Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Havlicek, Hans

doi:10.3842/sigma.2009.096

Cited by 27 publications

(53 citation statements)

References 33 publications

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“…5 of [3] and in example 5 of [11]. The 80 observables of the central quotient P q /Z(P q ) (with q = 3…”

Section: The Two-qutrit Systemmentioning

confidence: 99%

“…Further work was published to clarify this earlier work dealing with symplectic polar spaces of multiple qudits [10,11,12] and, in what concerns multiple qubits, its link to units in Clifford algebras [13], to Lie algebras [14] and to a class of singular curves in phase space [15]. Prior to the advent of quantum information science, the incidence properties of the q-dimensional geometry and the relations to Clifford algebras were published in [16,17].…”

Section: Introductionmentioning

confidence: 99%

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

Planat¹

2010

J. Phys. A: Math. Theor.

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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 = 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 Z 2 q , that arrange into the projective line P 1 (Z 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 = 22 ) 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 2n−1 (p) of cardinality σ(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 ψ(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.

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“…5 of [3] and in example 5 of [11]. The 80 observables of the central quotient P q /Z(P q ) (with q = 3…”

Section: The Two-qutrit Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Planat¹

2010

J. Phys. A: Math. Theor.

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“…Interestingly enough, this nested sequence of binomial configurations is identical with part of that found to be associated with Cayley-Dickson algebras [7]. Moreover, given the fact that PG(5, 2) is the natural embedding space for the symplectic polar space W (5, 2) that geometrizes the structure of the three-qubit Pauli group [8,9], this particular sequence of configurations may lead to further intriguing insights into the physical relevance of this group. …”

mentioning

confidence: 61%

The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

Санига

2015

Mathematics

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Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (28 6 , 56 3 )-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G 2 (8). It is also pointed out that a set of seven points of G 2 (8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (28 6 , 56 3 )-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).Keywords: combinatorial Grassmannian; binary Klein quadric; Conwell heptad; three-qubit Pauli group Let Q + (5, 2) be a hyperbolic quadric in a five-dimensional projective space PG(5, 2). As it is well known (see, e.g., [1,2]), there are 28 points lying off this quadric as well as 56 lines skew (or, external) to it. If the equation of the quadric is taken in a canonical form Q 0 : x 1 x 2 + x 3 x 4 + x 5 x 6 = 0, then the 28 off-quadric points are those listed in Table 1 and the 56 external lines are those given in Table 2. In Table 2, the "+" symbol indicates which point lies on a given line; for example, line 1 consists of points 1, 4 and 9. As it is obvious from this table, each line has three points and through each point there are six lines; hence, these points and lines form a (28 6 , 56 3 )-configuration. Next, a combinatorial Grassmannian G k (|X|) (see, e.g., [3,4]), where k is a positive integer and X is a finite set, |X| = N , is a point-line incidence structure whose points are all k-element subsets of X

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“…It is worth noting that the authors of [5] took their motivation from physics. They were looking for finite geometries potentially allowing physical applications, similar to the ones in [6,9] and [10], where a class of finite symplectic polar spaces and certain finite generalized polygons were successfully linked with quantum information theory (commuting and non-commuting elements of Pauli groups) and the theory of black holes and black strings (symmetry properties of entropy formulae). The cited papers contain a wealth of further references on related work.…”

Section: Lemma 1 (See [14 Theorem 11]) If |ψ| Is Odd Then Q Has Pomentioning

confidence: 98%

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

Shaw

Gordon

Havlicek

2011

Des. Codes Cryptogr.

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We consider various aspects of the Segre variety S := S 1,1,1 (2) in PG(7, 2), whose stabilizer group G S < GL(8, 2) has the structure N Sym(3), where N := GL(2, 2)× GL(2, 2) × GL(2, 2). In particular we prove that S determines a distinguished Z 3 -subgroup Z < GL(8, 2) such that AZ A −1 = Z, for all A ∈ G S , and in consequence S determines a G S -invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties S 1,1,1 (2) in PG(7, 2) come along in triplets {S, S , S } which share the same distinguished Z 3 -subgroup Z < GL(8, 2). We conclude by determining all fifteen G S -invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five G S -orbits of points in PG(7, 2).

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Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Cited by 27 publications

References 33 publications

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

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

The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

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