124,254 research outputs found
Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability
We recently presented a constructive solution to the N-representability
problem of the two-electron reduced density matrix (2-RDM)---a systematic
approach to constructing complete conditions to ensure that the 2-RDM
represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev.
Lett. 108, 263002 (2012)]. In this paper we provide additional details and
derive further N-representability conditions on the 2-RDM that follow from the
constructive solution. The resulting conditions can be classified into a
hierarchy of constraints, known as the (2,q)-positivity conditions where the q
indicates their derivation from the nonnegativity of q-body operators. In
addition to the known T1 and T2 conditions, we derive a new class of
(2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity
conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of
(2,6)-positivity conditions. The constraints obtained can be divided into two
general types: (i) lifting conditions, that is conditions which arise from
lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity
conditions and (ii) pure conditions, that is conditions which cannot be derived
from a simple lifting of the lower conditions. All of the lifting conditions
and the pure (2,q)-positivity conditions for q>3 require tensor decompositions
of the coefficients in the model Hamiltonians. Subsets of the new
N-representability conditions can be employed with the previously known
conditions to achieve polynomially scaling calculations of ground-state
energies and 2-RDMs of many-electron quantum systems even in the presence of
strong electron correlation
Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduced-density-matrix method
Minimizing the energy of an -electron system as a functional of a
two-electron reduced density matrix (2-RDM), constrained by necessary
-representability conditions (conditions for the 2-RDM to represent an
ensemble -electron quantum system), yields a rigorous lower bound to the
ground-state energy in contrast to variational wavefunction methods. We
characterize the performance of two sets of approximate constraints,
(2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at
capturing correlation in one-dimensional and quasi-one-dimensional (ladder)
Hubbard models. We find that, while both the DQG and DQGT conditions capture
both the weak and strong correlation limits, the more stringent DQGT conditions
improve the ground-state energies, the natural occupation numbers, the pair
correlation function, the effective hopping, and the connected (cumulant) part
of the 2-RDM. We observe that the DQGT conditions are effective at capturing
strong electron correlation effects in both one- and quasi-one-dimensional
lattices for both half filling and less-than-half filling
Analyticity of Entropy Rate of Hidden Markov Chains
We prove that under mild positivity assumptions the entropy rate of a hidden
Markov chain varies analytically as a function of the underlying Markov chain
parameters. A general principle to determine the domain of analyticity is
stated. An example is given to estimate the radius of convergence for the
entropy rate. We then show that the positivity assumptions can be relaxed, and
examples are given for the relaxed conditions. We study a special class of
hidden Markov chains in more detail: binary hidden Markov chains with an
unambiguous symbol, and we give necessary and sufficient conditions for
analyticity of the entropy rate for this case. Finally, we show that under the
positivity assumptions the hidden Markov chain {\em itself} varies
analytically, in a strong sense, as a function of the underlying Markov chain
parameters.Comment: The title has been changed. The new main theorem now combines the old
main theorem and the remark following the old main theorem. A new section is
added as an introduction to complex analysis. General principle and an
example to determine the domain of analyticity of entropy rate have been
added. Relaxed conditions for analyticity of entropy rate and the
corresponding examples are added. The section about binary markov chain
corrupted by binary symmetric noise is taken out (to be part of another
paper
The H-Covariant Strong Picard Groupoid
The notion of H-covariant strong Morita equivalence is introduced for
*-algebras over C = R(i) with an ordered ring R which are equipped with a
*-action of a Hopf *-algebra H. This defines a corresponding H-covariant strong
Picard groupoid which encodes the entire Morita theory. Dropping the positivity
conditions one obtains H-covariant *-Morita equivalence with its H-covariant
*-Picard groupoid. We discuss various groupoid morphisms between the
corresponding notions of the Picard groupoids. Moreover, we realize several
Morita invariants in this context as arising from actions of the H-covariant
strong Picard groupoid. Crossed products and their Morita theory are
investigated using a groupoid morphism from the H-covariant strong Picard
groupoid into the strong Picard groupoid of the crossed products.Comment: LaTeX 2e, 50 pages. Revised version with additional examples and
references. To appear in JPA
On Energy Conditions and Stability in Effective Loop Quantum Cosmology
In isotropic loop quantum cosmology, non-perturbatively modified dynamics of
a minimally coupled scalar field violates weak, strong and dominant energy
conditions when they are stated in terms of equation of state parameter. The
violation of strong energy condition helps to have non-singular evolution by
evading singularity theorems thus leading to a generic inflationary phase.
However, the violation of weak and dominant energy conditions raises concern,
as in general relativity these conditions ensure causality of the system and
stability of vacuum via Hawking-Ellis conservation theorem. It is shown here
that the non-perturbatively modified kinetic term contributes negative pressure
but positive energy density. This crucial feature leads to violation of energy
conditions but ensures positivity of energy density, as scalar matter
Hamiltonian remains bounded from below. It is also shown that the modified
dynamics restricts group velocity for inhomogeneous modes to remain sub-luminal
thus ensuring causal propagation across spatial distances.Comment: 29 pages, revtex4; few clarifications, references added, to appear in
CQ
Inviolable energy conditions from entanglement inequalities
Via the AdS/CFT correspondence, fundamental constraints on the entanglement
structure of quantum systems translate to constraints on spacetime geometries
that must be satisfied in any consistent theory of quantum gravity. In this
paper, we investigate such constraints arising from strong subadditivity and
from the positivity and monotonicity of relative entropy in examples with
highly-symmetric spacetimes. Our results may be interpreted as a set of energy
conditions restricting the possible form of the stress-energy tensor in
consistent theories of Einstein gravity coupled to matter.Comment: 25 pages, 3 figures, v2: refs added, expanded discussion of strong
subadditivity constraints in sections 2.1 and 4.
Quantum Semi-Markov Processes
We construct a large class of non-Markovian master equations that describe
the dynamics of open quantum systems featuring strong memory effects, which
relies on a quantum generalization of the concept of classical semi-Markov
processes. General conditions for the complete positivity of the corresponding
quantum dynamical maps are formulated. The resulting non-Markovian quantum
processes allow the treatment of a variety of physical systems, as is
illustrated by means of various examples and applications, including quantum
optical systems and models of quantum transport.Comment: 4 pages, revtex, no figures, to appear in Phys. Rev. Let
A vicinal surface model for epitaxial growth with logarithmic free energy
We study a continuum model for solid films that arises from the modeling of
one-dimensional step flows on a vicinal surface in the
attachment-detachment-limited regime. The resulting nonlinear partial
differential equation, , gives the evolution
for the surface slope as a function of the local height in a monotone
step train. Subject to periodic boundary conditions and positive initial
conditions, we prove the existence, uniqueness and positivity of global strong
solutions to this PDE using two Lyapunov energy functions. The long time
behavior of converging to a constant that only depends on the initial data
is also investigated both analytically and numerically.Comment: 18 pages, 7 figure
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