Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U_3 with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant J, defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on U(3) and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding |J| smaller than the value observed for the CKM matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the MNS matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.Comment: 33 pages, 6 figures version 2 - misprints corrected, explicit formulae for phases provide

The Stern-Gerlach Experiment Revisited

The Stern-Gerlach-Experiment (SGE) of 1922 is a seminal benchmark experiment of quantum physics providing evidence for several fundamental properties of quantum systems. Based on today's knowledge we illustrate the different benchmark results of the SGE for the development of modern quantum physics and chemistry. The SGE provided the first direct experimental evidence for angular momentum quantization in the quantum world and thus also for the existence of directional quantization of all angular momenta in the process of measurement. It measured for the first time a ground state property of an atom, it produced for the first time a `spin-polarized' atomic beam, it almost revealed the electron spin. The SGE was the first fully successful molecular beam experiment with high momentum-resolution by beam measurements in vacuum. This technique provided a new kinematic microscope with which inner atomic or nuclear properties could be investigated. The original SGE is described together with early attempts by Einstein, Ehrenfest, Heisenberg, and others to understand directional quantization in the SGE. Heisenberg's and Einstein's proposals of an improved multi-stage SGE are presented. The first realization of these proposals by Stern, Phipps, Frisch and Segr\`e is described. The set-up suggested by Einstein can be considered an anticipation of a Rabi-apparatus. Recent theoretical work is mentioned in which the directional quantization process and possible interference effects of the two different spin states are investigated. In full agreement with the results of the new quantum theory directional quantization appears as a general and universal feature of quantum measurements. One experimental example for such directional quantization in scattering processes is shown. Last not least, the early history of the `almost' discovery of the electron spin in the SGE is revisited.Comment: 50pp, 17 fig

Quantum-like Representation of Extensive Form Games: Wine Testing Game

We consider an application of the mathematical formalism of quantum mechanics (QM) outside physics, namely, to game theory. We present a simple game between macroscopic players, say Alice and Bob (or in a more complex form - Alice, Bob and Cecilia), which can be represented in the quantum-like (QL) way -- by using a complex probability amplitude (game's ``wave function'') and noncommutative operators. The crucial point is that games under consideration are so called extensive form games. Here the order of actions of players is important, such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information which is available to e.g. Bob about the previous action of Alice. In general one could not construct a classical probability space underlying a QL-game. This can happen even in a QL-game with two players. In a QL-game with three players Bell's inequality can be violated. The most natural probabilistic description is given by so called contextual probability theory completed by the frequency definition of probability

Explanation of the Gibbs paradox within the framework of quantum thermodynamics

The issue of the Gibbs paradox is that when considering mixing of two gases within classical thermodynamics, the entropy of mixing appears to be a discontinuous function of the difference between the gases: it is finite for whatever small difference, but vanishes for identical gases. The resolution offered in the literature, with help of quantum mixing entropy, was later shown to be unsatisfactory precisely where it sought to resolve the paradox. Macroscopic thermodynamics, classical or quantum, is unsuitable for explaining the paradox, since it does not deal explicitly with the difference between the gases. The proper approach employs quantum thermodynamics, which deals with finite quantum systems coupled to a large bath and a macroscopic work source. Within quantum thermodynamics, entropy generally looses its dominant place and the target of the paradox is naturally shifted to the decrease of the maximally available work before and after mixing (mixing ergotropy). In contrast to entropy this is an unambiguous quantity. For almost identical gases the mixing ergotropy continuously goes to zero, thus resolving the paradox. In this approach the concept of ``difference between the gases'' gets a clear operational meaning related to the possibilities of controlling the involved quantum states. Difficulties which prevent resolutions of the paradox in its entropic formulation do not arise here. The mixing ergotropy has several counter-intuitive features. It can increase when less precise operations are allowed. In the quantum situation (in contrast to the classical one) the mixing ergotropy can also increase when decreasing the degree of mixing between the gases, or when decreasing their distinguishability. These points go against a direct association of physical irreversibility with lack of information.Comment: Published version. New title. 17 pages Revte

Contribution to understanding the mathematical structure of quantum mechanics

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, probability density current, momentum operator, rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrodinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of the delta-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many particle systems and the postulates of quantum mechanics are also discussed.Comment: 21 page

