arXiv subjects

Maurice de Gosson

Publications and source records attributed to Maurice de Gosson.

At least 19 recordsLinked to original sources

arXiv2026

On Quantum Indeterminacy and the Uncertainty Principle

We propose a geometric formulation of quantum indeterminacy based on polar duality between convex bodies representing position and momentum data. A pair (X,P) of centrally symmetric convex bodies is said to satisfy the indeterminacy condition when the h-polar of X is included in P. We show that this condition naturally arises from the geometry of quantum blobs and is closely related to Hardy's uncertainty principle and the Donoho--Stark inequalities. Using symplectic capacities and John ellipsoids, we associate a canonical covariance ellipsoid with every quantum polar pair and prove that it satisfies the quantum condition. The Robertson--Schrödinger inequalities follow as a consequence. This provides a non-statistical formulation of quantum indeterminacy from which the usual uncertainty relations emerge. which the usual uncertainty relations emerge.

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arXiv2026

Generalized Marginals of tie -Wigner Distribution nand Lagrangian Tomography

We develop a symplectic formulation of the Lagrangian Radon transform based on the transitive action of the symplectic group on the Lagrangian Grassmannian. This geometric framework naturally extends the classical notion of the marginals of the Wigner transform, viewed as elementary tomograms, to arbitrary Lagrangian subspaces. The resulting generalized tomography provides a unified treatment of position and momentum measurements related by symplectic rotations. As applications, we revisit the Pauli reconstruction problem and establish reconstruction formulas for density operators from generalized Lagrangian tomograms.

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arXiv2025

A Phase Space Representation of the Metaplectic Group

The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.

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arXiv2025

Quantum Monads in Phase Space and Related Toeplitz Operators

In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in configuration space. To each of these states, we associate a classs of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.

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arXiv2025

Geometric Representation of Generalized Coherent States and their Symplectic Capacities: A Synthetic Approach

In this work we review, complete, and synthesize results linking generalized coherent stages (nondegradable Gaussian wavefunctions) to the notions of Fermi ellipsoids, quantum blobs, and microlocal pairs introduced in previous work. These geometric objects are Fermi ellipsoids, quantum blobs, and microlocal pairs. In addition we study various symplectic capacities associated with these objects.

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arXiv2025

Polar Duality and the Donoho--Stark Uncertainty Principle

Polar duality is a fundamental geometric concept that can be interpreted as a form of Fourier transform between convex sets. Meanwhile, the Donoho-Stark uncertainty principle in harmonic analysis provides a framework for comparing the relative concentrations of a function and its Fourier transform. Combining the Blaschke--Santaló inequality from convex geometry with the Donoho--Stark principle, we establish estimates for the trade-off of concentration between a square integrable function in a symmetric convex body and that of its Fourier transform in the polar dual of that body. In passing, we use the Donoho-Stark uncertainty principle to establish a new concentration result for the Wigner function.

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arXiv2024

Quantum Indeterminacy and Polar Duality: a Probabilistic Approach

We present a probabilistic argument supporting the application of polar duality, as discussed in our previous work, to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a convex body with the Donoho--Stark uncertainty principle from harmonic analysis, which characterizes the concentration of a function and its Fourier transform. The central result demonstrates that the sum of the probabilities of position concentration near a convex body and momentum concentration near its polar dual is equal to one, with an error term that diminishes rapidly as the number of degrees of freedom increases. This result motivates the interpretation of polar duality as a kind of geometric Fourier transform.

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arXiv2024

Phase Space Representation of the Density Operator: Bopp Pseudodifferential Calculus and Moyal Product

Bopp shifts, introduced in 1956, played a pivotal role in the statistical interpretation of quantum mechanics. As demonstrated in our previous work, Bopp's construction provides a phase-space perspective of quantum mechanics that is closely connected to the Moyal star product and its role in deformation quantization. In this paper, we both review and expand on our exploration of Bopp quantization, emphasizing its relationship with the Moyal product and its applications in elementary deformation quantization. Notably, we apply these constructions to the density operator, which represents mixed states in quantum mechanics, offering novel insights into its role within this framework.

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arXiv2024

Polar Duality and Quasi-States: a Geometric Picture of Quantum Indeterminacy

The aim of this paper is to suggest a new interpretation of quantum indeterminacy using the notion of polar duality from convex geometry. Our approach does not involve the usual variances and covariances, whose use to describe quantum uncertainties has been questioned by Uffink and Hilgevoord. We introduce instead the geometric notion of "quasi-states" which are related in a way that will be explained to the notion of "quantum blob" we have introduced in previous work. Considering the symmetries of the quasi-states leads to the definition of the canonical group of a quasi-state, which allows to classify them.

