46,063 research outputs found
A composite parameterization of unitary groups, density matrices and subspaces
Unitary transformations and density matrices are central objects in quantum
physics and various tasks require to introduce them in a parameterized form. In
the present article we present a parameterization of the unitary group
of arbitrary dimension which is constructed in a composite
way. We show explicitly how any element of can be composed of
matrix exponential functions of generalized anti-symmetric -matrices
and one-dimensional projectors. The specific form makes it considerably easy to
identify and discard redundant parameters in several cases. In this way,
redundancy-free density matrices of arbitrary rank can be formulated. Our
construction can also be used to derive an orthonormal basis of any
-dimensional subspaces of with the minimal number of
parameters. As an example it will be shown that this feature leads to a
significant reduction of parameters in the case of investigating distillability
of quantum states via lower bounds of an entanglement measure (the
-concurrence).Comment: 13 pages, 1 figur
Optimal Control Theory for Continuous Variable Quantum Gates
We apply the methodology of optimal control theory to the problem of
implementing quantum gates in continuous variable systems with quadratic
Hamiltonians. We demonstrate that it is possible to define a fidelity measure
for continuous variable (CV) gate optimization that is devoid of traps, such
that the search for optimal control fields using local algorithms will not be
hindered. The optimal control of several quantum computing gates, as well as
that of algorithms composed of these primitives, is investigated using several
typical physical models and compared for discrete and continuous quantum
systems. Numerical simulations indicate that the optimization of generic CV
quantum gates is inherently more expensive than that of generic discrete
variable quantum gates, and that the exact-time controllability of CV systems
plays an important role in determining the maximum achievable gate fidelity.
The resulting optimal control fields typically display more complicated Fourier
spectra that suggest a richer variety of possible control mechanisms. Moreover,
the ability to control interactions between qunits is important for delimiting
the total control fluence. The comparative ability of current experimental
protocols to implement such time-dependent controls may help determine which
physical incarnations of CV quantum information processing will be the easiest
to implement with optimal fidelity.Comment: 39 pages, 11 figure
Minimal paths in the commuting graphs of semigroups
Let be a finite non-commutative semigroup. The commuting graph of ,
denoted \cg(S), is the graph whose vertices are the non-central elements of
and whose edges are the sets of vertices such that and
. Denote by the semigroup of full transformations on a finite set
. Let be any ideal of such that is different from the ideal
of constant transformations on . We prove that if , then, with a
few exceptions, the diameter of \cg(J) is 5. On the other hand, we prove that
for every positive integer , there exists a semigroup such that the
diameter of \cg(S) is . We also study the left paths in \cg(S), that is,
paths such that and for all
i\in \{1,\ldot, m\}. We prove that for every positive integer ,
except , there exists a semigroup whose shortest left path has length .
As a corollary, we use the previous results to solve a purely algebraic old
problem posed by B.M. Schein.Comment: 23 pages; v.2: Lemma 2.1 corrected; v.3: final version to appear in
European J. of Combinatoric
Equivalence of Tripartite Quantum States under Local Unitary Transformations
The equivalence of tripartite pure states under local unitary transformations
is investigated. The nonlocal properties for a class of tripartite quantum
states in \Cb^K \otimes \Cb^M \otimes \Cb^N composite systems are
investigated and a complete set of invariants under local unitary
transformations for these states is presented. It is shown that two of these
states are locally equivalent if and only if all these invariants have the same
values.Comment: 7 page
Convex bodies of states and maps
We give a general solution to the question when the convex hulls of orbits of
quantum states on a finite-dimensional Hilbert space under unitary actions of a
compact group have a non-empty interior in the surrounding space of all density
states. The same approach can be applied to study convex combinations of
quantum channels. The importance of both problems stems from the fact that,
usually, only sets with non-vanishing volumes in the embedding spaces of all
states or channels are of practical importance. For the group of local
transformations on a bipartite system we characterize maximally entangled
states by properties of a convex hull of orbits through them. We also compare
two partial characteristics of convex bodies in terms of largest balls and
maximum volume ellipsoids contained in them and show that, in general, they do
not coincide. Separable states, mixed-unitary channels and k-entangled states
are also considered as examples of our techniques.Comment: 18 pages, 1 figur
2d Gauge Theories and Generalized Geometry
We show that in the context of two-dimensional sigma models minimal coupling
of an ordinary rigid symmetry Lie algebra leads naturally to the
appearance of the "generalized tangent bundle" by means of composite fields. Gauge transformations of the composite
fields follow the Courant bracket, closing upon the choice of a Dirac structure
(or, more generally, the choide of a "small
Dirac-Rinehart sheaf" ), in which the fields as well as the symmetry
parameters are to take values. In these new variables, the gauge theory takes
the form of a (non-topological) Dirac sigma model, which is applicable in a
more general context and proves to be universal in two space-time dimensions: A
gauging of of a standard sigma model with Wess-Zumino term
exists, \emph{iff} there is a prolongation of the rigid symmetry to a Lie
algebroid morphism from the action Lie algebroid
into (or the algebraic analogue of the morphism in the case of
). The gauged sigma model results from a pullback by this morphism
from the Dirac sigma model, which proves to be universal in two-spacetime
dimensions in this sense.Comment: 22 pages, 2 figures; To appear in Journal of High Energy Physic
Field-Dependent BRST-antiBRST Lagrangian Transformations
We continue our study of finite BRST-antiBRST transformations for general
gauge theories in Lagrangian formalism, initiated in [arXiv:1405.0790[hep-th]
and arXiv:1406.0179[hep-th]], with a doublet , , of
anticommuting Grassmann parameters and prove the correctness of the explicit
Jacobian in the partition function announced in [arXiv:1406.0179[hep-th]],
which corresponds to a change of variables with functionally-dependent
parameters induced by a finite Bosonic functional
and by the anticommuting generators of
BRST-antiBRST transformations in the space of fields and auxiliary
variables . We obtain a Ward identity depending on the
field-dependent parameters and study the problem of gauge
dependence, including the case of Yang--Mills theories. We examine a
formulation with BRST-antiBRST symmetry breaking terms, additively introduced
to the quantum action constructed by the Sp(2)-covariant Lagrangian rules,
obtain the Ward identity and investigate the gauge-independence of the
corresponding generating functional of Green's functions. A formulation with
BRST symmetry breaking terms is developed. It is argued that the gauge
independence of the above generating functionals is fulfilled in the BRST and
BRST-antiBRST settings. These concepts are applied to the average effective
action in Yang--Mills theories within the functional renormalization group
approach.Comment: 20+7 pages, no figures, presentation improved, typos corrected,
reference added, remarks on composite field approach added in Sec. 4 and App.
- …