Employing the fact that the geometry of the N -qubit (N ≥ 2) Pauli group is embodied in the structure of the symplectic polar space W(2N -1, 2) and using properties of the Lagrangian Grassmannian LGr(N, 2N ) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N -qubit Pauli group and a certain subset of elements of the 2 N -1qubit Pauli group.In order to reveal finer traits of this correspondence, the cases N = 3 (also addressed recently by Lévay, Planat and Saniga [J.High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and N = 4 are discussed in detail.As an apt application of our findings, we use the stratification of the ambient projective space PG(2 N -1, 2) of the 2 N -1 -qubit Pauli group in terms of G-orbits, where G ≡ SL(2, 2)×SL(2, 2)ו • •×SL(2, 2) S N , to decompose π(LGr(N, 2N )) into non-equivalent orbits.This leads to a partition of LGr(N, 2N ) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.
Employing the fact that the geometry of the N -qubit (N ≥ 2) Pauli group is embodied in the structure of the symplectic polar space W(2N -1, 2) and using properties of the Lagrangian Grassmannian LGr(N, 2N ) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N -qubit Pauli group and a certain subset of elements of the 2 N -1qubit Pauli group.In order to reveal finer traits of this correspondence, the cases N = 3 (also addressed recently by Lévay, Planat and Saniga [J.High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and N = 4 are discussed in detail.As an apt application of our findings, we use the stratification of the ambient projective space PG(2 N -1, 2) of the 2 N -1 -qubit Pauli group in terms of G-orbits, where G ≡ SL(2, 2)×SL(2, 2)ו • •×SL(2, 2) S N , to decompose π(LGr(N, 2N )) into non-equivalent orbits.This leads to a partition of LGr(N, 2N ) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.
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.
A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding {\it Pauli graph} are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i.\,e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4,3), the dual of W(3).
It is surmised that the algebra of the Pauli operators on the Hilbert space of N-qubits is embodied in the geometry of the symplectic polar space of rank N and order two, W_{2N - 1}(2). The operators (discarding the identity) answer to the points of W_{2N - 1}(2), their partitionings into maximally commuting subsets correspond to spreads of the space, a maximally commuting subset has its representative in a maximal totally isotropic subspace of W_{2N - 1}(2) and, finally, "commuting" translates into "collinear" (or "perpendicular").
A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3).
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
Following the spirit of a recent work of one of the authors (2011 J. Phys. A: Math. Theor. 44 045301), the essential structure of the generalized Pauli group of a qubit–qudit, where d = 2 k and an integer k ⩾ 2, is recast in the language of a finite geometry. A point of such geometry is represented by the maximum set of mutually commuting elements of the group and two distinct points are regarded as collinear if the corresponding sets have exactly 2 k − 1 elements in common. The geometry comprises 2 k − 1 copies of the generalized quadrangle of order 2 ('the doily') that form 2 k − 1 − 1 pencils arranged into a remarkable nested configuration. This nested structure reflects the fact that maximum sets of mutually commuting elements are of two different kinds (ordinary and exceptional) and exhibits an intriguing alternating pattern: the subgeometry of the exceptional points of the ( k + 2) case is found to be isomorphic to the full geometry of the k -case. It should be stressed, however, that these generic properties of the qubit–qudit geometry were inferred from purely computer-handled cases of k = 2, 3, 4 and 5 only and, therefore, their rigorous, computer-free proof for k ⩾ 6 still remains a mathematical challenge.