Vol.11
No.1&2 January 1, 2011
Research Articles:
Macroscopic
multispecies entanglement near quantum phase transitions
(pp00010007)
V.
Subrahmanyam
MultiSpecies entanglement, defined for a manyparticle system as the
entanglement between different species of particles, is shown to exist
in the thermodynamic limit of the system size going to infinity. This
macroscopic entanglement, as it can exhibit singular behavior, is
capable of tracking quantum phase transitions. The entanglement between
up and down spins has been analytically calculated for the
onedimensional Ising model in a transverse magnetic field. As the
coupling strength is varied, the first derivative of the entanglement
shows a jump discontinuity and the second derivative diverges near the
quantum critical point.
Surface code
quantum error correction incorporating accurate error propagation
(pp00080018)
Austin G. Fowler,
David S. Wang, and Lloyd C. L. Hollenberg
The surface code is a powerful quantum error correcting code that can be
defined on a 2D square lattice of qubits with only nearest neighbor
interactions. Syndrome and data qubits form a checkerboard pattern.
Information about errors is obtained by repeatedly measuring each
syndrome qubit after appropriate interaction with its four nearest
neighbor data qubits. Changes in the measurement value indicate the
presence of chains of errors in space and time. The standard method of
determining operations likely to return the code to its errorfree state
is to use the minimum weight matching algorithm to connect pairs of
measurement changes with chains of corrections such that the minimum
total number of corrections is used. Prior work has not taken into
account the propagation of errors in space and time by the twoqubit
interactions. We show that taking this into account leads to a quadratic
improvement of the logical error rate.
Characterization
of universal twoqubit Hamiltonians
(pp00190039)
Andrew
M. Childs, Debbie Leung, Laura Mancinska, and Maris Ozols
Suppose we can apply a given 2qubit Hamiltonian H to any
(ordered) pair of qubits. We say H is nuniversal if it
can be used to approximate any unitary operation on n qubits.
While it is well known that almost any 2qubit Hamiltonian is
2universal, an explicit characterization of the set of nonuniversal
2qubit Hamiltonians has been elusive. Our main result is a complete
characterization of 2nonuniversal 2qubit Hamiltonians. In particular,
there are three ways that a 2qubit Hamiltonian $H$ can fail to be
universal:
(1) H shares an eigenvector with the gate that swaps two qubits,
(2) H acts on the two qubits independently (in any of a certain
family of bases), or
(3) H has zero trace
(with the third condition relevant only when the global phase of the
unitary matters).
A 2nonuniversal 2qubit Hamiltonian can still be nuniversal
for some n \geq 3. We give some partial results on
3universality.
Nonlocal box
complexity and secure function evaluation
(pp00400069)
Marc Kaplan,
Sophie Laplante, Iordanis Kerenidis, and J\'er\'emie Roland
A nonlocal box is an abstract device into which Alice and Bob input
bits $x$ and $y$ respectively and receive outputs $a$ and $b$, where $a,b$
are uniformly distributed and $a \oplus b = x \wedge y$. Such boxes have
been central to the study of quantum or generalized nonlocality, as
well as the simulation of nonsignaling distributions. In this paper, we
start by studying how many nonlocal boxes Alice and Bob need in order
to compute a Boolean function $f$. We provide tight upper and lower
bounds in terms of the communication complexity of the function both in
the deterministic and randomized case. We show that nonlocal box
complexity has interesting applications to classical cryptography, in
particular to secure function evaluation, and study the question posed
by Beimel and Malkin \cite{BM} of how many Oblivious Transfer calls
Alice and Bob need in order to securely compute a function $f$. We show
that this question is related to the nonlocal box complexity of the
function and conclude by greatly improving their bounds. Finally,
another consequence of our results is that traceless twooutcome
measurements on maximally entangled states can be simulated with 3 \nlbs,
while no finite bound was previously known.
A new
entanglement measure: Dconcurrence
(pp00700078)
Zhihao Ma, Weigang
Yuan, Minli Bao, and XiaoDong Zhang
A new entanglement measure, which is called Dconcurrence, is proposed.
Then the upper bound for Dconcurrence is obtained. In addition,
Dconcurrence has some special merits, such as it is an entanglement
monotone, and is subadditive.
