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Journal of Physics A Mathematical and Theoretical

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

Michel R. P. Planat

Published Mar 24, 2011• 9 citations
Publisher

Abstract

The Journal of Physics A: Mathematical and Theoretical publishing team would like to apologise to the author of the above paper. Due to an oversight, the article was published with an incorrect publication date. The correct date is: 20 December 2010.
DOI
10.1088/1751-8121/44/16/169601

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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 $q$, containing a square, into its factors. Illustrative low dimensional examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems, the octit ($q=8$), qubit/quartit ($q=2\times 4$) and three-qubit ($q=2^3$) systems, and so on. In the single qudit case, e.g. $q=4,8,12,...$, one defines a bijection between the $σ(q)$ maximal commuting sets [with $σ[q)$ the sum of divisors of $q$] of Pauli observables and the maximal submodules of the modular ring $\mathbb{Z}_q^2$, that arrange into the projective line $P_1(\mathbb{Z}_q)$ and a independent set of size $σ(q)-ψ(q)$ [with $ψ(q)$ the Dedekind psi function]. In the multiple qudit case, e.g. $q=2^2, 2^3, 3^2,...$, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if $q=2^2$) and GQ(3,3) (if $q=3^2$). More precisely, in dimension $p^n$ ($p$ a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the $2n$-dimensional vector space over the field $\mathbb{F}_p$. In this space, one makes use of the commutator to define a symplectic polar space $W_{2n-1}(p)$ of cardinality $σ(p^{2n-1})$, that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of $W_{2n-1}(p)$ 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 $ψ(p^{2n-1})$. 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.

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Modular geometry of the symplectic group attached to aq-level system and to multipleq-dit mixtures

Michel R. P. Planat

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q , containing a square, into its factors. Illustrative low dimensional examples are the quartit ( q = 4) and two-qubit ( q = 2 2 ) systems, the octit ( q = 8), qubit/quartit ( q = 2 × 4) and three-qubit ( q = 2 3 ) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12,..., one defines a bijection between the σ ( q ) maximal commuting sets [with σ[ q )the sum of divisors of q ] of Pauli observables and the maximal submodules of the modular ring q 2 , that arrange into the projective line P 1 ( q ) and a independent set of size σ( q ) − ψ( q ) [with ψ( q ) the Dedekind psi function]. In the multiple qudit case, e.g. 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 = 2 2 ) and GQ (3,3) (if q = 3 2 ). More precisely, in dimension p n ( p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field F p . In this space, one makes use of the commutator to define a symplectic polar space W 2 n −1 (p) of cardinality σ ( p 2 n −1 ), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W 2 n −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 ( p 2 N −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.

Journal of Physics Conference Series•Mar 1, 2011•0 citations
We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension qqq, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q=4q=4q=4) and two-qubit (q=22q=2^2q=22) systems, the octit (q=8q=8q=8), qubit/quartit (q=2×4q=2\times 4q=2×4) and three-qubit (q=23q=2^3q=23) systems, and so on. In the single qudit case, e.g. q=4,8,12,...q=4,8,12,...q=4,8,12,..., one defines a bijection between the σ(q)σ(q)σ(q) maximal commuting sets [with σ[q)σ[q)σ[q) the sum of divisors of qqq] of Pauli observables and the maximal submodules of the modular ring Zq2\mathbb{Z}_q^2Zq2​, that arrange into the projective line P1(Zq)P_1(\mathbb{Z}_q)P1​(Zq​) and a independent set of size σ(q)−ψ(q)σ(q)-ψ(q)σ(q)−ψ(q) [with ψ(q)ψ(q)ψ(q) the Dedekind psi function]. In the multiple qudit case, e.g. q=22,23,32,...q=2^2, 2^3, 3^2,...q=22,23,32,..., the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if q=22q=2^2q=22) and GQ(3,3) (if q=32q=3^2q=32). More precisely, in dimension pnp^npn (ppp a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n2n2n-dimensional vector space over the field Fp\mathbb{F}_pFp​. In this space, one makes use of the commutator to define a symplectic polar space W2n−1(p)W_{2n-1}(p)W2n−1​(p) of cardinality σ(p2n−1)σ(p^{2n-1})σ(p2n−1), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n−1(p)W_{2n-1}(p)W2n−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(p2n−1)ψ(p^{2n-1})ψ(p2n−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.
We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q , containing a square, into its factors. Illustrative low dimensional examples are the quartit ( q = 4) and two-qubit ( q = 2 2 ) systems, the octit ( q = 8), qubit/quartit ( q = 2 × 4) and three-qubit ( q = 2 3 ) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12,..., one defines a bijection between the σ ( q ) maximal commuting sets [with σ[ q )the sum of divisors of q ] of Pauli observables and the maximal submodules of the modular ring q 2 , that arrange into the projective line P 1 ( q ) and a independent set of size σ( q ) − ψ( q ) [with ψ( q ) the Dedekind psi function]. In the multiple qudit case, e.g. 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 = 2 2 ) and GQ (3,3) (if q = 3 2 ). More precisely, in dimension p n ( p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field F p . In this space, one makes use of the commutator to define a symplectic polar space W 2 n −1 (p) of cardinality σ ( p 2 n −1 ), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W 2 n −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 ( p 2 N −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.