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

Abstract

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 $\sigma (q)$ maximal commuting sets [with $\sigma [q)$ 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 $\sigma (q)-\psi (q)$ [with $\psi (q)$ 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 $\sigma (p^{2n-1})$, 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 $\psi (p^{2n-1})$. 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.

18 pages, version submiited to J. Phys. A: Math. Theor

References in corpus (13)

• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Geometrical approach to mutually unbiased bases
• Black Hole Entropy and Finite Geometry
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line of an Arbitrary Single Qudit
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Feynman's path integral and mutually unbiased bases
• Projective Ring Line of a Specific Qudit
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• The isotropic lines of Z_{d}^{2}
• Nanometre-scale nuclear-spin device for quantum information processing
• Qudits of composite dimension, mutually unbiased bases and projective ring geometry
• Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

Cited by in corpus (18)

• On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
• Will a physicist prove the Riemann Hypothesis?
• Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
• Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon
• Informationally complete characters for quark and lepton mixings
• Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs
• Weak mutually unbiased bases
• Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
• Riemann hypothesis and Quantum Mechanics
• The Poincaré half-plane for informationally complete POVMs
• Group geometrical axioms for magic states of quantum computing
• The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary $\mathrm{LGr}(N,2N)$
• 'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
• Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2)
• Twelve-dimensional Pauli group contextuality
• Pauli graphs, Riemann hypothesis, Goldbach pairs