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## Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/12/2012

Relevância na Pesquisa

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In this paper, we give a new construction of resonant normal forms with a
small remainder for near-integrable Hamiltonians at a quasi-periodic frequency.
The construction is based on the special case of a periodic frequency, a
Diophantine result concerning the approximation of a vector by independent
periodic vectors and a technique of composition of periodic averaging. It
enables us to deal with non-analytic Hamiltonians, and in this first part we
will focus on Gevrey Hamiltonians and derive normal forms with an exponentially
small remainder. This extends a result which was known for analytic
Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As
applications, we obtain an exponentially large upper bound on the stability
time for the evolution of the action variables and an exponentially small upper
bound on the splitting of invariant manifolds for hyperbolic tori, generalizing
corresponding results for analytic Hamiltonians.

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## Limitations on the simulation of non-sparse Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.87%

The problem of simulating sparse Hamiltonians on quantum computers is well
studied. The evolution of a sparse N x N Hamiltonian H for time t can be
simulated using O(||Ht||poly(log N)) operations, which is essentially optimal
due to a no--fast-forwarding theorem. Here, we consider non-sparse Hamiltonians
and show significant limitations on their simulation. We generalize the
no--fast-forwarding theorem to dense Hamiltonians, ruling out generic
simulations taking time o(||Ht||), even though ||H|| is not a unique measure of
the size of a dense Hamiltonian $H$. We also present a stronger limitation
ruling out the possibility of generic simulations taking time poly(||Ht||,log
N), showing that known simulations based on discrete-time quantum walk cannot
be dramatically improved in general. On the positive side, we show that some
non-sparse Hamiltonians can be simulated efficiently, such as those with graphs
of small arboricity.; Comment: v1: 12 pages. v2: 15 pages; strengthened main result

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## Higher-order-in-spin interaction Hamiltonians for binary black holes from Poincar\'e invariance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.87%

The fulfillment of the space-asymptotic Poincar\'e algebra is used to derive
new higher-order-in-spin interaction Hamiltonians for binary black holes in the
Arnowitt-Deser-Misner canonical formalism almost completing the set of the
formally $1/c^4$ spin-interaction Hamiltonians involving nonlinear spin terms.
To linear order in $G$, the expressions for the $S^3p$- and the
$S^2p^2$-Hamiltonians are completed. It is also shown that there are no quartic
nonlinear $S^4$-Hamiltonians to linear order in $G$.; Comment: REVTeX4, 14 pages; center-of-mass-vector corrected Eq. (2.25) and
modified coefficients of the Hamiltonian Eq. (7.3) and corresponding source
terms Eqs. (7.5) and (7.6) following hereof; version to appear in Phys Rev D

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## A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization II

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We give a characterization of the class of gapped Hamiltonians introduced in
PartI [O]. The Hamiltonians in this class are given as MPS (Matrix product
state) Hamiltonians. In [O], we list up properties of ground state structures
of Hamiltonians in this class. In this Part II, we show the converse. Namely,
if a (not necessarily MPS) Hamiltonian $H$ satisfies five of the listed
properties, there is a Hamiltonian $H'$ from the class in [O], satisfying the
followings: The ground state spaces of the two Hamiltonians on the infinite
intervals coincide. The spectral projections onto the ground state space of $H$
on each finite intervals are approximated by that of $H'$ exponentially well,
with respect to the interval size. The latter property has an application to
the classification problem.

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## Weak KAM theoretic aspects for nonregular commuting Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Analysis of PDEs#Computer Science - Systems and Control#Mathematics - Optimization and Control

In this paper we consider the notion of commutation for a pair of continuous
and convex Hamiltonians, given in terms of commutation of their Lax- Oleinik
semigroups. This is equivalent to the solvability of an associated multi- time
Hamilton-Jacobi equation. We examine the weak KAM theoretic aspects of the
commutation property and show that the two Hamiltonians have the same weak KAM
solutions and the same Aubry set, thus generalizing a result recently obtained
by the second author for Tonelli Hamiltonians. We make a further step by
proving that the Hamiltonians admit a common critical subsolution, strict
outside their Aubry set. This subsolution can be taken of class C^{1,1} in the
Tonelli case. To prove our main results in full generality, it is crucial to
establish suitable differentiability properties of the critical subsolutions on
the Aubry set. These latter results are new in the purely continuous case and
of independent interest.; Comment: 37 pages. Third version. Presentation of the commutation property
changed. Proof of the main theorem made clearer

