In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. These uncertainty principles say that it is impossible to determine simultaneously the range (resp. position) and range rate (resp. momentum) of a moving target (resp. non-relativistic quantum-mechanical particle) with an arbitrarily high accuracy at a given instant of time. This analogy suggests that there should be a common mathematical structure behind that explains both of them. The concept of real Heisenberg nilpotent group lies at the foundations of signal theory as well as of quantum mechanics. Adopting the view point of nilpotent harmonic analysis it becomes obvious that the radar auto-ambiguity functions are of pure positive type on the symplectic time-frequency plane which will be identified with the tangent space to the Schrödinger coad-joint orbit of the Heisenberg group. Combining the classical Bochner theorem with Hudson’s characterization of the state vectors that give rise to non-negative Wigner quasi-probability densities we identify the Gabor functions as pulse envelopes of those radar auto-ambiguity functions on the time-frequency plane that are of positive type simultaneously with respect to its usual vector group structure as well as to its symplectic structure.
In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. These uncertainty principles say that it is impossible to determine simultaneously the range (resp. position) and range rate (resp. momentum) of a moving target (resp. non-relativistic quantum-mechanical particle) with an arbitrarily high accuracy at a given instant of time. This analogy suggests that there should be a common mathematical structure behind that explains both of them. The concept of real Heisenberg nilpotent group lies at the foundations of signal theory as well as of quantum mechanics. Adopting the view point of nilpotent harmonic analysis it becomes obvious that the radar auto-ambiguity functions are of pure positive type on the symplectic time-frequency plane which will be identified with the tangent space to the Schrödinger coad-joint orbit of the Heisenberg group. Combining the classical Bochner theorem with Hudson’s characterization of the state vectors that give rise to non-negative Wigner quasi-probability densities we identify the Gabor functions as pulse envelopes of those radar auto-ambiguity functions on the time-frequency plane that are of positive type simultaneously with respect to its usual vector group structure as well as to its symplectic structure.
In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. Quantum mechanics stands here for the quantum-mechanical description, at a given instant of time, of a non-relativistic particle. These uncertainty principles suggest that there should exist a common mathematical structure behind both quantum mechanics and the theory of signals. It is the aim of the present article to establish that the concept of real Heisenberg nilpotent group Ã(ℝ) lies at the foundations of both of these fields. The crucial point is to endow the time-frequency plane ℝ ⊕ ℝ with the structure of a two dimensional real symplectic vector space so that it gets symplectomorphic to the tangent plane to the “Schrodinger coadjoint orbit” at the point 1 in the Kirillov orbit picture for the unitary dual of Ã(ℝ). Various applications of this geometric relationship between signal theory and nil potent harmonic analysis are pointed out. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on Rd which may be written as P(x)exp(Ax,x), with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x)f^(y). We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.
A function of positive type can be defined as a positive functional on a convolution algebra of a locally compact group. In the case where the group is abelian, by Bochner's theorem a function of positive type is, up to normalization, the Fourier transform of a probability measure. Therefore, considering the group of translations on phase space, a suitably normalized phase-space function of positive type can be regarded as a realization of a classical state. Thus, it may be called a function of classical positive type . Replacing the ordinary convolution on phase space with the twisted convolution, one obtains a noncommutative algebra of functions whose positive functionals we may call functions of quantum positive type . In fact, by a quantum version of Bochner's theorem, a continuous function of quantum positive type is, up to normalization, the (symplectic) Fourier transform of a Wigner quasi-probability distribution; hence, it can be regarded as a phase-space realization of a quantum state. Playing with functions of positive type—classical and quantum—one is led in a natural way to consider a class of semigroups of operators, the classical-quantum semigroups . The physical meaning of these mathematical objects is unveiled via quantization, so obtaining a class of quantum dynamical semigroups that, borrowing terminology from quantum information science, may be called classical-noise semigroups .
We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.