2 research outputs found
The curious nonexistence of Gaussian 2-designs
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.Comment: 9 pages, no pretty figures (sorry!
Qutrit squeezing via semiclassical evolution
We introduce a concept of squeezing in collective qutrit systems through a geometrical picture connected to the deformation of the isotropic fluctuations of su(3) operators when evaluated in a coherent state. This kind of squeezing can be generated by Hamiltonians nonlinear in the generators of su(3) algebra. A simplest model of such a nonlinear evolution is analyzed in terms of semiclassical evolution of the SU(3) Wigner function