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Polar Duality and the Reconstruction of Quantum Covariance Matrices from Partial Data

Maurice A. de Gosson

Abstract

We address the problem of the reconstruction of quantum covariance matrices using the notion of Lagrangian and symplectic polar duality introduced in previous work. We apply our constructions to Gaussian quantum states which leads to a non-trivial generalization of Pauli's reconstruction problem and we state a simple tomographic characterization of such states.

Polar Duality and the Reconstruction of Quantum Covariance Matrices from Partial Data

Abstract

We address the problem of the reconstruction of quantum covariance matrices using the notion of Lagrangian and symplectic polar duality introduced in previous work. We apply our constructions to Gaussian quantum states which leads to a non-trivial generalization of Pauli's reconstruction problem and we state a simple tomographic characterization of such states.
Paper Structure (16 sections, 12 theorems, 118 equations)

This paper contains 16 sections, 12 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. The RSUP for Gaussians
  4. Multivariate Correlated Gaussians
  5. The multivariate RSUP
  6. Orthogonal projections of the covariance ellipsoid
  7. Pauli's Problem and its Generalizations
  8. Paulis' reconstruction problem
  9. Lagrangian frames
  10. Gaussian reconstruction by partial tomography
  11. Geometric Interpretation
  12. Polar Duality and Covariance Ellipsoid
  13. Symplectic polar duality
  14. A tomographic result
  15. The case of mixed states
  16. ...and 1 more sections

Key Result

Proposition 1

The metaplectic group $\operatorname*{Mp}(n)$ acts transitively on Gaussians $\psi_{X,Y}=\psi_{X,Y}^{0}$ up to a unimodular factor:

Theorems & Definitions (18)

  • Proposition 1
  • Claim 2
  • Claim 3
  • Claim 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • Example 9
  • Lemma 10
  • ...and 8 more