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## Linked $\mathcal{PT }$-symmetry to Supersymmetry in a class of non-Hermitian Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

27.14%

We introduce and study a class of non-Hermitian Hamiltonians which have
velocity dependent potentials. Since stability can not be advocated directly
from the classical potential, we show that the energy spectra are real and
bounded from below which proves the stability of the spectra of all members in
the class. We find that the introduced class of non-Hermitian Hamiltonians do
have a corresponding superpartner class of non-Hermitian Hamiltonians. We were
able to introduce supercharges which in conjunction with the corresponding
super Hamiltonians constitute a closed super algebra. Among the introduced
Hamiltonians, we show that non-$\mathcal{PT }$-symmetric Hamiltonians can be
transformed into their corresponding superpartner Hamiltonians via a specific
canonical transformation while the $\mathcal{PT }$-symmetric ones failed to be
mapped to their corresponding superpartner Hamiltonians via the same canonical
transformation. Since canonical transformations preserve the spectrum, we
conclude that non-$\mathcal{PT }$-symmetric Hamiltonians out of the introduced
class of Hamiltonians have the same spectrum as the corresponding superpartner
Hamiltonians and thus Susy is broken for such Hamiltonians. This kind of
intertwining of $\mathcal{PT }$-symmetry and Supersymmetry is new as all the so
far discussed cases concentrate on Hamiltonians of broken $\mathcal{PT }%
$-symmetry that have broken Supersymmetry too while we showed that Susy can be
also broken for non-$\mathcal{PT }$-symmetric and non-Hermitian Hamiltonians .; Comment: 11 pages...

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## Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/08/2008

Relevância na Pesquisa

26.93%

Consider the set of all Hamiltonians whose largest and smallest energy
eigenvalues, E_max and E_min, differ by a fixed energy \omega. Given two
quantum states, an initial state |\psi_I> and a final state |\psi_F>, there
exist many Hamiltonians H belonging to this set under which |\psi_I> evolves in
time into |\psi_F>. Which Hamiltonian transforms the initial state to the final
state in the least possible time \tau? For Hermitian Hamiltonians, $\tau$ has a
nonzero lower bound. However, among complex non-Hermitian PT-symmetric
Hamiltonians satisfying the same energy constraint, \tau can be made
arbitrarily small without violating the time-energy uncertainty principle. The
minimum value of \tau can be made arbitrarily small because for PT-symmetric
Hamiltonians the evolution path from the vector |\psi_I> to the vector
|\psi_F>, as measured using the Hilbert-space metric appropriate for this
theory, can be made arbitrarily short. The mechanism described here resembles
the effect in general relativity in which two space-time points can be made
arbitrarily close if they are connected by a wormhole. This result may have
applications in quantum computing.; Comment: 20 pages. To appear in: Time in Quantum Mechanics II, edited by J.G.
Muga

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## Time Dependent Quadratic Hamiltonians,SU(1,1), SU(2), SU(2,1) and SU(3)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.87%

The properties of SU(1,1) SU(2),SU(2,1) and SU(3) have often been used in
quantum optics. In this paper we demonstrate the use of these symmetries. The
group properties of SU(1,1) SU(2), and SU(2,1) are used to find the transition
probabilities of various time dependent quadratic Hamiltonians. We consider
Hamiltonians representing the frequency converter,parametric amplifier and
raman scattering.These Hamiltonians are used to describe optical coupling in
nonlinear crystals.; Comment: This paper is a replacement, new, appendix, sections and references
have been added. A version of this paper has been accepted by 'Physics
Letters A' on 17th of febuary 2006. It is available from
http://www.sciencedirect.com. Title of paper 'Time dependent quadratic
Hamiltonians SU(2) and SU(2,1)' author Paul Croxson 19 pages

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## Hyperbolic Deformation on Quantum Lattice Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.93%

