Formal Aspects of the Uncertainty Relations.

J. R. Croca

Jonas Maziero

The existence of incompatible observables constitutes one of the most prominent characteristics of quantum mechanics (QM) and can be revealed and formalized through uncertainty relations. The Heisenberg-Robertson-Schrödinger uncertainty relation (HRSUR) was proved at the dawn of quantum formalism and is ever-present in the teaching and research on QM. Notwithstanding, the HRSUR possess the so called triviality problem. That is to say, the HRSUR yields no information about the possible incompatibility between two observables if the system was prepared in a state which is an eigenvector of one of them. After about 85 years of existence of the HRSUR, this problem was solved recently by Lorenzo Maccone and Arun K. Pati. In this article, we start doing a brief discussion of general aspects of the uncertainty principle in QM and recapitulating the proof of HRSUR. Afterwards we present in simple terms the proof of the Maccone-Pati uncertainty relation, which can be obtained basically via the application of the parallelogram law and Cauchy-Schwarz inequality.

C. G. DARWIN

Abstract 1. In the general synthesis of classical dynamics with the quantum theory, the Uncertainty Principle plays a most useful part. It is of course only one aspect of the new mechanics, but it is a very helpful one since by its means it becomes easy to see where the old classical ideas broke down. The state of affairs in the quantum theory is not unlike that of the early days of relativity, when most of those who studied the subject felt the need of supporting the formal theory by seeing how the old ideas failed in specific cases. Here the formal theory is very abstract and is not easy to follow intuitively, and the Uncertainty Principle plays much the same rôle as did the examples of clocks and rods in relativity theory. For this reason it is more appropriate for illustrative examples, than for any extreme generality, and though a number of examples have been already given by Heisenberg and Bohr, it may not be amiss to have some more. There are probably some who will have shared my experience that it is often by no means easy to detect how the uncertainty enters into a given experiment, though once detected the arguments are usually very simple. In a recent conversation Professor Bohr criticised some rather careless remarks that I had made, and the present work was undertaken to clear matters up. It may be shortly described as a study of the Uncertainty Principle in connection with electrometers and magnetometers. It may be of interest to point out that the Uncertainty Principle can be regarded from a rather different aspect. The "resolving power" of optical instruments was discussed very fully by Rayleigh on wave principles, but as long as matter was regarded in the classical manner, a mechanical instrument could be considered as capable of measuring quantities with absolute accuracy. Now that we know that matter also has wave properties, there is need of a theory of the resolving power of mechanical instruments, and this is exactly what the Uncertainty Principle supplies.

Lubomír Skála, V. Kapsa, Martina Lužová

Marcos L. W. Basso

Bohr’s complementarity principle was introduced as a qualitative statement about quantum systems, which have properties that are equally real, but mutually exclusive. This principle, together with the uncertainty principle, is at the origin of Quantum Mechanics (QM), following the development of the theory since then. Another intriguing aspect of QM is entanglement, which is a type of correlation that is only possible within the mathematical formalism of QM. Its central importance in the Quantum foundations, as well as its important role in the fields of quantum information and quantum computation, has made the theory of entanglement achieve great progress in recent decades. In this dissertation we discuss the aspects and relations among complementarity relations, uncertainty, and entanglement. Such aspects are also investigated in relativistic scenarios, since the interest on how entanglement behaves in relativistic scenarios has grown more and more. First, by exploring the properties of the density operator, we obtain complementarity relations, using coherence measures well known in the literature. Next, we show that it is possible to obtain complete complementarity relations, i.e., relations that also take into account entanglement. For this, we explore the purity of a multipartite quantum system. In addition, we discuss the relation between complementarity and uncertainty of an observable, since it is possible to decompose uncertainty into its classical and quantum parts. We also obtain complementarity relations for uncertainty. Finally, we study the complete complementary relations and its Lorentz invariance. Although it is known that entanglement entropy does not remain invariant under Lorentz transformations, and neither does the measures that quantifies complementarity, we show that theses measures when taken together, in a complete complementarity relation, are Lorentz invariant.