There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG($n, p$), $n$ being odd and $p$ a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of $n$-simplices, i. e., pairs of $n$-simplices which are \emph{mutually inscribed and circumscribed} to each other. For group elements representing an $n$-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension $n$ of the associated polar space in group theoretic terms. Any Moebius pair of $n$-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for $p=2$ and $n=5$.
There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG(n,p), n being odd and p a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of n-simplices, i. e., pairs of n-simplices which are \emph{mutually inscribed and circumscribed} to each other. For group elements representing an n-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension n of the associated polar space in group theoretic terms. Any Moebius pair of n-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for p=2 and n=5.
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).
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.
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.
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).
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.
The n-qubit Pauli group is an essential ingredient to most quantum applications, from computing and error correction to benchmarking and simulation. Despite comprising merely a discrete set of operators, it exhibits many quintessential features of quantum theory, including contextuality, which has been identified as a key resource to quantum advantage in a variety of different flavours. Here, we extend this analysis, introducing the notion of a `noncontextual property' whose nonexistence proves the Kochen-Specker theorem, similarly to and generalising common arguments based on the nonexistence of valuations. We relate this notion formally to the existence of Boolean-valued frame functions, and characterise all such frame functions in the case of the n-qubit Pauli group.
For two qubits, we show that the Pauli group admits noncontextual properties, despite admitting no valuations. We then establish this as the only nontrivial such case with n≥2, by proving that any Boolean-valued frame function on stabiliser states is constant for more than two qubits. We also perform a similar analysis for the symplectic theory underlying the n-qubit Pauli group, for which nonconstant Boolean-valued frame functions exist for all n, yet only in restricted form. By comparison, this shows that contextuality in the n-qubit Pauli group is not only a consequence of the projective nature of the Pauli group as a representation of its underlying symplectic vector space, but of the geometry of symplectic polar spaces itself. In geometric terms, our result determines all Cameron-Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space.