A group of non-uniform quantum lattice Hamiltonians in one dimension is
introduced, which is related to the hyperbolic $1 + 1$-dimensional space. The
Hamiltonians contain only nearest neighbor interactions whose strength is
proportional to $\cosh j \lambda$, where $j$ is the lattice index and where
$\lambda \ge 0$ is a deformation parameter. In the limit $\lambda \to 0$ the
Hamiltonians become uniform. Spacial translation of the deformed Hamiltonians
is induced by the corner Hamiltonians. As a simple example, we investigate the
ground state of the deformed $S = 1/2$ Heisenberg spin chain by use of the
density matrix renormalization group (DMRG) method. It is shown that the ground
state is dimerized when $\lambda$ is finite. Spin correlation function show
exponential decay, and the boundary effect decreases with increasing $\lambda$.; Comment: 5 pages, 4 figures

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## Graph states as ground states of two-body frustration-free Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.93%

The framework of measurement-based quantum computation (MBQC) allows us to
view the ground states of local Hamiltonians as potential resources for
universal quantum computation. A central goal in this field is to find models
with ground states that are universal for MBQC and that are also natural in the
sense that they involve only two-body interactions and have a small local
Hilbert space dimension. Graph states are the original resource states for
MBQC, and while it is not possible to obtain graph states as exact ground
states of two-body Hamiltonians here we construct two-body frustration-free
Hamiltonians that have arbitrarily good approximations of graph states as
unique ground states. The construction involves taking a two-body
frustration-free model that has a ground state convertible to a graph state
with stochastic local operations, then deforming the model such that its ground
state is close to a graph state. Each graph state qubit resides in a subspace
of a higher dimensional particle. This deformation can be applied to two-body
frustration-free Affleck-Kennedy-Lieb-Tasaki (AKLT) models, yielding
Hamiltonians that are exactly solvable with exact tensor network expressions
for ground states. For the star-lattice AKLT model...

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## Classification of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

27.07%

We classify all Hamiltonians with rank 1 symmetry, acting on a periodic
three-state spin chain, and solvable through (generalisation of) the coordinate
Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of
the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev,
Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians,
there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as
subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a
generalization of the genus 5 special branch found by Martins, plus three new
ones. We get also two 14-vertex Hamiltonians.
We solve all these Hamiltonians using CBA, and provide their spectrum,
eigenfunctions and Bethe equations. A special attention is made to provide the
specifications of our multi-parametric Hamiltonians that give back known
Hamiltonians.; Comment: 30 pages; web page: http://www.coulomb.univ-montp2.fr/3Ham

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## Hubbard-like Hamiltonians for interacting electrons in s, p and d orbitals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.93%

Hubbard-like Hamiltonians are widely used to describe on-site Coulomb
interactions in magnetic and strongly-correlated solids, but there is much
confusion in the literature about the form these Hamiltonians should take for
shells of p and d orbitals. This paper derives the most general s, p and d
orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and
presents them in ways convenient for practical calculations. We use the full
configuration interaction method to study p and d orbital dimers and compare
results obtained using the correct Hamiltonian and the collinear and vector
Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the
nature of the ground state, the time evolution of excited states, and the
electronic heat capacity.; Comment: Updated the paper to make some clarifications and include colour
figures

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## A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.98%

We have proposed previously a method for constructing self-conjugate
Hamiltonians H_eta in the eta-representation with a flat scalar product to
describe the dynamics of Dirac particles in arbitrary gravitational fields. In
this paper, we prove that, for block-diagonal metrics, the Hamiltonians H_eta
can be obtained, in particular, using "reduced" parts of Dirac Hamiltonians,
i.e. expressions for Dirac Hamiltonians derived using tetrad vectors in the
Schwinger gauge without or with a few summands with bispinor connectivities.
Based on these results, we propose a modified method for constructing
Hamiltonians in the eta-representation with a significantly smaller amount of
required calculations. Using this method, here we for the first time find
self-conjugate Hamiltonians for a number of metrics, including the Kerr metric
in the Boyer-Lindquist coordinates, the Eddington-Finkelstein,
Finkelstein-Lemaitre, Kruskal, Clifford torus metrics and for non-stationary
metrics of open and spatially flat Friedmann models.; Comment: 24 pages

