76 research outputs found
Qubits from extra dimensions
We link the recently discovered black hole-qubit correspondence to the
structure of extra dimensions. In particular we show that for toroidal
compactifications of type IIB string theory simple qubit systems arise
naturally from the geometrical data of the tori parametrized by the moduli. We
also generalize the recently suggested idea of the attractor mechanism as a
distillation procedure of GHZ-like entangled states on the event horizon, to
moduli stabilization for flux attractors in F-theory compactifications on
elliptically fibered Calabi-Yau four-folds. Finally using a simple example we
show that the natural arena for qubits to show up is an embedded one within the
realm of fermionic entanglement of quantum systems with indistinguishable
constituents.Comment: 32 pages Late
On the geometry of four qubit invariants
The geometry of four-qubit entanglement is investigated. We replace some of
the polynomial invariants for four-qubits introduced recently by new ones of
direct geometrical meaning. It is shown that these invariants describe four
points, six lines and four planes in complex projective space . For
the generic entanglement class of stochastic local operations and classical
communication they take a very simple form related to the elementary symmetric
polynomials in four complex variables. Moreover, their magnitudes are
entanglement monotones that fit nicely into the geometric set of -qubit ones
related to Grassmannians of -planes found recently. We also show that in
terms of these invariants the hyperdeterminant of order 24 in the four-qubit
amplitudes takes a more instructive form than the previously published
expressions available in the literature. Finally in order to understand two,
three and four-qubit entanglement in geometric terms we propose a unified
setting based on furnished with a fixed quadric.Comment: 19 page
On the geometry of a class of N-qubit entanglement monotones
A family of N-qubit entanglement monotones invariant under stochastic local
operations and classical communication (SLOCC) is defined. This class of
entanglement monotones includes the well-known examples of the concurrence, the
three-tangle, and some of the four, five and N-qubit SLOCC invariants
introduced recently. The construction of these invariants is based on bipartite
partitions of the Hilbert space in the form with . Such partitions can be given
a nice geometrical interpretation in terms of Grassmannians Gr(L,l) of l-planes
in that can be realized as the zero locus of quadratic polinomials
in the complex projective space of suitable dimension via the Plucker
embedding. The invariants are neatly expressed in terms of the Plucker
coordinates of the Grassmannian.Comment: 7 pages RevTex, Submitted to Physical Review
Mermin's Pentagram as an Ovoid of PG(3,2)
Mermin's pentagram, a specific set of ten three-qubit observables arranged in
quadruples of pairwise commuting ones into five edges of a pentagram and used
to provide a very simple proof of the Kochen-Specker theorem, is shown to be
isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective
space of order two, PG(3,2). This demonstration employs properties of the real
three-qubit Pauli group embodied in the geometry of the symplectic polar space
W(5,2) and rests on the facts that: 1) the four observables/operators on any of
the five edges of the pentagram can be viewed as points of an affine plane of
order two, 2) all the ten observables lie on a hyperbolic quadric of the
five-dimensional projective space of order two, PG(5,2), and 3) that the points
of this quadric are in a well-known bijective correspondence with the lines of
PG(3,2).Comment: 5 pages, 4 figure
A study of two-qubit density matrices with fermionic purifications
We study 12 parameter families of two qubit density matrices, arising from a
special class of two-fermion systems with four single particle states or
alternatively from a four-qubit state with amplitudes arranged in an
antisymmetric matrix. We calculate the Wooters concurrences and the
negativities in a closed form and study their behavior. We use these results to
show that the relevant entanglement measures satisfy the generalized
Coffman-Kundu-Wootters formula of distributed entanglement. An explicit formula
for the residual tangle is also given. The geometry of such density matrices is
elaborated in some detail. In particular an explicit form for the Bures metric
is given.Comment: 21 pages, 1 figur
The geometry of entanglement: metrics, connections and the geometric phase
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on
the quaternionic Hopf fibration of over the quaternionic projective space
with an fiber the geometry of
entanglement for two qubits is investigated. The relationship between base and
fiber i.e. the twisting of the bundle corresponds to the entanglement of the
qubits. The measure of entanglement can be related to the length of the
shortest geodesic with respect to the Mannoury-Fubini-Study metric on between an arbitrary entangled state, and the separable state nearest to
it. Using this result an interpretation of the standard Schmidt decomposition
in geometric terms is given. Schmidt states are the nearest and furthest
separable ones lying on, or the ones obtained by parallel transport along the
geodesic passing through the entangled state. Some examples showing the
correspondence between the anolonomy of the connection and entanglement via the
geometric phase is shown. Connections with important notions like the
Bures-metric, Uhlmann's connection, the hyperbolic structure for density
matrices and anholonomic quantum computation are also pointed out.Comment: 42 page
Exact solutions for semirelativistic problems with non-local potentials
It is shown that exact solutions may be found for the energy eigenvalue
problem generated by the class of semirelativistic Hamiltonians of the form H =
sqrt{m^2+p^2} + hat{V}, where hat{V} is a non-local potential with a separable
kernel of the form V(r,r') = - sum_{i=1}^n v_i f_i(r)g_i(r'). Explicit examples
in one and three dimensions are discussed, including the Yamaguchi and Gauss
potentials. The results are used to obtain lower bounds for the energy of the
corresponding N-boson problem, with upper bounds provided by the use of a
Gaussian trial function.Comment: 13 pages, 3 figure
Correlation induced non-Abelian quantum holonomies
In the context of two-particle interferometry, we construct a parallel
transport condition that is based on the maximization of coincidence intensity
with respect to local unitary operations on one of the subsystems. The
dependence on correlation is investigated and it is found that the holonomy
group is generally non-Abelian, but Abelian for uncorrelated systems. It is
found that our framework contains the L\'{e}vay geometric phase [2004 {\it J.
Phys. A: Math. Gen.} {\bf 37} 1821] in the case of two-qubit systems undergoing
local SU(2) evolutions.Comment: Minor corrections; journal reference adde
A Notable Relation between n-Qubit and 2ⁿ⁻¹-Qubit Pauli Groups via Binary LGr(n,2n)
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 2ⁿ⁻¹-qubit 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(2n−1,2) of the 2ⁿ⁻¹-qubit Pauli group in terms of G-orbits, where G≡SL(2,2)×SL(2,2)×⋯×SL(2,2)⋊Sn, 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
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