The curious nonexistence of Gaussian 2-designs

TL;DR

This paper proves that Gaussian states in continuous-variable quantum systems cannot form 2-designs, revealing a fundamental difference from finite-dimensional systems and impacting quantum optical tomography techniques.

Contribution

It demonstrates the nonexistence of Gaussian 2-designs in continuous-variable systems, contrasting with finite-dimensional cases where irreducibility guarantees their existence.

Findings

Gaussian states do not form 2-designs in continuous variables

The ensemble of Gaussian states lacks a well-defined average state

This difference affects optical state and process tomography

Abstract

2-designs -- ensembles of quantum pure states whose 2nd moments equal those of the uniform Haar ensemble -- are optimal solutions for several tasks in quantum information science, especially state and process tomography. We show that Gaussian states cannot form a 2-design for the continuous-variable (quantum optical) Hilbert space L2(R). This is surprising because the affine symplectic group HWSp (the natural symmetry group of Gaussian states) is irreducible on the symmetric subspace of two copies. In finite dimensional Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such as mutually unbiased bases in prime dimensions) are always 2-designs. This property is violated by continuous variables, for a subtle reason: the (well-defined) HWSp-invariant ensemble of Gaussian states does not have an average state because the averaging integral does not converge. In fact, no Gaussian ensemble is even close (in a precise sense) to being a 2-design. This surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography.