scholarly journals Relaxed uncertainty relations and information processing

2009 ◽  
Vol 9 (9&10) ◽  
pp. 801-832 ◽  
Author(s):  
G. Ver Steeg ◽  
S. Wehner
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Random Access ◽  
The Other ◽  
Uncertainty Relations ◽  
Expectation Values ◽  
Chsh Inequality ◽  
Local Correlations ◽  
Non Local

We consider a range of "theories'' that violate the uncertainty relation for anti-commuting observables derived. We first show that Tsirelson's bound for the CHSH inequality can be derived from this uncertainty relation, and that relaxing this relation allows for non-local correlations that are stronger than what can be obtained in quantum mechanics. We continue to construct a hierarchy of related non-signaling theories, and show that on one hand they admit superstrong random access encodings and exponential savings for a particular communication problem, while on the other hand it becomes much harder in these theories to learn a state. We show that the existence of these effects stems from the absence of certain constraints on the expectation values of commuting measurements from our non-signaling theories that are present in quantum theory.

Time in Quantum Mechanics

2011 ◽  
Author(s):  
Jan Hilgevoord ◽  
David Atkinson
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Classical Mechanics ◽  
Special Problem ◽  
Uncertainty Relations ◽  
Operator Formalism ◽  
Heisenberg Uncertainty ◽  
Time In Quantum Mechanics ◽  
Time Operators

Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.

scholarly journals BLACK HOLE UNCERTAINTIES

1993 ◽  
Vol 08 (20) ◽  
pp. 1925-1941
Author(s):  
ULF H. DANIELSSON
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Wave Functions ◽  
Uncertainty Relations ◽  
Two Dimensional ◽  
Black Hole Mass ◽  
Dilaton Black Holes

In this work the quantum theory of two-dimensional dilaton black holes is studied using the Wheeler-De Witt equation. The solutions correspond to wave functions of the black hole. It is found that for an observer inside the horizon, there are uncertainty relations for the black hole mass and a parameter in the metric determining the Hawking flux. Only for a particular value of this parameter can both be known with arbitrary accuracy. In the generic case there is instead a relation that is very similar to the so-called string uncertainty relation.

scholarly journals Observability, Unobservability and the Copenhagen Interpretation in Dirac's Methodology of Physics

Quanta ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 68-87 ◽  
Author(s):  
Andrea Oldofredi ◽  
Michael Esfeld
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Close Contact ◽  
Copenhagen Interpretation ◽  
The Other ◽  
Interpretation Of Quantum Mechanics ◽  
Present Essay ◽  
Other Hand ◽  
Methodological Principles ◽  
The One

Paul Dirac has been undoubtedly one of the central figures of the last century physics, contributing in several and remarkable ways to the development of quantum mechanics; he was also at the centre of an active community of physicists, with whom he had extensive interactions and correspondence. In particular, Dirac was in close contact with Bohr, Heisenberg and Pauli. For this reason, among others, Dirac is generally considered a supporter of the Copenhagen interpretation of quantum mechanics. Similarly, he was considered a physicist sympathetic with the positivistic attitude which shaped the development of quantum theory in the 1920s. Against this background, the aim of the present essay is twofold: on the one hand, we will argue that, analyzing specific examples taken from Dirac's published works, he can neither be considered a positivist nor a physicist methodologically guided by the observability doctrine. On the other hand, we will try to disentangle Dirac's figure from the mentioned Copenhagen interpretation, since in his long career he employed remarkably different—and often contradicting—methodological principles and philosophical perspectives with respect to those followed by the supporters of that interpretation.Quanta 2019; 8: 68–87.

scholarly journals Uncertainty Relations in Hydrodynamics

2020 ◽  
Author(s):  
Gyell Gonçalves de Matos ◽  
Takeshi Kodama ◽  
Tomoi Koide
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Quantum Particle ◽  
Navier Stokes ◽  
Pitaevskii Equation ◽  
Uncertainty Relations ◽  
Derived Relations ◽  
Stochastic Variational Method ◽  
Virtual Trajectory ◽  
Fourier Equation

Uncertainty relations in hydrodynamics are numerically studied. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys.\ Lett.\ A\textbf{382}, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schr\"{o}dinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid.

scholarly journals Uncertainty relations: An operational approach to the error-disturbance tradeoff

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 20 ◽  
Author(s):  
Joseph M. Renes ◽  
Volkher B. Scholz ◽  
Stefan Huber
Keyword(s):  
Quantum Theory ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
Actual Measurement ◽  
Quantum Uncertainty ◽  
Joint Measurability ◽  
Finite Dimensional ◽  
Wave Particle Duality ◽  
Heisenberg Type ◽  
Conceptual Difficulties

The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.

scholarly journals Uncertainty Relations: Curiosities and Inconsistencies

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1640
Author(s):  
Krzysztof Urbanowski
Keyword(s):  
Quantum Theory ◽  
Lower Bound ◽  
Uncertainty Relation ◽  
Quantum Particle ◽  
Uncertainty Relations ◽  
Rigorous Analysis ◽  
Special Cases ◽  
The Status ◽  
Symmetric Quantum ◽  
Standard Deviations

Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.

The basis of statistical quantum mechanics

1929 ◽  
Vol 25 (1) ◽  
pp. 62-66 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Dynamical System ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Degrees Of Freedom ◽  
Present Note ◽  
Classical Mechanics ◽  
The Other ◽  
Statistical Nature ◽  
Actual System ◽  
Quantum Treatment

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.

XX.—Aristotle, Newton, Einstein

1943 ◽  
Vol 61 (3) ◽  
pp. 231-246
Author(s):  
E. T. Whittaker
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
External World ◽  
The Other ◽  
Isaac Newton ◽  
Hundredth Anniversary ◽  
Essential Nature ◽  
Great Discovery ◽  
Eighteenth And Nineteenth Centuries ◽  
Almost All

IT falls to us this year to commemorate the greatest of men of science, Isaac Newton, on the occasion of the three-hundredth anniversary of his birth. The centuries have not dimmed his fame, and the passage of time is unlikely ever to displace him from the supreme position. His discoveries, however—and this is part of their glory—have not persisted unchanged, but in the hands of his successors have been continually unfolding into fresh evolutions. During the eighteenth and nineteenth centuries there was an immense expansion of knowledge, springing directly from his work, and forming ultimately a vast superstructure based on the Newtonian concepts of space, mass, and force. Since 1900 the progress of science has continued, but the development of physics has changed in character: it has become subversive and radical, questioning the traditional assumptions and uprooting the old foundations. In 1915 the Newtonian doctrine of gravitation was superseded by that of Einstein: the divergence between the results of the two theories, so far as concerns the calculation of the movements of the planets, is extremely slight, and indeed, in almost all cases, too small to be detected by observation; but on the question of the essential nature of gravitation, the two conceptions differ completely and are associated with opposite philosophies of the external world. The other great discovery of the present century is the quantum theory, which in its perfected form of quantum-mechanics appeared in 1925: this also is completely irreconcilable with the postulates of Newtonian science.

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