Vol.12
No.9&10,
September 1, 2012
Research Articles:
Criteria for measures of quantum correlations
(pp07210742)
Aharon Brodutch and Kavan Modi
Entanglement does not describe all quantum correlations and several
authors have shown the need to go beyond entanglement when dealing with
mixed states. Various different measures have sprung up in the
literature, for a variety of reasons, to describe bipartite and
multipartite quantum correlations; some are known under the collective
name {\it quantum discord}. Yet, in the same sprit as the criteria for
entanglement measures, there is no general mechanism that determines
whether a measure of quantum and classical correlations is a proper
measure of correlations. This is partially due to the fact that the
answer is a bit muddy. In this article we attempt tackle this muddy
topic by writing down several criteria for a ``good" measure of
correlations. We breakup our list into \emph{necessary}, \emph{reasonable},
and \emph{debatable} conditions. We then proceed to prove several of
these conditions for generalized measures of quantum correlations.
However, not all conditions are met by all measures; we show this via
several examples. The reasonable conditions are related to continuity of
correlations, which has not been previously discussed. Continuity is an
important quality if one wants to probe quantum correlations in the
laboratory. We show that most types of quantum discord are continuous
but none are continuous with respect to the measurement basis used for
optimization.
Numberphase uncertainty relations in terms of generalized entropies
(pp07430762)
Alexey E. Rastegin
Numberphase uncertainty relations are formulated in terms of unified
entropies which form a family of twoparametric extensions of the
Shannon entropy. For two generalized measurements, unifiedentropy
uncertainty relations are given in both the statedependent and
stateindependent forms. A few examples are discussed as well. Using the
PeggBarnett formalism and the Riesz theorem, we obtain a nontrivial
inequality between normlike functionals of generated
probability distributions in finite dimensions. The principal point is
that we take the infinitedimensional limit right for this inequality.
Hence numberphase uncertainty relations with finite phase resolutions
are expressed in terms of the unified entropies. Especially important
case of multiphoton coherent states is separately considered. We also
give some entropic bounds in which the corresponding integrals of
probability density functions are involved.
Application of indirect Hamiltonian tomography to complex systems
with short coherence times
(pp07630774)
Koji Maruyama, Daniel Burgarth, Akihito Ishizaki, Takeji Takui, and K. Birgitta Whaley
The identification of parameters in the Hamiltonian that describes
complex manybody quantum systems is generally a very hard task. Recent
attention has focused on such problems of Hamiltonian tomography for
networks constructed with twolevel systems. For open quantum systems,
the fact that injected signals are likely to decay before they
accumulate sufficient information for parameter estimation poses
additional challenges. In this paper, we consider use of the gateway
approach to Hamiltonian tomography \cite{Burgarth2009,Burgarth2009a} to
complex quantum systems with a limited set of state preparation and
measurement probes. We classify graph properties of networks for which
the Hamiltonian may be estimated under equivalent conditions on state
preparation and measurement. We then examine the extent to which the
gateway approach may be applied to estimation of Hamiltonian parameters
for network graphs with nontrivial topologies mimicking biomolecular
systems.
Local solutions of maximum likelihood estimation in quantum state tomography
(pp07750790)
Douglas S. Goncalves, Marcia A. GomesRuggiero, Carlile Lavor, Osvaldo
J. Farias, and P. H. Souto Ribeiro
Maximum likelihood estimation is one of the most used methods in quantum
state tomography, where the aim is to reconstruct the density matrix of
a physical system from measurement results. One strategy to deal with
positivity and unit trace constraints is to parameterize the matrix to
be reconstructed in order to ensure that it is physical. In this case,
the negative loglikelihood function in terms of the parameters, may
have several local minima. In various papers in the field, a source of
errors in this process has been associated to the possibility that most
of these local minima are not global, so that optimization methods could
be trapped in the wrong minimum, leading to a wrong density matrix. Here
we show that, for convex negative loglikelihood functions, all local
minima of the unconstrained parameterized problem are global, thus any
minimizer leads to the maximum likelihood estimation for the density
matrix. We also discuss some practical sources of errors.
Blind reconciliation
(pp07910812)
Jesus MartinezMateo, David Elkouss, and Vicente Martin
Information reconciliation is a crucial procedure in the classical
postprocessing of quantum key distribution (QKD). Poor reconciliation
efficiency, revealing more information than strictly needed, may
compromise the maximum attainable distance, while poor performance of
the algorithm limits the practical throughput in a QKD device.
