Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on non-commutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.
We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.
Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for bipartite quantum systems. It is shown that this criterion can be better than the previous ones in detecting entanglement. The results are generalized to multipartite quantum states.
We study entanglement and genuine entanglement of tripartite and four-partite quantum states by using Heisenberg-Weyl (HW) representation of density matrices. Based on the correlation tensors in HW representation, we present criteria to detect entanglement and genuine tripartite and four-partite entanglement. Detailed examples show that our method can detect more entangled states than previous criteria.
Finite dimensional representations of extended Weyl-Heisenberg algebra are studied both from mathematical and applied viewpoints. They are used to define unitary phase operator and the corresponding eigenstates (phase states). It is also shown that the unitary depolarizers can be constructed in a general setting in terms of phase operators. Generation of generalized Bell states using the phase operator is presented and their expressions in terms of the elements of mutually unbiased bases are given.
We present a list of formulae useful for Weyl-Heisenberg integral quantizations, with arbitrary weight, of functions or distributions on the plane. Most of these formulae are known, others are original. The list encompasses particular cases like Weyl-Wigner quantization (constant weight) and coherent states (CS) or Berezin quantization (Gaussian weight). The formulae are given with implicit assumptions on their validity on appropriate space(s) of functions (or distributions). One of the aims of the document is to accompany a work in progress on Weyl-Heisenberg integral quantization of dynamics…
This paper uses deformed coherent states, based on a deformed Weyl-Heisenberg algebra that unifies the well-known SU(2), Weyl-Heisenberg, and SU(1,1) groups, through a common parameter. We show that deformed coherent states provide the theoretical foundation of a meta-kernel function, that is a kernel which in turn defines kernel functions. Kernel functions drive developments in the field of machine learning and the meta-kernel function presented in this paper opens new theoretical avenues for the definition and exploration of kernel functions. The meta-kernel function applies associated revol…
It is the aim of this paper to show how to construct Perelomov and Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite d…
We consider a reducible unitary representation of Heisenberg-Weyl group in a tensor product of two Hilbert spaces. A non-commutative operator graph generated by this representation is introduced. It is shown that spectral projections of unitaries in the representation are anticliques (quantum error-correcting codes) for this graph. The obtained codes are appeared to be linear envelopes of entangled vectors.
In a previous Note we established a necessary and sufficient condition for a multivariate Weyl--Heisenberg system G(ϕ,Λ) to be a frame when the window is a generalized Gaussian (squeezed coherent state) and Λ a rectangular lattice. In this Note we extend this result to lattices of the type $Λ=MZ^{2n}$ where M is a positive definite symmetric matrix.
The article explores a new formalism for describing motion in quantum mechanics. The construction is based on generalized coherent states with evolving fiducial vector. Weyl-Heisenberg coherent states are utilised to split quantum systems into `classical' and `quantum' degrees of freedom. The decomposition is found to be equivalent to quantum mechanics perceived from a semi-classical frame. The split allows for introduction of a new definition of classical state and is a convenient starting point for approximate analysis of quantum dynamics. An example of a meta-stable state is given as a prac…
We study deformations of the harmonic oscillator algebra known as polynomial Heisenberg algebras (PHAs), and establish a connection between them and extended affine Weyl groups of type $A^{(1)}_m$, where $m$ is the degree of the PHA. To establish this connection, we employ supersymmetric quantum mechanics to first connect a polynomial Heisenberg algebra to symmetric systems of differential equations. This connection has been previously used to relate quantum systems to non-linear differential equations; most notably, the fourth and fifth Painlevé equations. Once this is done, we use previous s…
We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple interpretation involving, for example, the natural composition of graphs. This provides a deeper insight into the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorial underpinning of its abstract formalism.
We investigate the Weyl-Wigner-Gröenewold-Moyal, the Stratonovich and the Berezin group quantization schemes for the space-space noncommutative Heisenberg-Weyl group. We show that the $\star$-product for the deformed algebra of Weyl functions for the first scheme is different than that for the other two, even though their respective quantum mechanics' are equivalent as far as expectation values are concerned, provided that some additional criteria are imposed on the implementation of this process. We also show that it is the $\star$-product associated with the Stratonovich and the Berezin form…
This paper is concerned with the construction of phase operators, phase states, vector phase states, and coherent states for a generalized Weyl-Heisenberg algebra. This polynomial algebra (that depends on real parameters) is briefly described. The various states are defined on a finite- or infinite-dimensional space depending on the parameters. This report constitutes an introduction to three papers published by the authors in J. Phys. A [43 (2010) 115303 and 45 (2012) 244036] and J. Math. Phys. [52 (2011) 082101]. See these three papers for the relevant references.
Let ϕ be an arbitrary generalized Gaussian (squeezed coherent state), Λ_{αβ}=(α_1 Z \times\cdot\cdot\cdot\times α_{n}\mathbb{Z)\times}(β_{1}\mathbb{Z}\times\cdot\cdot\cdot \timesβ_{n}\mathbb{Z)}$ a rectangular lattice. We show that there exists a positive definite symplectic matrix M (depending on ϕ) such that the multivariate Weyl--Heisenberg system G(ϕ,MΛ_{αβ}) is a frame. In a forthcoming Note we will prove a converse to this result.
Coherent-state superpositions are of great importance for many quantum subjects, ranging from foundational to technological, e.g., from tests of collapse models to quantum metrology. Here we explore various aspects of these states, related to the connection between sub-Planck structures present in their Wigner function and their sensitivity to displacements (ultimately determining their metrological potential). We review this for the usual Heisenberg-Weyl algebra associated to a harmonic oscillator, and extend it to find analogous results for the $\mathfrak{su}(2)$ algebra, typically associate…
Photons carrying orbital angular momentum (OAM) are excellent qudits and are widely used in several applications, such as long distance quantum communication, $d$-dimensional teleportation and high-resolution imaging and metrology. All these protocols rely on quantum tomography to characterise the OAM state, which currently requires complex measurements involving spatial light modulators and mode filters. To simplify the measurement and characterisation of OAM states, here we apply a recent tomography protocol [Asadian et al., \pra {\bf 94}, 010301 (2016)]. Our scheme for OAM tomography in $d$…
In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the `striped' Riordan subgroups; further, a striped quasigroup and a semigroup…
The aim of this article is to construct à la Perelomov and à la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This generalized Weyl-Heisenberg algebra, noted A(x), depends on r real parameters and is an extension of the one-parameter algebra introduced in Daoud M and Kibler MR 2010 J. Phys. A: Math. Theor. 43 115303 which covers the cases of the su(1,1) algebra (for x > 0), the su(2) algebra (for x < 0) and the h(4) ordinary Weyl-Heisenberg algebra (for x = 0). For finite-dimensional representations of A(x) and A(x,s), where A(x,s) is a truncation of order s of A(x…
The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum m…
A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl-Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of N = d-1 symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl-Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are r…
Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependenci…
Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and applied to the inhomogeneous symplectic group ISp(2n). The projective representations are given in terms of the ordinary unitary representations of the central extension of the group with respect to its first homotopy group direct product with its second cohomology group. The second cohomology group of ISp(2n) is the one dimensional abelian group and the ce…
Deformation of the Heisenberg-Weyl algebra and the Lie superalgebra $\mathfrak{osp}\left( {1|2} \right)$: exact solution for the quantum harmonic oscillator with a position-dependent mass
