arXiv subjects

J. Tolar

Publications and source records attributed to J. Tolar.

18 recordsLinked to original sources

arXiv2018

On Clifford groups in quantum computing

The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, $π/4$-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras -- covering any single as well as composite finite quantum systems -- directly correspond to Clifford groups defined as quotients with respect to $\U(1)$.

quant-ph↗

arXiv2014

A classification of finite quantum kinematics

Quantum mechanics in Hilbert spaces of finite dimension $N$ is reviewed from the number theoretic point of view. For composite numbers $N$ possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order $N$ as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of $N$. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.

quant-ph↗

arXiv2012

Symmetries of finite Heisenberg groups for multipartite systems

A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Z_{n_{i}}, i=1,...,k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces of dimensions n_{1},...,n_{k}. Symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.

math-ph↗

arXiv2012

Symmetries of finite Heisenberg groups for k-partite systems

Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in Z_{n_i}, i=1,...,k. Their Hilbert spaces are given by k-fold tensor products of Hilbert spaces of dimensions n_1,...,n_k. Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.

math-ph↗

arXiv2012

Quantizations on the circle and coherent states

We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S^1 including the Aharonov-Bohm type quantum description. The coherent states are constructed by Perelomov's method as group related coherent states generated by Weyl operators on the quantum phase space Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was Z_M x Z_M. The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, also some of their further properties are studied demonstrating similarities as well as substantial differences.

quant-ph↗

arXiv2011

Coherent states on the circle

We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization over a finite periodic chain where the phase space was Z_M x Z_M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.

quant-ph↗

arXiv2010

Symmetries of the finite Heisenberg group for composite systems

Symmetries of the finite Heisenberg group represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. As is well known, these symmetries are properly expressed in terms of certain normalizer. This paper extends previous investigations to composite quantum systems consisting of two subsystems - qudits - with arbitrary dimensions n and m. In this paper we present detailed descriptions - in the group of inner automorphisms of GL(nm,C) - of the normalizer of the Abelian subgroup generated by tensor products of generalized Pauli matrices of orders n and m. The symmetry group is then given by the quotient group of the normalizer.

quant-ph↗

arXiv2009

Intrinsic properties of quantum systems

A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be attributed to individual quantum systems without violating standard quantum mechanics, which are called intrinsic. The intrinsic properties are classified into structural and conditional. A systematic and self-consistent account is given. Much more statements become meaningful than any version of Copenhagen interpretation would allow. A new approach to classical properties, quantum cosmology and measurement problem is suggested. A quantum definition of classical states is proposed.

quant-ph↗

arXiv2008

Intrinsic and extrinsic properties of quantum systems

The paper attempts to convince that the orthodox interpretation of quantum mechanics does not contradict philosophical realism by throwing light onto certain properties of quantum systems that seem to have escaped attention as yet. The exposition starts with the philosophical notions of realism. Then, the quantum mechanics as it is usually taught is demoted to a mere part of the theory called phenomenology of observations, and the common impression about its contradiction to realism is explained. The main idea of the paper, the physical notion of intrinsic properties, is introduced and many examples thereof are given. It replaces the irritating dichotomy of quantum and classical worlds by a much softer difference between intrinsic and extrinsic properties, which concern equally microscopic and macroscopic systems. Finally, the classicality and the quantum measurement are analyzed and found to present some still unsolved problems. A possible way of dealing with the Schrödinger cat is suggested that is based on the intrinsic properties. A simple quantum model of one classical property illustrates how our philosophy may work.

quant-ph↗

arXiv2007

Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions

Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of N+1 mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of SL(2,Z_N) on finite phase space Z_N x Z_N implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime N, Z_N is also a finite field.

quant-ph↗

arXiv2007

Dihedral symmetry of periodic chain: quantization and coherent states

Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the non-Abelian dihedral group D_n - is taken as its symmetry group. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.

quant-ph↗

arXiv2006

Graded contractions of the Pauli graded sl(3,C)

