About the Dedekind psi function in Pauli graphs

作者：Planat M^*

来源：Revista Mexicana de Fisica, 2011, 57(3): 107-112.

• Pauli graphs
• discrete phase space
• graph theory
• finite groups
• number theory

We study the commutation structure within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. The simplest illustrative examples are the quartit (q = 4) and two-qubit (q = 2(2)) systems. It is shown how the sum of divisor function sigma(q) and the Dedekind psi function psi(q) = q Pi(p vertical bar g)(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 = pm and p a prime), the arithmetical functions sigma(p(2n-1)) and psi(p(2n-1)) count the cardinality of the symplectic polar space W2n-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.

• 出版日期2011-7