Historically, reconciliation has been mainly done using close to minimal
information disclosure but heavily interactive procedures, like \textit{Cascade},
or using less efficient but also less interactive just one message is
exchanged procedures, like the ones based in lowdensity paritycheck
(LDPC) codes. The price to pay in the LDPC case is that good efficiency
is only attained for very long codes and in a very narrow range centered
around the quantum bit error rate (QBER) that the code was designed to
reconcile, thus forcing to have several codes if a broad range of QBER
needs to be catered for. Real world implementations of these methods are
thus very demanding, either on computational or communication resources
or both, to the extent that the last generation of GHz clocked QKD
systems are finding a bottleneck in the classical part. In order to
produce compact, high performance and reliable QKD systems it would be
highly desirable to remove these problems. Here we analyse the use of
shortlength LDPC codes in the information reconciliation context using
a low interactivity, \textit{blind}, protocol that avoids an a priori
error rate estimation. We demonstrate that $2 \times 10^3$ bits length
LDPC codes are suitable for blind reconciliation. Such codes are of high
interest in practice, since they can be used for hardware
implementations with very high throughput.
Quantum codes from codes over Gaussian integers with respect to the Mannheim metric
(pp08130819)
Mehmet Ozen and Murat Guzeltepe
In this paper, some nonbinary quantum codes using classical codes over
Gaussian integers are obtained. Also, some of our quantum codes are
better than or comparable with those known before, (for instance
$[[8,2,5]]_{4+i}$).
On nonbinary quantum convolutional BCH codes
(pp08200842)
Giuliano G. La Guardia
Several new families of nonbinary quantum convolutional
BoseChaudhuriHocquenghem (BCH) codes are constructed in this paper.
These code constructions are performed algebraically and not by
computation search. The quantum convolutional codes constructed here
have parameters better than the ones available in the literature and
they have noncatastrophic encoders and encoder inverses. These new
families consist of unitmemory as well as multimemory convolutional
stabilizer codes.
Quantum algorithms for invariants of triangulated manifolds
(pp08430863)
Gorjan Alagic and Edgar A. Bering IV
One of the apparent advantages of quantum computers over their classical
counterparts is their ability to efficiently contract tensor networks.
In this article, we study some implications of this fact in the case of
topological tensor networks. The graph underlying these networks is
given by the triangulation of a manifold, and the structure of the
tensors ensures that the overall tensor is independent of the choice of
internal triangulation. This leads to quantum algorithms for additively
approximating certain invariants of triangulated manifolds. We discuss
the details of this construction in two specific cases. In the first
case, we consider triangulated surfaces, where the triangle tensor is
defined by the multiplication operator of a finite group; the resulting
invariant has a simple closedform expression involving the dimensions
of the irreducible representations of the group and the Euler
characteristic of the surface. In the second case, we consider
triangulated 3manifolds, where the tetrahedral tensor is defined by the
socalled Fibonacci anyon model; the resulting invariant is the
wellknown TuraevViro invariant of 3manifolds.
Quantum phase estimation with arbitrary constantprecision phase shift operators
(pp08640875)
Hamed Ahmadi and ChenFu Chiang
While Quantum phase estimation (QPE) is at the core of many
quantum algorithms known to date, its physical implementation
(algorithms based on quantum Fourier transform (QFT) ) is highly
constrained by the requirement of highprecision controlled phase shift
operators, which remain difficult to realize. In this paper, we
introduce an alternative approach to approximately implement QPE with
arbitrary constantprecision controlled phase shift operators. The new
quantum algorithm bridges the gap between QPE algorithms based on QFT
and Kitaev's original approach. For approximating the eigenphase precise
to the nth bit, Kitaev's original approach does not require any
controlled phase shift operator. In contrast, QPE algorithms based on
QFT or approximate QFT require controlled phase shift operators with
precision of at least Pi/2n. The new approach fills the gap and requires
only arbitrary constantprecision controlled phase shift operators. From
a physical implementation viewpoint, the new algorithm outperforms
Kitaev's approach.
Systematic
distillation of composite Fibonacci anyons using one mobile
quasiparticle (pp08760892)
Ben W.
Reichardt
A topological quantum computer should allow intrinsically faulttolerant
quantum computation, but there remains uncertainty about how such a
computer can be implemented. It is known that topological quantum
computation can be implemented with limited quasiparticle braiding
capabilities, in fact using only a single mobile quasiparticle, if the
system can be properly initialized by measurements. It is also known
that measurements alone suffice without any braiding, provided that the
measurement devices can be dynamically created and modified. We study a
model in which both measurement and braiding capabilities are limited.
Given the ability to pull nontrivial Fibonacci anyon pairs from the
vacuum with a certain success probability, we show how to simulate
universal quantum computation by braiding one quasiparticle and with
only one measurement, to read out the result. The difficulty lies in
initializing the system. We give a systematic construction of a family
of braid sequences that initialize to arbitrary accuracy nontrivial
composite anyons. Instead of using the SolovayKitaev theorem, the
sequences are based on a quantum algorithm for convergent search.
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