Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns out that many measurement assemblages used in quantum information theory as well as for studying the foundations of quantum mechanics are entirely determined by symmetry; moreover, starting from a certain symmetry group, novel types of measurement sets can be constructed. The insight gained from symmetry allows us to easily determine whether the measurements in the set are incompatible under noisy conditions, i.e., whether they can be regarded as genuinely distinct ones. In addition, symmetry enables us to identify finite sets of measurements having a high sensitivity to reveal the quantumness of distributed quantum states.
Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows Ω$ associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global".The microscopic point of view leads to the introduction of an additional layer over the grupoid $G$, giving rise to a suitable algebraic structure of 2-groupoid.On the other hand, taking advantage of the notion of group of bisections of a given groupoid…
The concept of quantum superposition is reconsidered and discussed from the viewpoint of Bohmian mechanics, the hydrodynamic formulation of quantum mechanics, in order to elucidate some physical consequences that go beyond the simple mathematical idea of linearly combining vectors in a Hilbert space. Specifically, the discussion turns around the connection between symmetries characterizing the wave function and the behavior in time displayed by the quantum flux when the latter is analyzed in terms of streamlines (Bohmian trajectories). This is illustrated with a series of analytical results an…
We study the spectrum generating closed nonlinear superconformal algebra that describes $\mathcal{N}=2$ super-extensions of rationally deformed quantum harmonic oscillator and conformal mechanics models with coupling constant $g=m(m+1)$, $m\in {\mathbb N}$. It has a nature of a nonlinear finite $W$ superalgebra being generated by higher derivative integrals, and generally contains several different copies of either deformed superconformal $\mathfrak{osp}(2|2)$ algebra in the case of super-extended rationally deformed conformal mechanics models, or deformed super-Schrodinger algebra in the case…
Determining the measurement uncertainty region is a difficult problem for generic sets of observables. For this reason the literature on exact measurement uncertainty regions is focused on symmetric sets of observables, where the symmetries are used to simplify the calculation. We provide a framework to systematically exploit available symmetries, formulated in terms of covariance, to simplify problems of measurement uncertainty. Our key result is that for a wide range figures of merit the optimal compatible approximations of covariant target observables are themselves covariant. This substant…
Rubik's Cube is one of the most famous combinatorial puzzles involving nearly $4.3 \times 10^{19}$ possible configurations. Its mathematical description is expressed by the Rubik's group, whose elements define how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the Cube from its geometrical constraints. Cubies are describedby single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the Cube, as a geometrical object, shows symmetrieswhich are broken whe…
We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being…
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dim…
The four-bar linkage is a basic arrangement of mechanical engineering and represents the simplest movable system formed by a closed sequence of bar-shaped bodies. Although the mechanism can have in general a spatial arrangement, we focus here on the prototypical planar case, starting however from a spatial viewpoint. The classification of the mechanism relies on the angular range spanned by the rotational motion of the bars allowed by the ratios among their lengths and is established by conditions for the existence of either one or more bars allowed to move as cranks, namely to be permitted to…
We consider symmetry as a foundational concept in quantum mechanics and rewrite quantum mechanics and measurement axioms in this description. We argue that issues related to measurements and physical reality of states can be better understood in this view. In particular, the abstract concept of symmetry provides a basis-independent definition for observables. Moreover, we show that the apparent projection/collapse of the state as the final step of measurement or decoherence is the result of breaking of symmetries. This phenomenon is comparable with a phase transition by spontaneous symmetry br…
Statistical Mechanics and Symmetries of Non-Abelian Anyon Proliferation: From Deformation to Decoherence
California Institute of Technology（加州理工学院）；Ghent university（根特大学）；Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；University of California, Berkeley（加利福尼亚大学伯克利分校）
We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review the construction of a semiheap on a Hilbert space and the set of bounded operators on a Hilbert space. The new aspect of this work is a discussion of how symmetries of a quantum system induce homomorphisms of the relevant semiheaps and ternary algebras.