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## Trivial Low Energy States for Commuting Hamiltonians, and the Quantum PCP Conjecture

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

27.01%

We consider whether or not Hamiltonians which are sums of commuting
projectors have "trivial" ground states which can be constructed by a local
quantum circuit of bounded depth and range acting on a product state. While the
toric code only has nontrivial ground states, commuting projector Hamiltonians
which are sums of two-body interactions have trivial ground states.
We define an "interaction complex" for a Hamiltonian, generalizing the
interaction graph, and we show that if this complex can be continuously mapped
to a 1-complex using a map with bounded diameter of pre-images then the
Hamiltonian has a trivial ground state assuming one technical condition on the
Hamiltonian (this condition holds for all stabilizer Hamiltonians, and we also
prove the result for all Hamiltonians under an assumption on the 1-complex).
While this includes cases considered by Ref., it also includes other
Hamiltonians whose interaction complexes cannot be coarse-grained into the case
of Ref.
One motivation for this is the quantum PCP conjecture. Many commonly studied
interaction complexes can be mapped to a 1-complex after removing a small
fraction of sites. For commuting projector Hamiltonians on such complexes, a
trivial ground state for the Hamiltonian with those sites removed is a low
energy trivial state for the original Hamiltonian. Such states can act as a
classical witness to the existence of a low energy state. While this result
applies only to commuting Hamiltonians...

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## Spherical-separablility of non-Hermitian Hamiltonians and pseudo-PT-symmetry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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Non-Hermitian but P(phi)T(phi)-symmetrized spherically-separable Dirac and
Schrodinger Hamiltonians are considered. It is observed that the descendant
Hamiltonians H(r), H(theta), and H(phi) play essential roles and offer some
"user-feriendly" options as to which one (or ones) of them is (or are)
non-Hermitian. Considering a P(phi)T(phi)-symmetrized H(phi), we have shown
that the conventional Dirac (relativistic) and Schrodinger (non-relativistic)
energy eigenvalues are recoverable. We have also witnessed an unavoidable
change in the azimuthal part of the general wavefunction. Moreover, setting a
possible interaction V(theta) in the descendant Hamiltonian H(theta) would
manifest a change in the angular theta-dependent part of the general solution
too. Whilst some P(phi)T(phi)-symmetrized H(phi) Hamiltonians are considered, a
recipe to keep the regular magnetic quantum number m, as defined in the regular
traditional Hermitian settings, is suggested. Hamiltonians possess properties
similar to the PT-symmetric ones (here the non-Hermitian P(phi)T(phi)-symmetric
Hamiltonians) are nicknamed as pseudo-PT-symmetric.; Comment: 13 pages, 2 figures. This article is a combination of arXiv:0711.3887
and arXiv:0710.5814

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## Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematical Physics#High Energy Physics - Theory#Mathematics - Spectral Theory#Nuclear Theory#Physics - Atomic Physics#Quantum Physics

We study complex potentials and related non-diagonalizable Hamiltonians with
special emphasis on formal definitions of associated functions and Jordan
cells. The nonlinear SUSY for complex potentials is considered and the theorems
characterizing its structure are presented. We present the class of potentials
invariant under SUSY transformations for non-diagonalizable Hamiltonians and
formulate several results concerning the properties of associated functions .
We comment on the applicability of these results for softly non-Hermitian
PT-symmetric Hamiltonians. The role of SUSY (Darboux) transformations in
increasing/decreasing of Jordancells in SUSY partner Hamiltonians is thoroughly
analyzed and summarized in the Index Theorem. The properties of
non-diagonalizable Hamiltonians as well as the Index Theorem are illustrated in
the solvable examples of non-Hermitian reflectionless Hamiltonians . The
rigorous proofs are relegated to the Part II of this paper. At last, some
peculiarities in resolution of identity for discrete and continuous spectra
with a zero-energy bound state at threshold are discussed.; Comment: 31 pp., detailed discussion of PT symmetry and more of relevant refs
are added