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arXiv2024

A metaplectic perspective of uncertainty principles in the Linear Canonical Transform domain

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.

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arXiv2023

Symplectic and Lagrangian Polar Duality; Applications to Quantum Information Geometry

Polar duality is a well-known concept from convex geometry and analysis. In the present paper, we study two symplectically covariant versions of polar duality keeping in mind their applications to quantum mechanics. The first variant makes use of the symplectic form on phase space and allows a precise study of the covariance matrix of a density operator. The latter is a fundamental object in quantum information theory., The second variant is a symplectically covariant version of the usual polar duality highlighting the role played by Lagrangian planes. It allows us to define the notion of "geometric quantum states" with are in bijection with generalized Gaussians.

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arXiv2023

Pointillisme à la Signac and Construction of a Quantum Fiber Bundle Over Convex Bodies

We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane.. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of "quantum blobs" introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets.

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arXiv2023

Reconstruction of Gaussian Quantum States from Ideal Position Measurements: Beyond Pauli's Problem, I

We show that the covariance matrix of a quantum state can be reconstructed from position measurements using the simple notion of polar duality, familiar from convex geometry. In particular, all multidimensional Gaussian states (pure or mixed) can in principle be reconstructed if the quantum system is well localized in configuration space. The main observation which makes this possible is that the John ellipsoid of the Cartesian product of the position localization by its polar dual contains a quantum blob, and can therefore be identified with the covariance ellipsoid of a quantum state.

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arXiv2022

On the Wigner Distribution of the Reduced Density Matrix

CConsider a bipartite quantum system consisting of two subsystems A and B. The reduced density matrix ofA a is obtained by taking the partial trace with respect to B. In this work, we will show that the Wigner distribution of this reduced density matrix is obtained by integrating the total Wigner distribution with respect to the phase space variables corresponding to subsystem B. The proof we give is rigorous (as opposed to those found in the literature) and makes use of the Weyl--Wigner--Moyal phase space formalism. Our main result is applied to general Gaussian mixed states, of which it gives a particularly simple and precise description. We also briefly discuss the purification of a mixed state from the Wigner formalism point of view.

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arXiv2022

Toeplitz Density Operators and their Separability Properties

Toeplitz operators (also called localization operators) are a generalization of the well-known anti-Wick pseudodifferential operators studied by Berezin and Shubin. When a Toeplitz operator is positive semi-definite and has trace one we call it a density Toeplitz operator. Such operators represent physical states in quantum mechanics. In the present paper we study several aspects of Toeplitz operators when their symbols belong to some well-known functional spaces (e.g. the Feichtinger algebra) and discuss (tentatively) their separability properties with an emphasis on the Gaussian case.

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arXiv2022

Entanglement of Bipartite Gaussian States: a Simple Criterion and its Geometric Interpretation

Werner and Wolf have proven in Phys. Rev. Lett. 86(16) (2001) a very elegant necessary and sufficient condition for a bosonic continuous variable bipartite Gaussian mixed quantum state to be separable. This condition is, however, difficult to implement in practice. In the present Letter, we propose a simpler condition which only involves the calculation of the symplectic matrix in the Williamson diagonalization of the covariance matrix of the state under consideration. The main tool in our construction is the observation, proved in previous work, that the Wigner transform is covariant only under symplectic or antisymplectic linear transformations. We also give a geometric interpretation of our condition in terms of the orthogonal projections of "quantum blobs"..

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arXiv2022

Phase Spaces, Parity Operators, and the Born-Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time-frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions [A. Grossmann, Comm. Math. Phys. 48(3), 191 (1976)], parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class [L. Cohen, J. Math. Phys. 7(5), 781 (1966)] and recover various quantization schemes and distribution functions from the literature. The parity-operator approach is also applied to the Born-Jordan distribution which originates from the Born-Jordan quantization [M. Born, P. Jordan, Z. Phys. 34(1), 858 (1925)]. The corresponding parity operator is written as a weighted average of both displacements and squeezing operators and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born-Jordan parity operator in the number-state basis and example quantum states reveal unique features of the Born-Jordan distribution.

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arXiv2022

Symplectic Polar Duality, Quantum Blobs, and Generalized Gaussians

We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of "quantum blobs" introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson--Schrödinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results. We apply these geometric constructions to the characterization of pure Gaussian states in terms of partial information on the covariance ellipsoid, which allows us to formulate statements related to symplectic tomography.

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