Weyl Operators Research Articles

Uncertainty relations on the joint numerical range of operators

The expectation values of operators drawn from a single quantum state cannot be outside of a particular region, called their allowed region or the joint numerical range of the operators. We present a method to obtain all necessary and sufficient constraints—from Hermiticity, normalization, and positivity of a state and through the Born rule—that analytically defines the allowed region. Then, we present the allowed regions for the Heisenberg–Weyl operators, the angular momentum operators, and for their functions in dimension two to infinity. Especially, we consider three kinds of functions—combinations of powers of the ladder operators, powers of the angular momentum operators, and their anticommutators—and discover the allowed regions of different shapes. Here we also introduce uncertainty measures on the joint numerical range that are different from the standard deviation and the Shannon entropy. With the measures, we achieve a new kind of tight uncertainty relations for the Weyl- and the angular-momentum-operators. Overall, we demonstrate how the joint numerical range and the uncertainty relations change as the dimension grows. We apply the quantum de Finetti theorem to attain the allowed regions and tight uncertainty relations in the limit where the dimension goes to infinity.

On the regime of localized excitations for disordered oscillator systems

We study quantum oscillator lattice systems with disorder, in arbitrary dimension, requiring only partial localization of the associated effective one-particle Hamiltonian. This leads to a many-body localized regime of excited states with arbitrarily large energy density. We prove zero-velocity Lieb-Robinson bounds for the dynamics of Weyl operators as well as for position and momentum operators restricted to this regime. Dynamical localization is also shown in the form of quasi-locality of the time evolution of local Weyl operators and through exponential clustering of the dynamic correlations of states with localized excitations.

High-dimensional quantum teleportation under noisy environments

We study the protocol of qudit teleportation using quantum systems subjected to several kinds of noise for arbitrary dimensionality $d$. We consider four classes of noise: dit-flip, $d$-phase-flip, dit-phase-flip and depolarizing, each of them corresponding to a family of Weyl operators, introduced via Kraus formalism. We derive a general expression for the average fidelity of teleportation in arbitrary dimension $d$ for any combination of noise on the involved qudits. Under a different approach we derive the average fidelity of teleportation for a more general scenario involving the $d$-dimensional generalization of amplitude damping noise as well. We show that all possible scenarios may be classified in four different behaviours and discuss the cases in which it is possible to improve the fidelity by increasing the associated noise fractions. All our results are in agreement with previous analysis by Fortes and Rigolin for the case of qubits (Phys. Rev. A, 92 012338, 2015).

Weyl asymptotics for perturbed functional difference operators

We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = e−bP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with N∈N and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 2N − 1. This class of potentials W includes polynomials of even degree with leading coefficient 1, which have recently been considered in Grassi and Mariño [SIGMA Symmetry Integrability Geom. Methods Appl. 15, 025 (2019)]. In this paper, we show that such operators have discrete spectrum and obtain Weyl-type asymptotics for the Riesz means and for the number of eigenvalues. This is an extension of the result previously obtained in Laptev et al. [Geom. Funct. Anal. 26, 288–305 (2016)] for W = V + ζV−1, where V = e2πbx, ζ > 0.

On Weyl completions of partial operator matrices

Let H and K be complex separable Hilbert spaces. Given the operators A∈B(H) and B∈B(K,H), we define MX,Y:=[ABXY], where X∈B(H,K) and Y∈B(K) are unknown elements. In this article, we give a necessary and sufficient condition for MX,Y to be a (right) Weyl operator for some X∈B(H,K) and Y∈B(K). Moreover, we show that if dimK<∞, then MX,Y is a left Weyl operator for some X∈B(H,K) and Y∈B(K) if and only if [AB] is a left Fredholm operator and ind([AB])≤dimK; if dimK=∞, then MX,Y is a left Weyl operator for some X∈B(H,K) and Y∈B(K).

Wigner Functions and Weyl Operators on the Euclidean Motion Group

The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl–Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.

