Browsing by Author "Lévay, Péter"

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    Grassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Lévay, Péter; Planat, Michel; Saniga, Metod
    We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three -qubit observables into 135 symmetric four -qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently- discovered structural and cardinality properties of an aggregate of three-qubit Mermin’s “magic” pentagrams. Moreover, some intriguing connections with the so-called black-hole– qubit correspondence are also pointed out.
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    A relation between N-qubit and 2N−1-qubit Pauli groups via binary LGr(N,2N)
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Holweck, Frédéric; Saniga, Matod; Lévay, Péter
    Employing the fact that the geometry of the N-qubit (N≥2) Pauli group is embodied in the structure of the symplectic polar space W(2N−1,2) and using properties of the Lagrangian Grassmannian LGr(N,2N) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N-qubit Pauli group and a certain subset of elements of the 2N−1-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N=3 (also addressed recently by Lévay, Planat and Saniga (JHEP 09 (2013) 037)) and N=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2N−1,2) of the 2N−1-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋅⋅⋅×SL(2,2)⋊SN, to decompose π––(LGr(N,2N)) into non-equivalent orbits. This leads to a partition of LGr(N,2N) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.