Generalized Marginals of tie -Wigner Distribution nand Lagrangian Tomography

Abstract

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.

The paper studies continuous-variable quantum-state reconstruction from integrals of Wigner distributions over affine Lagrangian subspaces. It formulates these tomograms using the compact symplectic subgroup $U(n)=\operatorname{Sp}(n)\cap O(2n)$ and its metaplectic representation, relating the geometry of integration planes to transformed position measurements. For a Lagrangian plane $\ell=\{(x,p):Ax+Bp=0\}$ represented by a symplectic rotation $U$, it derives $R_\ell W\psi(x,p)=|\widehat U\psi(Ax+Bp)|^2$. The argument uses symplectic covariance of the Wigner transform and its usual marginal identities; independence from the orthonormal parametrization of the plane is also established. An extension incorporating matrix rescalings is introduced, followed by a proposed inversion formula integrating over the tomographic coordinate and matrix parameters $A,B$. For centered pure Gaussian states, the paper discusses the ambiguity remaining after position and momentum measurements. In one mode, the undetermined covariance has two possible values, $\sigma_{xp}=\pm\sqrt{\sigma_{xx}\sigma_{pp}-\hbar^2/4}$. It then gives a reconstruction construction based on Bloch–Messiah diagonalization: a tomogram in a diagonalizing frame determines the squeezing eigenvalues. The stated limitation is that this frame must already be known, making the construction primarily theoretical rather than practical. Finally, the formalism is extended to mixed states through spectral decompositions and cross-Wigner distributions. Using the orthonormal-basis property of appropriately normalized cross-Wigner functions, the paper expresses tomograms in terms of density-operator coefficients and outlines coefficient-based reconstruction. No experimental data or numerical reconstruction study is presented.