702 research outputs found

    About the Dedekind psi function in Pauli graphs

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    We study the commutation structure within the Pauli groups built on all decompositions of a given Hilbert space dimension qq, containing a square, into its factors. The simplest illustrative examples are the quartit (q=4q=4) and two-qubit (q=22q=2^2) systems. It is shown how the sum of divisor function σ(q)\sigma(q) and the Dedekind psi function ψ(q)=qpq(1+1/p)\psi(q)=q \prod_{p|q} (1+1/p) enter into the theory for counting the number of maximal commuting sets of the qudit system. In the case of a multiple qudit system (with q=pmq=p^m and pp a prime), the arithmetical functions σ(p2n1)\sigma(p^{2n-1}) and ψ(p2n1)\psi(p^{2n-1}) count the cardinality of the symplectic polar space W2n1(p)W_{2n-1}(p) that endows the commutation structure and its punctured counterpart, respectively. Symmetry properties of the Pauli graphs attached to these structures are investigated in detail and several illustrative examples are provided.Comment: Proceedings of Quantum Optics V, Cozumel to appear in Revista Mexicana de Fisic

    Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?

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    We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension qq, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q=4q=4) and two-qubit (q=22q=2^2) systems, the octit (q=8q=8), qubit/quartit (q=2×4q=2\times 4) and three-qubit (q=23q=2^3) systems, and so on. In the single qudit case, e.g. q=4,8,12,...q=4,8,12,..., one defines a bijection between the σ(q)\sigma (q) maximal commuting sets [with σ[q)\sigma[q) the sum of divisors of qq] of Pauli observables and the maximal submodules of the modular ring Zq2\mathbb{Z}_q^2, that arrange into the projective line P1(Zq)P_1(\mathbb{Z}_q) and a independent set of size σ(q)ψ(q)\sigma (q)-\psi(q) [with ψ(q)\psi(q) the Dedekind psi function]. In the multiple qudit case, e.g. q=22,23,32,...q=2^2, 2^3, 3^2,..., the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if q=22q=2^2) and GQ(3,3) (if q=32q=3^2). More precisely, in dimension pnp^n (pp a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n2n-dimensional vector space over the field Fp\mathbb{F}_p. In this space, one makes use of the commutator to define a symplectic polar space W2n1(p)W_{2n-1}(p) of cardinality σ(p2n1)\sigma(p^{2n-1}), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n1(p)W_{2n-1}(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p2n1)\psi(p^{2n-1}). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo

    Krausz dimension and its generalizations in special graph classes

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    A {\it krausz (k,m)(k,m)-partition} of a graph GG is the partition of GG into cliques, such that any vertex belongs to at most kk cliques and any two cliques have at most mm vertices in common. The {\it mm-krausz} dimension kdimm(G)kdim_m(G) of the graph GG is the minimum number kk such that GG has a krausz (k,m)(k,m)-partition. 1-krausz dimension is known and studied krausz dimension of graph kdim(G)kdim(G). In this paper we prove, that the problem "kdim(G)3""kdim(G)\leq 3" is polynomially solvable for chordal graphs, thus partially solving the problem of P. Hlineny and J. Kratochvil. We show, that the problem of finding mm-krausz dimension is NP-hard for every m1m\geq 1, even if restricted to (1,2)-colorable graphs, but the problem "kdimm(G)k""kdim_m(G)\leq k" is polynomially solvable for (,1)(\infty,1)-polar graphs for every fixed k,m1k,m\geq 1
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