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GENERAL PROBLEMS OF METROLOGY AND MEASUREMENT TECHNIQUES

Z. I. Sharagina, S. K. Ivanov

Published Jan 1, 1979• 0 citations
OpenAlex

Abstract

(i) is therefore a good reflection of the crux of the measurement process. Here H(X) is the which characterizes the a priori of the quantity being measured, and H(X/Y) is the total conditional which characterizes the of the quantity being measured which remains after the measurement result Y has been obtained. Let us examine the application of the concept in the case of continuous distributions. The value of the as a numerical characteristic of a continuous random variable is rejected in [12] on the basis that the equals ~ for continuous random variables. The introduction of the probability density p(x) could be rejected on approximately the same basis by considering that this quantity does not reflect the probabilistic characteristic by not being a probability; and only the product p(x)dx possesses the probabilistic interpretation, but equals 0 for continuous quantities and hence loses meaning. Wherever required and possible, the probability is approximated by the expression p(x)Ax, where &x, albeit small, is a finite quantity. The concepts of reduced (differential) entropy and quantization step have an analogous (but not identical) meaning [16]. If is understood to be the mathematical expectation of the logarithm of the probability of random variable values, then it is indeed impossible to relate this concept to continuous random variables. On the other hand, the initial physical meaning, uncertainty, vanishes in the reduced entropy concept. The initial concept uncertainty not only remains real for continuous random variables in the examination of the quantization step concept, but stands out more, exposing most completely the content of the Heisenberg principle and the negative principle of information [5, 7], according to which no quantity being observed can be measured with arbitrarily high accuracy. The limit is not set on this accuracy by the imperfection of the instruments but by objective laws of nature.

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(i) is therefore a good reflection of the crux of the measurement process. Here H(X) is the which characterizes the a priori of the quantity being measured, and H(X/Y) is the total conditional which characterizes the of the quantity being measured which remains after the measurement result Y has been obtained. Let us examine the application of the concept in the case of continuous distributions. The value of the as a numerical characteristic of a continuous random variable is rejected in [12] on the basis that the equals ~ for continuous random variables. The introduction of the probability density p(x) could be rejected on approximately the same basis by considering that this quantity does not reflect the probabilistic characteristic by not being a probability; and only the product p(x)dx possesses the probabilistic interpretation, but equals 0 for continuous quantities and hence loses meaning. Wherever required and possible, the probability is approximated by the expression p(x)Ax, where &x, albeit small, is a finite quantity. The concepts of reduced (differential) entropy and quantization step have an analogous (but not identical) meaning [16]. If is understood to be the mathematical expectation of the logarithm of the probability of random variable values, then it is indeed impossible to relate this concept to continuous random variables. On the other hand, the initial physical meaning, uncertainty, vanishes in the reduced entropy concept. The initial concept uncertainty not only remains real for continuous random variables in the examination of the quantization step concept, but stands out more, exposing most completely the content of the Heisenberg principle and the negative principle of information [5, 7], according to which no quantity being observed can be measured with arbitrarily high accuracy. The limit is not set on this accuracy by the imperfection of the instruments but by objective laws of nature.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
The usual conjectures of quantum measurements approaches, inspired from the traditional interpretation of Heisenberg's ("uncertainty") relations, are proved as being incorrect. A group of reconsidered conjectures and a corresponding new approach are set forth. The quantum measurements, regarded experimentally as statistical samplings, are described theoretically by means of linear integral transforms of quantum probability density and current, from intrinsic into recorded readings. Accordingly, the quantum observables appear as random variables, valuable, in both readings, through probabilistic numerical parameters (characteristics). The measurements uncertainties (errors) are described by means of the intrinsic-recorded changes for the alluded parameters or for the informational entropies. The present approach, together with other author's investigations, give a natural and unified reconsideration of primary problems of Heisenberg's relations and quantum measurements. The respective reconsideration can offer some nontrivial elements for an expected re-examination of some disputed subsequent questions regarding the foundations and interpretation of quantum mechanics.
What does it take for real-deterministic ccc-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic ccc-valued variables out of the weak values obtained via a nonperturbing weak measurement of quantum operators with a postselection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to convert the incompatibility between the quantum operator and the basis into the fluctuation of an ``error term,'' and then superimpose it onto the real part of the weak value. We show that this class of ``ccc-valued physical quantities'' provides a real-deterministic contextual hidden-variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the ccc-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two ccc-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty principle is captured by a specific irreducible epistemic restriction between the two ccc-valued physical quantities, foreign in classical mechanics, constraining the allowed form of their joint distribution. We then suggest an epistemic interpretation of the two terms decomposing the ccc-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.