The d² Pauli operators attached to a composite qudit in dimension d may be mapped to the vectors of the symplectic module Z² d (Zd the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions d = 6, 10, 15, 12, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the
The d² Pauli operators attached to a composite qudit in dimension d may be mapped to the vectors of the symplectic module Z² d (Zd the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions d = 6, 10, 15, 12, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the
The d2 Pauli operators attached to a composite qudit in dimension d may be mapped to the vectors of the symplectic module Zd2 (Zd the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions d=6,10,15,12, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings Z6, Z10, Z15, Z6×F4 and Z6×Z3, respectively.
As a continuation of our previous work ( Havlicek and Saniga 2007 J. Phys. A: Math. Gen. 40 F943–52 ( Preprint 0708.4333)) an algebraic geometrical study of a single d -dimensional qudit is made, with d being any positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring . Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of . We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes. The operators are also seen to be structured into disjoint ‘layers’ according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.
A very particular connection between the commutation relations of the elements of the generalized Pauli group of a d -dimensional qudit, d being a product of distinct primes, and the structure of the projective line over the (modular) ring is established, where the integer exponents of the generating shift ( X ) and clock ( Z ) operators are associated with submodules of . Under this correspondence, the set of operators commuting with a given one—a perp-set—represents a submodule of . A crucial novel feature here is that the operators are also represented by non -admissible pairs of . This additional degree of freedom makes it possible to view any perp-set as a set-theoretic union of the corresponding points of the associated projective line.
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.
The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their bases/maximal sets of commuting operators, follow a nice graph theoretical/geometrical pattern. One may identify vertices of a graph with the operators so that edges join commuting pairs of them to form the so-called Pauli graph PpN . One may identify points of a geometry with the operators so that lines correspond to the maximal commuting sets of them. As per two-qubits (p = 2, N = 2) all basic properties and partitionings of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W(2). The structure of the two-qutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal with its dual to see the relation with the geometry of generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized adjacency graph for multiple (N > 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. These mathematical concepts relate to mutually unbiased bases and to quantum entanglement. Michel Planat† and Metod Saniga‡ The N-qudit fabric: Pauli graph and finite geometries
Qudits with local dimension d>2 can have unique structure and uses that qubits ( d=2 ) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, d in several ways. To cover composite values of d , we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.