On the positivity of trace class operators

The characterization of positivity properties of Weyl operators is a notoriously difficult problem, and not much progress has been made since the pioneering work of Kastler, Loupias, and Miracle-Sole (KLM). In this paper we begin by reviewing and giving simpler proofs of some known results for trace-class Weyl operators; the latter play an essential role in quantum mechanics. We then apply time-frequency analysis techniques to prove a phase space version of the KLM condition; the main tools are Gabor frames and the Wigner formalism. Finally, discrete approximations of the KLM condition, which are tractable numerically, are provided.

Almost Diagonalization of $$\tau $$ τ -Pseudodifferential Operators with Symbols in Wiener Amalgam and Modulation Spaces

In this paper we focus on the almost-diagonalization properties of $\tau$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sj\"ostrand class, for which Gr\"ochenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $\tau$-pseudodifferential operator, for $\tau \in [0,1]$, also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for $\tau$-pseudodifferential operators on Wiener amalgam and modulation spaces.

Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits

For a connected simply connected nilpotent Lie group $$\textsf {G}$$ with Lie algebra $${{\mathfrak {g}}}$$ and unitary dual $$\widehat{{\textsf {G}}}$$ one has (a) a global quantization of operator-valued symbols defined on $$\textsf {G}\times \widehat{{\textsf {G}}}$$ , involving the representation theory of the group, (b) a quantization of scalar-valued symbols defined on $$\textsf {G}\times {{\mathfrak {g}}}^*$$ , taking the group structure into account and (c) Weyl-type quantizations of all the coadjoint orbits $$\big \{\Omega _\xi \mid \xi \in \widehat{{\textsf {G}}}\big \}$$ . We show how these quantizations are connected, in the case when flat coadjoint orbits exist. This is done by a careful new analysis of the composition of two different types of Fourier transformations, interesting in itself. We also describe the concrete form of the operator-valued symbol quantization, by using Kirillov theory and the Euclidean version of the unitary dual and Plancherel measure. In the case of the Heisenberg group, this corresponds to the known picture, presenting the representation theoretical pseudo-differential operators in terms of families of Weyl operators depending on a parameter. For illustration, we work out a couple of examples and put into evidence some specific features of the case of Lie algebras with one-dimensional center. When $$\textsf {G}$$ is also graded, we make a short presentation of the symbol classes $$S^m_{\rho ,\delta }$$ , transferred from $$\textsf {G}\times \widehat{{\textsf {G}}}$$ to $$\textsf {G}\times {{\mathfrak {g}}}^*$$ by means of the connection mentioned above.

Weyl and Browder S-spectra in a right quaternionic Hilbert space

In this note first we study the Weyl operators and Weyl S-spectrum of a bounded right quaternionic linear operator, in the setting of the so-called S-spectrum, in a right quaternionic Hilbert space. In particular, we give a characterization for the S-spectrum in terms of the Weyl operators. In the same space we also study the Browder operators and introduce the Browder S-spectrum.

SO(4) Landau models and matrix geometry

We develop an in-depth analysis of the SO(4) Landau models on S3 in the SU(2) monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the SU(2) monopole are introduced to provide a concrete coordinate representation of SO(4) operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the SO(4) covariance of the eigenfunctions. With the spin connection of S3, we construct an SO(4) invariant Weyl–Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac–Landau and supersymmetric Landau models, are investigated too. With the developed SO(4) technologies, we derive the three-dimensional matrix geometry in the Landau models. By applying the level projection method to the Landau models, we identify the matrix elements of the S3 coordinates as the fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy three-sphere geometry emerges in each of the Landau levels and only in the degenerate lowest energy sub-bands. We also point out that Dirac–Landau operator accommodates two fuzzy three-spheres in each Landau level and the mass term induces interaction between them.

A Schatten–von Neumann class criterion for the magnetic Weyl calculus

We prove a criterion for a “magnetic” Weyl operator to be trace-class by extending a method developed by Cordes, Kato and Arsu. Using the Calderon–Vaillancourt type Theorem for magnetic Weyl operators and an interpolation argument we also give a criterion for a “magnetic” Weyl operator to be in a Schatten–von Neumann class with .

