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Special fermionic systems entered the realm of quantum chemistry in the seventies in the work of Borland and Dennis in the form of a toy model. This work was leading to a detailed study of the N-representability problem by Klyachko. The topic then has been reconsidered in the light of entanglement theory boiling down to the notion of entanglement polytopes. Recently building on certain properties of such special fermionic systems, a connection between the coupled cluster method and entanglement has been established. In this paper we show that precisely such a special class of systems also provides an interesting physical realization for structures related to the Lie algebras of exceptional groups. This result draws such exotic symmetry structures under the umbrella of entangled systems of physical relevance. [ABSTRACT FROM AUTHOR]

The Coupled Cluster (CC) and full CI expansions are studied for three fermions with six and seven modes. Surprisingly the CC expansion is tailor made to characterize the usual stochastic local operations and classical communication (SLOCC) entanglement classes. It means that the notion of a SLOCC transformation shows up quite naturally as a one relating the CC and CI expansions, and going from the CI expansion to the CC one is equivalent to obtaining a form for the state where the structure of the entanglement classes is transparent. In this picture, entanglement is characterized by the parameters of the cluster operators describing transitions from occupied states to singles, doubles, and triples of non-occupied ones. Using the CC parametrization of states in the seven-mode case, we give a simple formula for the unique SLOCC invariant J. Then we consider a perturbation problem featuring a state from the unique SLOCC class characterized by J # 0. For this state with entanglement generated by doubles, we investigate the phenomenon of changing the entanglement type due to the perturbing effect of triples. We show that there are states with real amplitudes such that their entanglement encoded into configurations of clusters of doubles is protected from errors generated by triples. Finally we put forward a proposal to use the parameters of the cluster operator describing transitions to doubles for entanglement characterization. Compared to the usual SLOCC classes, this provides a coarse grained approach to fermionic entanglement. [ABSTRACT FROM AUTHOR]

We propose a generalization of the usual stochastic local operations and classical communications (SLOCC) and local unitary (LU) classifications of entangled pure state fermionic systems based on the spin group. Our generalization relies on the fact that there is a representation of this group acting on the fermionic Fock space, which, when restricted to fixed particle number subspaces, naturally recovers the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space, which change the particle number. The classification scheme built on the spin group naturally prohibits entanglement between states containing even and odd numbers of fermions. In our scheme the problem of the classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so-called pure spinors. We construct the basic invariants of the spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states; an intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to the configurations of wrapped membranes reinterpreted as qubits. [ABSTRACT FROM AUTHOR]

We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three-qubit observables into 135 symmetric four -qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently-discovered structural and cardinality properties of an aggregate of three-qubit Mermin’s “magic” pentagrams. Moreover, some intriguing connections with the so-called black-hole-qubit correspondence are also pointed out. [ABSTRACT FROM AUTHOR]

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 × 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2, 4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2, 2), whereas the 12 matrices without eigenvalues have their geometric counterpart in the associated double-six. The fine details of this correspondence, including the precise algebraic meaning/analogue of collinearity, are furnished by employing the representation of GQ(2, 4) as a quadric in PG(5, 2) of projective index one. An interesting physics application of our findings is also mentioned. [ABSTRACT FROM AUTHOR]

The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7, 2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7, 2), the seven-dimensional projective space of order 2. The quadric is selected in such a way that it contains all 135 symmetric elements of the group. Under such circumstances, the third element on the line defined by any two points of an ovoid is skew-symmetric, as is the nucleus of the conic defined by any three points of an ovoid. Each ovoid thus yields 36/84 elements of the former/latter type, accounting for all 120 skew-symmetric elements of the group. There are a number of notable types of ovoid-associated subgeometries of the group, of which we mention the following: a subset of 12 skew-symmetric elements lying on four mutually skew lines that span the whole ambient space; a subset of 15 symmetric elements that corresponds to two ovoids sharing three points; a subset of 19 symmetric elements generated by two ovoids on a common point; a subset of 27 symmetric elements that can be partitioned into three ovoids in two unique ways; a subset of 27 skewsymmetric elements that exhibits a 15+2×6 split similar to that exhibited by an elliptic quadric of PG(5, 2); and a subset of seven skew-symmetric elements formed by the nuclei of seven conics having two points in common, which is an analog of a Conwell heptad of PG(5, 2). The strategy we employed is completely novel and unique in its nature, mainly due to the fact that hyperbolic quadrics in binary projective spaces of higher dimensions have no ovoids. Such a detailed dissection of the geometry of the group in questionmay, for example, be crucial in getting further insights into the still-puzzling black-hole-qubit correspondence/analogy. [ABSTRACT FROM AUTHOR]

Having in mind their potential quantum physical applications, we classify all geometric hyperplanes of the near hexagon that is a direct product of a line of size three and the generalized quadrangle of order two. There are eight different kinds of them, totalling to 1,023 = 210 − 1 = |PG(9, 2)|, and they form two distinct families intricately related with the points and lines of the Veldkamp space of the quadrangle in question. [ABSTRACT FROM AUTHOR]

The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang’s SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov’s spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess–Zumino term unlike the one in the complex case. [ABSTRACT FROM AUTHOR]

Using the idea that the symmetry generators commuting with a Landau-like Hamiltonian containing non-Abelian gauge fields will be matrix-valued differential operators, we reconsider the eigenvalue problem of the five-dimensional (5-D) Kepler problem on a SU(2) instanton background. We quickly reproduce the result of Pletyukhov and Tolkachev [J. Math. Phys. 40, 93-100 (1999)], obtained for the energy spectrum. The eigenstates can be expressed in terms of the SU(2) monopole harmonics. The relevance of the theory of induced representations for solving similar problems is emphasized. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]