Heisenberg-Weyl Group Research Articles

Reciprocal relativity of noninertial frames: quantum mechanics

Noninertial transformations on time–position–momentum–energy space {t, q, p, e} with invariant Born–Green metric and the symplectic metric −de ∧ dt + dp ∧ dq are studied. This group of transformations contains the Lorentz group as the inertial special case and, in the limit of small forces and velocities, reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds2 = −dt2. The transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. In the limit of b → ∞, spacetime is invariant. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary representations of its central extension. The same method of projective representations for the inhomogeneous group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous group is the cover of the quaplectic group . is the Weyl–Heisenberg group. The group, and the associated Heisenberg commutation relations central to quantum mechanics, results directly from requiring projective representations. A set of second-order wave equations result from the representations of the Casimir operators.

Born Reciprocity and the Granularity of Spacetime

The Schr\"odinger-Robertson inequality for relativistic position and momentum operators X^\mu, P_\nu, \mu, \nu = 0,1,2,3, is interpreted in terms of Born reciprocity and `non-commutative' relativistic phase space geometry. For states which saturate the Schr\"odinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called `quaplectic' group U(3,1)xH(3,1) (the semi-direct product of the unitary relativistic dyamical symmetry U(3,1) with the Weyl-Heisenberg group H(3,1)). The example of the `scalar' case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case,it is suggested that the semiclassical limit corresponds to the separate emergence of space-time and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.

Factorization of the Wigner distribution in prime power dimensions

There are different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g., to perform full tomography of this state. For instance, quasi-distributions such as the Weyl or Wigner distributions provide complete information about a quantum state which is equivalent to the information contained in the density matrix. In the case of composite systems, of which the subsystems are not necessarily located at the same place, the experimental feasibility of the tomographic process is considerably simplified whenever it can be realized through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions, which is the subject of the present paper. The discrete Heisenberg-Weyl group is an essential ingredient of our approach.

The sampling theorem and coherent state systems in quantum mechanics

The well-known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg–Weyl group. In particular, it is shown that the Poisson Summation Formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalized coherent state systems of the Heisenberg–Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on the properties of von Neumann and finer lattices based on standard and generalized coherent state systems.

Hofstadter problem on the honeycomb and triangular lattices: Bethe ansatz solution

We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with $2\ensuremath{\pi}p∕q$ flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as a Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows us to consider them further in the limit $p,q\ensuremath{\rightarrow}\ensuremath{\infty}$ by the technique of thermodynamic Bethe ansatz and analyze the Hofstadter problem for the irrational flux.

ABOUT THE MEAN KING'S PROBLEM AND DISCRETE WIGNER DISTRIBUTIONS

When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations. These Bell states are in one to one correspondence with the N2elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function. Thanks to the properties of generalised Bell states and of quadratic extensions of finite fields, we derive a particular solution for the Mean King's problem. This solution is in turn shown to be in one to one correspondence with a set of N2self-adjoint operators that provides a discrete realisation of the Wigner quasi-distribution.

QUANTUM INFORMATION, ENTANGLEMENT AND RELATIONSHIPS

We present several physical applications that were generated in the framework of quantum information science. We emphasize the crucial role played, in this approach, by a group of unitary transformations, the generalized Pauli or Heisenberg–Weyl group, and by a non-classical property, called entanglement, which appears to be a basic ingredient in Quantum Information Theory. We sketch the links between entanglement and non-locality, and discuss an analogy between entanglement and (human) relationships.

SQUARE INTEGRABLE PROJECTIVE REPRESENTATIONS AND SQUARE INTEGRABLE REPRESENTATIONS MODULO A RELATIVELY CENTRAL SUBGROUP

We introduce the notion of square integrable group representation modulo a relatively central subgroup and, establishing a link with square integrable projective representations, we prove a generalization of a classical theorem of Duflo and Moore. As an example, we apply the results obtained to the Weyl–Heisenberg group.

