Symmetric informationally complete–positive operator valued measures and the extended Clifford group

By David Marcus Appleby. 2004

306 citations recorded by the source corpus.

We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Fina...

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0d8b5536572560398a63a474709a1f2e9a16021c

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Citations in the corpus

On Weyl-Heisenberg orbits of equiangular lines (2008, methodology): “On page 17 in [2], Appleby also describes a method to find the stability group of a given fiducial vector z ∈ C , the set of all U ∈ EC(d)/I(d) for which z is eigenvector.”
Tight informationally complete quantum measurements (2006, background): “Analytical constructions, however, are known only for d ≤ 10 and d = 12, 13, 19 [24, 27, 47, 48, 49, 50].”
A note on equiangular tight frames (2008, background): “Proof: Without loss of generality we let l = n and set ’k := ’ ( 1 ) k = fk, for k = 1,...,n 1 (i.e., we remove the last element of the frame {fk}n k=1).”
Steiner equiangular tight frames (2010, background): “In the complex setting, much attention has focused on the maximal case of M(2) vectors in HM [2, 15, 18, 21, 23].”
Quantum-Bayesian Coherence (2009, methodology): “Among the various arguments the Quantum Bayesians use to come to this conclusion, not least in importance is a thoroughgoing personalist account of all probabilities (Bernardo, 1994; de Finetti, 1931, 1990; Jeffrey, 2004; Ramsey, 1931; Savage, 1954)—where the “all” in this sentence includes probabil”
Optimizing quantum process tomography with unitary 2-designs (2007, background): “These are sets of unitary operator bases which form subgroups of the projective Clifford group PC(d) [39, 40] and have cardinalities”
Spectra of Graphs (2011, background): “The binary [234, 12, 112] code derived from the Hill code has a [122, 11, 56] code as residue—also this is a world record.”
Introduction to Matrix Analysis and Applications (2014, background): “SIC-POVMs are conjectured to exist in all dimensions [80, 11].”
High-Resolution Radar via Compressed Sensing (2008, methodology): “ conjectured that for any Nthere exists an (equiangular tight) Gabor frame with N2 elements which achieves the bound 1= p N+1. However, explicit constructions are known only for a very few cases, cf. [19]. Therefore, and because the difference between 1= p N and 1= p N+1 is negligible for large N, w”
Geometry of Quantum States: Index (2007, background): “In Hilbert space this doubling of the group is achieved by representing the extra group elements by anti-unitary transformations [42].”