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## Solution of the problem of uniqueness and hermiticity of hamiltonians for Dirac particles in gravitational fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/07/2010

Relevância na Pesquisa

27.07%

The authors prove that the dynamics of spin 1/2 particles in stationary
gravitational fields can be described using an approach, which builds upon the
formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis
of three expressions for Hamiltonians, which are derived from the Dirac
equation and describe the dynamics of spin 1/2 particles in the gravitational
field of the Kerr solution. The Hamiltonians correspond to different choices of
tetrad vectors and differ from each other. The differences between the
Hamiltonians confirm the conclusion known from many studies that the
Hamiltonians derived from the Dirac equation are non-unique. Application of
standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians
produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the
resulting Hamiltonian is the same as that of the Hamiltonians derived from the
Dirac equation with any chosen system of tetrad vectors. For description of the
dynamics of spin 1/2 particles in stationary gravitational fields can be used
not only the formalism of pseudo-Hermitian Hamiltonians, but also an
alternative approach, which employs the Parker scalar product. The authors show
that the alternative approach is equivalent to the formalism of
pseudo-Hermitian Hamiltonians.; Comment: 20 pages

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## Realizable Hamiltonians for Universal Adiabatic Quantum Computers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.93%

It has been established that local lattice spin Hamiltonians can be used for
universal adiabatic quantum computation. However, the 2-local model
Hamiltonians used in these proofs are general and hence do not limit the types
of interactions required between spins. To address this concern, the present
paper provides two simple model Hamiltonians that are of practical interest to
experimentalists working towards the realization of a universal adiabatic
quantum computer. The model Hamiltonians presented are the simplest known
QMA-complete 2-local Hamiltonians. The 2-local Ising model with 1-local
transverse field which has been realized using an array of technologies, is
perhaps the simplest quantum spin model but is unlikely to be universal for
adiabatic quantum computation. We demonstrate that this model can be rendered
universal and QMA-complete by adding a tunable 2-local transverse XX coupling.
We also show the universality and QMA-completeness of spin models with only
1-local Z and X fields and 2-local ZX interactions.; Comment: Paper revised and extended to improve clarity; to appear in Physical
Review A

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## Finite-Dimensional PT-Symmetric Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/03/2003

Relevância na Pesquisa

26.98%

This paper investigates finite-dimensional representations of PT-symmetric
Hamiltonians. In doing so, it clarifies some of the claims made in earlier
papers on PT-symmetric quantum mechanics. In particular, it is shown here that
there are two ways to extend real symmetric Hamiltonians into the complex
domain: (i) The usual approach is to generalize such Hamiltonians to include
complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real
symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the
first approach the spectrum remains real, while in the second approach the
spectrum remains real if the PT symmetry is not broken. Both generalizations
give a consistent theory of quantum mechanics, but if D>2, a D-dimensional
Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional
PT-symmetric matrix Hamiltonian.; Comment: 8 pages, 1 figure

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## Local Hamiltonians with Approximation-Robust Entanglement

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.93%

Quantum entanglement is considered, by and large, to be a very delicate and
non-robust phenomenon that is very hard to maintain in the presence of noise,
or non-zero temperatures. In recent years however, and motivated, in part, by a
quest for a quantum analog of the PCP theorem researches have tried to
establish whether or not we can preserve quantum entanglement at "constant"
temperatures that are independent of system size. This would imply that any
quantum state with energy at most, say 0.05 of the total available energy of
the Hamiltonian, would be highly-entangled. To this date, no such systems were
found, and moreover, it became evident that even embedding local Hamiltonians
on robust, albeit "non-physical" topologies, namely expanders, does not
guarantee entanglement robustness. In this study, we indicate that such
robustness may be possible after all: We construct an infinite family of
O(1)-local Hamiltonians, corresponding to check terms of a quantum
error-correcting code with the following property of inapproximability: any
quantum state with energy at most 0.05 w.r.t. the total available energy cannot
be even approximately simulated by classical circuits of bounded
(sub-logarithmic) depth. In a sense, this implies that even providing a
"witness" to the fact that the local Hamiltonian can be "almost" satisfied...

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