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Communication Dans Un Congrès Année : 2011

Modular geometry of the symplectic group\\ associated to a q-level system

Résumé

A $q$-level quantum system (also denoted a $q$-dit) is a promising concept of quantum information processing. A relevant application is quantum computing based on solid state NMR with spins $\frac{3}{2}$ (or quartits), $\frac{7}{2}$ (or octits) or higher spins. The $q$-dit Pauli group, acting on a vector $\left|s\right\rangle$ of the $q$-dimensional Hilbert space is generated by two unitary $X$ (shift) and clock $Z$ operators via $X\left|s\right\rangle=\left|s+1\right\rangle$ and $Z\left|s\right\rangle=\omega^s \left|s\right\rangle$, with $\omega$ a primitive $q$-th root of unity. We investigate the commutation relations among the $q^2-1$ operators of the set $S=\{X^aZ^b,~a=0,\cdots,q-1;~b=0,\cdots,(q-1)~\mbox{and}~(a,b)\ne (0,0)\}$. We find a bijection between the $\sigma(q)$ maximal commuting sets [$\sigma(q)$: the sum of divisors of $q$] of $S$ and the maximal submodules of $\mathbb{Z}_q^2$. Among them we find $\psi(q)$ maximal commuting sets [with $\psi(q)\le \sigma(q)$ be the Dedekind psi function and equality holds if $q$ is free of square] corresponding to free cyclic submodules $\{(ua,ub):u \in \mathbb{Z}_q\}$ $\ne (0,0)$ of $\mathbb{Z}_q^2$, that form the projective line $\mathbb{P}_1(\mathbb{Z}_q)$ over $\mathbb{Z}_q$. Then, the number of admissible operators of the Pauli group involved in $\mathbb{P}_1(\mathbb{Z}_q)$ equals $J_2(q)=\psi(q)\phi(q)$ [with $\phi(q)$ the Euler totient function, and $J_2(q)$ is a Jordan totient function]. They are elements of the symplectic group $G=\mbox{Sp}(2,\mathbb{Z}_q)$ [i.e. the homogeneous principal congruence subgroup of level $q$ of the modular group $PSL(2,\mathbb{Z})$], of order $q J_2(q)$. Such $q$-dit representations of $G$ will be displayed for a few special cases. Previous work on this topic is in \lq\lq A Vourdas et al, J Phys A :Math Theor 43, 042001 (2010)" and \lq\lq H Havlicek et al, J Phys A: Math Theor 41, 015302 (2008)". The non-modular case departs considerably from the above structure. As an illustration, the qubit/quartit and qubit/octit systems are 3-doily and 7-doily arrangements, respectively [a doily is a (self-dual) generalized quadrangle $GQ(2,2)$, that controls the symmetries of observables/bases in the two-qubit system].
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Dates et versions

hal-00510584 , version 1 (19-08-2010)

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  • HAL Id : hal-00510584 , version 1

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Michel Planat. Modular geometry of the symplectic group\\ associated to a q-level system. GROUP28: The XXVIII International Colloquium on Group-Theoretical Methods in Physics, Jul 2010, United Kingdom. pp.012050. ⟨hal-00510584⟩
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