polarisationsverschränkte Photonen gezeigt, dass Bellsche Ungleichungen mit niedrigerer Verletzung höhere Signifikanz haben können. Die Frage nach den besten Observablen zur Unterscheidung eines Zustands von der Äquivalenzklasse eines anderen wird behandelt. Zwei Maße für die Unterscheidungskraft werden definiert, und für mehrere Beispiele werden optimale Familien von Observablen gefunden. Es wird eine Eigenschaft von Stabilisatorbasen gezeigt, die eine natürliche Verallgemeinerung der mutual unbiasedness darstellt. Für Familien aus mehreren dichotomen, paarweise antikommutierenden Observablen werden Unschärferelationen mit verschiedenen Entropien systematisch konstruiert. Exponentielle Familien ermöglichen eine Klassifikation von Zuständen nach den enthaltenen Korrelationen. Hierbei wird ein Zustand als k-korreliert angesehen, wenn er sich als thermischer Zustand eines k-Teilchen-Hamiltonoperators schreiben lässt. Es werden Zeugenoperatoren zur Detektion von Wechselwirkungen höherer Ordnung konstruiert, und ein Algorithmus zur Berechnung des nächsten k-korrelierten Zustands wird entwickelt.

The historical development of uncertainty relation, begining with first Heisenberg's thoughts of uncertainty principle is summed up in this thesis. After proving validity of Schwarz inequality general uncertainty relation for two hermitian operators is obtained, and from this general version the validity of Heisenberg uncertainty relation is than proved. The most important part of this work is the obtention of two new uncertainty relations, which are stronger then Heisenberg or Robertson-Schrödinger uncertainty relation, and their specific form for two examples - a free particle in a state discribed by the gaussian wave packet and the linear harmonic oscillator with a wave function in the shape of gaussian packet.

The main non-classical feature of quantum key distribution (QKD) is that it is characterized by a trade-off relation that limits the information possibly gained by a spy and the quality of the transmission line between the authorized users. In particular, perfect cloning is impossible, due to this trade-off, while optimal imperfect cloning saturates the trade-off relation. We investigate by numerical methods the deep nature of this trade-off relation, in the case of optimal cloning, and find that it reveals a subtle interplay between fidelity and entanglement.

Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of ${N+1}$ mutually unbiased bases in Hilbert spaces of prime-power dimension ${N=p^\m}$, with $p$ prime and $\m$ a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number ($N=6,10,12,...$). In this review, we present a unified approach in which the basis states are labeled by numbers ${0,1,2,...,N-1}$ that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two--q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the ...[rest deleted], Comment: Review article with 106 pages, 3 figures, 4 tables, 196 references

The scaling properties of various composite information-theoretic measures (Shannon and R\'enyi entropy sums, Fisher and Onicescu information products, Tsallis entropy ratio, Fisher-Shannon product and shape complexity) are studied in position and momentum spaces for the non-relativistic hydrogenic atoms in the presence of parallel magnetic and electric fields. Such measures are found to be invariant at the fixed values of the scaling parameters given by $s_1 = B \hbar^3(4\pi\epsilon_0)^2 / (Z^2m^2e^3)$ and $s_2 = F \hbar^4(4\pi\epsilon_0)^3 / (Z^3e^5m^2)$. Numerical results which support the validity of the scaling properties are shown by choosing the representative example of the position space shape complexity. Physical significance of the resulting scaling behaviour is discussed., Comment: 10 pages, 2 figures

The Einstein-Rupp experiments have been unduly neglected in the history of quantum mechanics. While this is to be explained by the fact that Emil Rupp was later exposed as a fraud and had fabricated the results, it is not justified, due to the importance attached to the experiments at the time. This paper discusses Rupp's fraud, the relation between Albert Einstein and Rupp, and the Einstein-Rupp experiments, and argues that these experiments were an influence on Niels Bohr's development of complementarity and Werner Heisenberg's formulation of the uncertainty relations.

Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators $x$, $p$. The resulting Hamiltonians contain a contribution proportional to $p^4$ and their $p$-dependent terms may also be functions of $x$. The theory is illustrated by considering P\"oschl-Teller and Morse potentials., Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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We investigate the relationship between mixedness and entanglement for Gaussian states of continuous variable systems. We introduce generalized entropies based on Schatten $p$-norms to quantify the mixedness of a state, and derive their explicit expressions in terms of symplectic spectra. We compare the hierarchies of mixedness provided by such measures with the one provided by the purity (defined as ${\rm tr} \varrho^2$ for the state $\varrho$) for generic $n$-mode states. We then review the analysis proving the existence of both maximally and minimally entangled states at given global and marginal purities, with the entanglement quantified by the logarithmic negativity. Based on these results, we extend such an analysis to generalized entropies, introducing and fully characterizing maximally and minimally entangled states for given global and local generalized entropies. We compare the different roles played by the purity and by the generalized $p$-entropies in quantifying the entanglement and the mixedness of continuous variable systems. We introduce the concept of average logarithmic negativity, showing that it allows a reliable quantitative estimate of continuous variable entanglement by direct measurements of global and marginal generalized $p$-entropies., Comment: 18 pages, 9 figures; expanded version; to be published in Phys. Rev. A