The Lie algebra $sl(3,\C)$ is considered in the basis of generalized Pauli matrices. Corresponding grading is the Pauli grading here. It is one of the four gradings of the algebra which cannot be further refined. The set $\es$ of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of $sl(3,\C)$, which is essentially the finite group $SL(2,\Z_3)$, and the induced symmetry of the system $\es$. A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classes are non-parametric and 13 solutions depend on one or two continuous parameters, providing a continuum of equivalence classes and subsequently continuum of non-isomorphic Lie algebras. Solutions of the contraction equations of Pauli graded $sl(3,\bC)$ are identified here as specific solvable Lie algebras of dimensions up to 8. Earlier algorithms for identification of Lie algebras, given by their structure constants, had to be made more efficient in order to distinguish non-isomorphic Lie algebras encountered here. Resulting Lie algebras are summarized in tabular form. There are 88 indecomposable solvable Lie algebras of dimension 8, 77 of them being nilpotent. There are 11 infinite sets of parametric Lie algebras which still deserve further study.

math-ph↗

arXiv2006

Borel quantization and nonlinear quantum mechanics. A review of developments in the series "Symmetries in Science" I - XIII

The results, different aspects and applications of our method of quantisation on configuration manifolds - called Borel Quantisation - were presented at meetings of the series `Symmetries in Science' and can be found in the published proceedings. The developments with numerous coauthors, on Borel quantisation and the related family of nonlinear Schrödinger equations called Doebner-Goldin equations, are reviewed and commented here.

quant-ph↗

arXiv2004

The fine gradings of sl(3,C) and their symmetries

We describe the normalizers for all non-conjugate maximal Abelian subgroups of diagonalizable automorphisms of sl(3,C) and show their relation to the symmetries of equations related to the graded contraction.

math-ph↗

arXiv2003

Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices

We consider the grading of $sl(n,\mathbb{C})$ by the group $Π_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $Π_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $Π_3$ and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in $Π_3$.

math-ph↗

arXiv2003

Telescopic system with a rotating objective element

The angular resolution is tha ability of a telescope to render detail: the higher the resolution the finer is the detail. it is, together with the aperture, the most important characteristic of telescopes. We propose a new construction of telescopes with improved ratio of angular resolution and area of the primary optical element (mirror or lense). For this purpose we use the rotation of the primary optical element with one dominating dimension. The length of the dominating dimension of the primary optical element determines the angular resolution. During the rotation a sequence of images is stored in a computer and the images of observed objects can be reconstructed using a relatively simple software. The angular resolution is determined by the maximal length of the primary optical element of the system. This construction of telescopic systems allows to construct telescopes of high resolution with lower weight and fraction of usual costs.

astro-ph↗

arXiv2001

Quantization of kinematics on configuration manifolds

The review is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables --- generalized positions and momenta. A broad class of quantum kinematics is rigorously constructed for systems, the configuration space of which is either a homogeneous space of a Lie group or a connected smooth finite-dimensional manifold without boundary. The class also includes systems in an external gauge field for an Abelian or a compact gauge group. Conditions for equivalence and irreducibility of generalized quantum kinematics are investigated with the aim of classification of possible quantizations. Complete classification theorems are given in two special cases. It is attempted to motivate the global approach based on a generalization of imprimitivity systems called {\em quantum Borel kinematics}. These are classified by means of global invariants --- quantum numbers of topological origin. Selected examples are presented which demonstrate the richness of applications of Borel quantization. The review aims to provide an introductory survey of the subject and to be sufficiently selfcontained as well, so that it can serve as a standard reference concerning Borel quantization for systems admitting localization on differentiable manifolds.

math-ph↗

arXiv1994

Group quantization of parametrized systems II. Pasting Hilbert spaces

The method of group quantization described in the preceeding paper I is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface. A simple completely solvable toy system is studied that admits a pair of maximal transversal surfaces intersecting all orbits. The corresponding two quantum mechanics are constructed. The similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum Hilbert spaces on the other hand suggests how the two Hilbert spaces are to be pasted together. The resulting quantum theory is checked to be equivalent to that constructed directly by means of Dirac's operator constraint method. The complete system of partial Hamiltonians for any of the two transversal surfaces is chosen and the quantum Schrödinger or Heisenberg pictures of time evolution are constructed.

gr-qc↗