The Finite Heisenberg-Weyl Groups in Radar and Communications

We investigate the theory of the finite Heisenberg-Weyl group in relation to the development of adaptive radar and to the construction of spreading sequences and error-correcting codes in communications. We contend that this group can form the basis for the representation of the radar environment in terms of operators on the space of waveforms. We also demonstrate, following recent developments in the theory of error-correcting codes, that the finite Heisenberg-Weyl groups provide a unified basis for the construction of useful waveforms/sequences for radar, communications, and the theory of error-correcting codes.

Extremal quantum cloning machines

We investigate the problem of cloning a set of states that is invariant under the action of an irreducible group representation. We then characterize the cloners that are extremal in the convex set of group covariant cloning machines, among which one can restrict the search for optimal cloners. For a set of states that is invariant under the discrete Weyl-Heisenberg group, we show that all extremal cloners can be unitarily realized using the so-called double-Bell states, whence providing a general proof of the popular ansatz used in the literature for finding optimal cloners in a variety of settings. Our result can also be generalized to continuous-variable optimal cloning in infinite dimensions, where the covariance group is the customary Weyl-Heisenberg group of displacements.

CONSTRUCTION OF GENERALIZED UNCERTAINTY PRINCIPLES AND WAVELETS IN ANISOTROPIC SOBOLEV SPACES

This paper deals with the construction of generalized uncertainty relations. Starting with a two-operator parabola ansatz, we derive a new set of uncertainties by extending the parabola ansatz to quadratic forms. This setting can be applied to the computation of uncertainties where more than two operators are involved. For affine Weyl–Heisenberg groups, some examples are explored.

Mode displacements in multimode systems and applications to entanglement

A $d$-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in ${Z}_{d}$ and the mode phase space is ${Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}$. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the $d$ modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the $d$ observers; and they also change the momentum with which the entanglement propagates.

Symmetric informationally complete–positive operator valued measures and the extended Clifford group

We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7,13,19,21,31,… . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.

Evolution of Orbits and Quantum Correlation Functions by Quadratic Hamiltonians

Quantum systems with quadratic Hamiltonians are considered. Some results about the time evolution of homogeneous polynomials and of quantum correlation functions are given. The image of arbitrary orbit of Weyl–Heisenberg group under this time evolution is shown to be again an orbit of this group. For quantum free particle it is shown that its time evolution intersects arbitrary such orbit at most once. A result about existence of more orbits having the same dispersion of some quantum position is presented.

Implementation of group-covariant positive operator valued measures by orthogonal measurements

We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given group-covariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.

Two-Wavelet Localization Operators on $$ L^p(\mathbb{R}^n) $$ for the Weyl-Heisenberg Group

We give results on the boundedness and compactness of localization operators with two admissible wavelets on \( L^p(\mathbb{R}^n) \) for the Weyl-Heisenberg group.

Representations of coherent states in non-orthogonal bases

Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl-Heisenberg group, which can then be thought of as characterizing the dual pair.

Lüders theorem for coherent-state POVMs

Lüders’ theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on Lie groups. In general, operators turn out to be invariant under the Lüders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lüders map generated by the coherent states of the Heisenberg–Weyl group, H3. The Lüders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.

Representation of semigroups in rigged Hilbert spaces: Subsemigroups of the Weyl–Heisenberg group

In this paper we study how differentiable representations of certain subsemigroups of the Weyl–Heisenberg group may be obtained in suitably constructed rigged Hilbert spaces. These semigroup representations are induced from a continuous unitary representation of the Weyl–Heisenberg group in a Hilbert space. Aspects of the rigged Hilbert space formulation of time asymmetric quantum mechanics are also investigated within the context of the results developed here.

Generalized coherent states and the diagonal representation for operators

We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch-Gordan theory and the reciprocity theorems associated with induced unitary group representation. Applications to several examples involving $SU(2),$ $SU(3),$ and the Heisenberg-Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase-space description of quantum mechanics and quantum state reconstruction problems.