2011/10/05 by Robin Blume-Kohout, Peter S. Turner · 20 citations
Computer Science · Mathematics · Physics and Astronomy · #Computability, Logic, AI Algorithms #Covariant transformation #Gaussian #Gaussian process #Hilbert space #Irreducibility #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum mechanics #Quantum state #Quantum tomography #Symplectic geometry #quant-ph
paper · pdf · doi:10.1007/s00220-014-1894-3
published in Communications in Mathematical Physics 326(3), 755-771 (Springer Science+Business Media) · 9 pages, no pretty figures (sorry!)
arxiv created 2011/10/05 · openalex publication_date 2014/02/19 · arxiv updated 2014/03/06 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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.