Quantum channels irreducibly covariant with respect to the finite group generated by the Weyl operators

In matrix algebras, we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the Weyl operators. In particular, we analyze the irreducibly covariant quantum channels, that is, the completely positive and trace-preserving linear maps. Interestingly, imposing additional symmetries leads to the so-called generalized Pauli channels, which were recently considered in the context of the non-Markovian quantum evolution. Finally, we provide examples of irreducibly covariant positive but not necessarily completely positive maps.

Inverse Weyl transform/operator

The Weyl procedure associates a function of two ordinary variables, called the c-function or symbol, with an operator, called the Weyl operator of the symbol. One generally formulates this association by defining the operator corresponding to a given symbol. In this paper we consider the reverse problem: Given the Weyl operator, what is the matching symbol? We give a number of explicit formulas for obtaining the symbol that would generate an arbitrary Weyl operator, and we illustrate each form with an example.

Weylness of 2 × 2 operator matrices

AbstractLet and be complex separable infinite‐dimensional Hilbert spaces. Given the operators and , we define where is an unknown element. In this paper, a necessary and sufficient condition is given for to be a right Weyl (left Weyl, or Weyl) operator for some . Moreover, some relevant properties and illustrating examples are also given.

Strong ultra-regularity properties for positive elements in the twisted convolutions

We show that positive elements with respect to the twisted convolutions, belonging to some ultra-test function space of certain order at origin, belong to the ultra-test function space of the same order everywhere. We apply the result to positive semi-definite Weyl operators.

Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces

This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.

Self-adjoint perturbations of spectra for upper triangular operator matrices

This paper is concerned with the self-adjoint perturbations of the spectra for the upper triangular partial operator matrix with given diagonal entries. A necessary and sufficient condition is given under which such operator matrix admits a Weyl (Fredholm) operator completion by choosing some bounded self-adjoint operator. It is shown that the self-adjoint perturbation of the Weyl (essential) spectrum can be the proper set of the general perturbation. Combining the spectral properties, we further characterize the perturbation of the Weyl (essential) spectrum for Hamiltonian operators.

Additive Bounds of Minimum Output Entropies for Unital Channels and an Exact Qubit Formula

We investigate minimum output (R\'enyi) entropy of qubit channels and unital quantum channels. We obtain an exact formula for the minimum output entropy of qubit channels, and bounds for unital quantum channels. Interestingly, our bounds depend only on the operator norm of the matrix representation of the channels on the space of trace-less Hermitian operators. Moreover, since these bounds respect tensor products, we get bounds for the capacity of unital quantum channels, which is saturated by the Werner-Holevo channel. Furthermore, we construct an orthonormal basis, besides the Gell-Mann basis, for the space of trace-less Hermitian operators by using discrete Weyl operators. We apply our bounds to discrete Weyl covariant channels with this basis, and find new examples in which the minimum output R\'enyi $2$-entropy is additive.

Generalized Kato Decomposition and Essential Spectra

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator T on a Banach space X we prove that $$T=T_M\oplus T_N$$ with $$T_M \in $$ R and $$T_N$$ quasinilpotent (nilpotent) if and only if T admits a generalized Kato decomposition (T is of Kato type) and 0 is not an interior point of the corresponding spectrum $$\sigma _\mathbf{R}(T) =\{\lambda \in \mathbb {C}: T-\lambda \notin \mathbf{R}\}$$ . Moreover, we prove that if $$T-\lambda _0$$ admits a generalized Kato decomposition, then $$\sigma _\mathbf{R}(T)$$ does not cluster at $$\lambda _0$$ if and only if $$\lambda _0$$ is not an interior point of $$\sigma _\mathbf{R}(T)$$ . As a consequence we get several results on cluster points of essential spectra. In that way we extend some results regarding the approximate point spectrum and the surjective spectrum given by Aiena and Rosas (J. Math. Anal. Appl. 279:180–188, 2003), as well as results given by Jiang and Zhong (J. Math. Anal. Appl. 356:322–327, 2009) to the cases of essential spectra.