Modular geometry of the symplectic group attached to a q-level system and to multiple q-dit mixtures

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 = 22) systems, the octit (q = 8), qubit/quartit (q = 2×4) and three-qubit (q = 23) 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 ℤq2, that arrange into the projective line P1(ℤq) 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 = 22, 23, 32,…, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if q = 22) and GQ(3,3) (if q = 32). More precisely, in dimension pn (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 Fp. In this space, one makes use of the commutator to define a symplectic polar space W2n−1(p) of cardinality σ(p2n−1), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of 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). 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.