The Pauli Problem for Gaussian Quantum States: Geometric Interpretation

Abstract

We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier transform and allows a geometric interpretation of the uncertainty principle and allows to apprehend the Pauli problem in a rather simple way.

Symplectic Radon Transform and the Metaplectic Representation

“…in §, which gives us the opportunity to shortly discuss the Pauli problem for Gaussians, thus generalizing results in [4].…”

A Few Almost Trivial Notes on the Symplectic Radon Transform and the Tomographic Picture of Quantum Mechanics

“…(ii) Let us denote A the right-hand side of the equality (5). Using the first marginal property (10) we have…”

“…Historically, this problem goes back to the famous question Pauli asked in [14], whether the probability densities |ψ(x)| 2 and | ψ(p)| 2 uniquely determine the wavefunction ψ(x). The answer is negative, as is seen on the following simple example [10]: of Gaussian wavefunction in one spatial dimension…”

Polar Duality, John Ellipsoid, and Generalized Gaussians

“…We are going to apply Theorem 9 to characterize pure Gaussian density operators without prior knowledge of the full covariance matrix. This is related to the so-called "Pauli reconstruction problem" [21] we have discussed in [14]. The latter can be reformulated in terms of the Wigner transform as follows: given a function…”