QUBIC: The QU Bolometric Interferometer for Cosmology

One of the major challenges of modern cosmology is the detection of B-mode polarization anisotropies in the CMB. These originate from tensor fluctuations of the metric produced during the inflationary phase. Their detection would therefore constitute a major step towards understanding the primordial Universe. The expected level of these anisotropies is however so small that it requires a new generation of instruments with high sensitivity and extremely good control of systematic effects. We propose the QUBIC instrument based on the novel concept of bolometric interferometry, bringing together the sensitivity advantages of bolometric detectors with the systematics effects advantages of interferometry. Methods: The instrument will directly observe the sky through an array of entry horns whose signals will be combined together using an optical combiner. The whole set-up is located inside a cryostat. Polarization modulation will be achieved using a rotating half-wave plate and interference fringes will be imaged on two focal planes (separated by a polarizing grid) tiled with bolometers. We show that QUBIC can be considered as a synthetic imager, exactly similar to a usual imager but with a synthesized beam formed by the array of entry horns. Scanning the sky provides an additional modulation of the signal and improve the sky coverage shape. The usual techniques of map-making and power spectrum estimation can then be applied. We show that the sensitivity of such an instrument is comparable with that of an imager with the same number of horns. We anticipate a low level of beam-related systematics thanks to the fact that the synthesized beam is determined by the location of the primary horns. Other systematics should be under good control thanks to an autocalibration technique, specific to our concept, that will permit the accurate determination of most of the systematics parameters.Comment: 12 pages, 10 figures, submitted to Astronomy and Astrophysic

Locality for quantum systems on graphs depends on the number field

Adapting a definition of Aaronson and Ambainis [Theory Comput. 1 (2005), 47--79], we call a quantum dynamics on a digraph "saturated Z-local" if the nonzero transition amplitudes specifying the unitary evolution are in exact correspondence with the directed edges (including loops) of the digraph. This idea appears recurrently in a variety of contexts including angular momentum, quantum chaos, and combinatorial matrix theory. Complete characterization of the digraph properties that allow such a process to exist is a long-standing open question that can also be formulated in terms of minimum rank problems. We prove that saturated Z-local dynamics involving complex amplitudes occur on a proper superset of the digraphs that allow restriction to the real numbers or, even further, the rationals. Consequently, among these fields, complex numbers guarantee the largest possible choice of topologies supporting a discrete quantum evolution. A similar construction separates complex numbers from the skew field of quaternions. The result proposes a concrete ground for distinguishing between complex and quaternionic quantum mechanics.Comment: 9 page

On the verge of Umdeutung in Minnesota: Van Vleck and the correspondence principle (Part One)

In October 1924, the Physical Review, a relatively minor journal at the time, published a remarkable two-part paper by John H. Van Vleck, working in virtual isolation at the University of Minnesota. Van Vleck combined advanced techniques of classical mechanics with Bohr's correspondence principle and Einstein's quantum theory of radiation to find quantum analogues of classical expressions for the emission, absorption, and dispersion of radiation. For modern readers Van Vleck's paper is much easier to follow than the famous paper by Kramers and Heisenberg on dispersion theory, which covers similar terrain and is widely credited to have led directly to Heisenberg's "Umdeutung" paper. This makes Van Vleck's paper extremely valuable for the reconstruction of the genesis of matrix mechanics. It also makes it tempting to ask why Van Vleck did not take the next step and develop matrix mechanics himself.Comment: 82 page

QUBIC: The QU Bolometric Interferometer for Cosmology

Context. One of the major challenges of modern cosmology is the detection of B-mode polarization anisotropies in the Cosmic Microwave Background. These originate from tensor fluctuations of the metric produced during the inflationary phase. Their detection would therefore constitute a major step towards understanding the primordial Universe. The expected level of these anisotropies is however so small that it requires a new generation of instruments with high sensitivity and extremely good control of systematic eects. Aims. We propose the QUBIC instrument based on the novel concept of bolometric interferometry, bringing together the sensitivity advantages of bolometric detectors with the systematics eects advantages of interferometry. Methods. The instrument will directly observe the sky through an array of entry horns whose signals will be combined together using an optical combiner. The whole set-up is located inside a cryostat. Polarization modulation will be achieved using a rotating half-wave plate and the images of the interference fringes will be formed on two focal planes (separated by a polarizing grid) tiled with bolometers. Results.We show that QUBIC can be considered as a synthetic imager, exactly similar to a usual imager but with a synthesized beam formed by the array of entry horns. Scanning the sky provides an additional modulation of the signal and improve the sky coverage shape. The usual techniques of map-making and power spectrum estimation can then be applied. We show that the sensitivity of such an instrument is comparable with that of an imager with the same number of horns. We anticipate a low level of beam-related systematics thanks to the fact that the synthesized beam is determined by the location of the primary horns. Other systematics should be under good control thanks to an autocalibration technique, specific to our concept, that will permit the accurate determination of most of the systematics parameters