Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On the use of minimum volume ellipsoids and symplectic capacities for studying classical uncertainties for joint position momentum measurements

Maurice A. de Gosson, Universität Wien Nuhag

Published Jan 1, 2010• 17 citations
OpenAlex

Abstract

We study the minimum volume ellipsoid estimator associates to a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics.

Related papers

On the use of minimum volume ellipsoids and symplectic capacities for studying classical uncertainties for joint position–momentum measurements

Maurice A de Gosson

We study the minimum volume ellipsoid estimator associated with a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics.

arXiv:1008.4877•Journal of Statistical Mechanics Theory and Experiment•Nov 2, 2010•8 citations

The Optimal Pure Gaussian State Associated with Joint Position-Momentum Measurements

Maurice A. de Gosson

We show that given experimental data obtained from joint position and momentum measurements one can construct an optimal pure state approximating the observed quantum state. For that purpose we use a tool from multivariate statistical analysis (the MVE method), which relies on the existence of a minimum volume ellipsoid (the John--Löwner ellipsoid) containing a given convex set. This method allows us to determine the shape of the covariance ellipsoid, which is thereafter calibrated using the expression of the uncertainty principle in terms of the notion of symplectic capacity. We finally use the Wigner formalism to produce the best approximating Gaussian state.

arXiv:1204.2956•Apr 13, 2012•0 citations

The non-commutative Robertson–Schrödinger uncertainty principle

Agapitos Hatzinikitas

Abstract We investigate properties of the covariance matrix in the framework of non-commutative quantum mechanics for an one-parameter family of transformations between the familiar Heisenberg–Weyl algebra and a particular extension of it. Employing as a measure of the Robertson–Schrödinger uncertainty principle the linear symplectic capacity of the Weyl ellipsoid (and its dual), we determine its corresponding bounds. Inequalities between the capacities for non-commutative phase-spaces are established. We also present a constructive example based on a simple model to justify our theoretical predictions.

arXiv:2208.05871•Quantum Studies Mathematics and Foundations•Oct 28, 2022•4 citations

Uncertainty Principle, Phase Space Ellipsoids and Weyl Calculus

Maurice A. de Gosson

We state a precise form of the uncertainty principle in terms of phase space ellipsoids, which we then express in terms of the symplectic capacity of phase space ellipsoids. We apply our approach to the study of the positivity of the Wigner transform of a pure quantum state, and of that of the Weyl operator associated to the average of a positive symbol over a phase space ellipsoid.

Birkhäuser Basel eBooks•Jan 1, 2006•13 citations

Symplectic coarse-grained dynamics: Chalkboard motion in classical and quantum mechanics

Maurice A. de Gosson

In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schrödinger). In the present work we reverse this paradigm and view the motions themselves as being the primary objects. This is made possible by studying arbitrary phase space motions, not of points, but of (small) ellipsoids with the requirement that the symplectic capacity of these ellipsoids is preserved. This allows us to guide and control these motions as we like. In the classical case these ellipsoids correspond to a symplectic coarse graining of phase space, and in the quantum case they correspond to the "quantum blobs" we defined in previous work, and which can be viewed as minimum uncertainty phase space cells which are in a one-to-one correspondence with Gaussian pure states.

arXiv:1901.06554•Advances in Theoretical and Mathematical Physics•Jan 1, 2020•7 citations

On Quantum Indeterminacy and the Uncertainty Principle

Maurice de Gosson

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.

quant-pharXiv:2605.01103•May 5, 2026•0 citations
We show that given experimental data obtained from joint position and momentum measurements one can construct an optimal pure state approximating the observed quantum state. For that purpose we use a tool from multivariate statistical analysis (the MVE method), which relies on the existence of a minimum volume ellipsoid (the John--Löwner ellipsoid) containing a given convex set. This method allows us to determine the shape of the covariance ellipsoid, which is thereafter calibrated using the expression of the uncertainty principle in terms of the notion of symplectic capacity. We finally use the Wigner formalism to produce the best approximating Gaussian state.
Abstract We investigate properties of the covariance matrix in the framework of non-commutative quantum mechanics for an one-parameter family of transformations between the familiar Heisenberg–Weyl algebra and a particular extension of it. Employing as a measure of the Robertson–Schrödinger uncertainty principle the linear symplectic capacity of the Weyl ellipsoid (and its dual), we determine its corresponding bounds. Inequalities between the capacities for non-commutative phase-spaces are established. We also present a constructive example based on a simple model to justify our theoretical predictions.
In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schrödinger). In the present work we reverse this paradigm and view the motions themselves as being the primary objects. This is made possible by studying arbitrary phase space motions, not of points, but of (small) ellipsoids with the requirement that the symplectic capacity of these ellipsoids is preserved. This allows us to guide and control these motions as we like. In the classical case these ellipsoids correspond to a symplectic coarse graining of phase space, and in the quantum case they correspond to the "quantum blobs" we defined in previous work, and which can be viewed as minimum uncertainty phase space cells which are in a one-to-one correspondence with Gaussian pure states.
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.