Measurable lower
bounds on concurrence (pp00790094)
Iman Sargolzahi,
Sayyed Yahya Mirafzali, and Mohsen Sarbishaei
We derive measurable lower bounds on concurrence of arbitrary mixed
states, for both bipartite and multipartite cases. First, we construct
measurable lower bonds on the \textit{purely algebraic} bounds of
concurrence [F. Mintert \textit{et al.} (2004), Phys. Rev. lett., 92,
167902]. Then, using the fact that the sum of the square of the
algebraic bounds is a lower bound of the squared concurrence, we sum
over our measurable bounds to achieve a measurable lower bound on
concurrence. With two typical examples, we show that our method can
detect more entangled states and also can give sharper lower bonds than
the similar ones.
Quantum
interpolation of polynomials (pp00950103)
Daniel M. Kane and
Samuel A. Kutin
Can a quantum computer efficiently interpolate polynomials? We consider
blackbox algorithms that seek to learn information about a polynomial
$f$ from input/output pairs $(x_i, f(x_i))$. We define a more general
class of \emph{$(d,S)$independent} function properties, where, outside
of a set $S$ of exceptions, knowing $d$ input values does not help one
predict the answer. There are essentially two strategies to computing
such a function: query $d+1$ random input values, or search for one of
the $S$ exceptions. We show that, up to constant factors, we cannot
beat these two approaches.
A family of norms
with applications in quantum information theory II
(pp01040123)
Nathaniel Johnston
and David W. Kribs
We consider the problem of computing the family of operator norms
recently introduced. We develop a family of semidefinite programs that
can be used to exactly compute them in small dimensions and bound them
in general. Some theoretical consequences follow from the duality theory
of semidefinite programming, including a new constructive proof that for
all r there are nonpositive partial transpose Werner states that
are rundistillable. Several examples are considered via a MATLAB
implementation of the semidefinite program, including the case of Werner
states and randomly generated states via the Bures measure, and
approximate distributions of the norms are provided. We extend these
norms to arbitrary convex mapping cones and explore their implications
with positive partial transpose states.
Generation of
clustertype entangled coherent states using weak nonlinearities
(pp01240141)
Nguyen B. An,
Kisik Kim, and Jaewan Kim
We propose a scheme to generate a recently introduced type of entangled
coherent states using realistic weak crossKerr nonlinearities and
intense laser beams. An intense laser can be filtered to make a faint
one to be used for production of a single photon which is necessary in
our scheme. The optical devices used are conventional ones such as
interferometer, mirrors, beamsplitters, phaseshifters and
photodetectors. We also provide a detailed analysis on the effects of
possible imperfections and decoherence showing that our scheme is robust
against such effects.
An efficient
conversion of quantum circuits to a linear nearest neighbor architecture
(pp01420166)
Yuichi Hirata,
Masaki Nakanishi, Shigeru Yamashita and Yasuhiko Nakashima
Several promising implementations of quantum computation rely on a
Linear Nearest Neighbor (LNN) architecture, which arranges quantum bits
on a line, and allows neighbor interactions only. Therefore, several
specific circuits have been designed on an LNN architecture. However, a
general and efficient conversion method for an arbitrary circuit has not
been established. Therefore, this paper gives an efficient conversion
technique to convert quantum circuits to an LNN architecture. When a
quantum circuit is converted to an LNN architecture, the objective is to
reduce the size of the additional circuit added by the conversion and
the time complexity of the conversion. The proposed method requires less
additional circuitry and time complexity compared with naive techniques.
To develop the method, we introduce two key theorems that may be
interesting on their own. In addition, the proposed method also achieves
less overhead than some known circuits designed from scratch on an LNN
architecture.
Mathematical
framework for detection and quantification of nonclassical correlation
(pp01670180)
Akira SaiToh,
Robabeh Rahimi, and Mikio Nakahara
Existing measures of bipartite nonclassical correlation that is
typically characterized by nonvanishing nonlocalizable information under
the zeroway CLOCC protocol are expensive in computational cost. We
define and evaluate economical measures on the basis of a new class of
maps, eigenvaluepreservingbutnotcompletelyeigenvaluepreserving
(EnCE) maps. The class is in analogy to the class of
positivebutnotcompletelypositive (PnCP) maps that have been commonly
used in the entanglement theories. Linear and nonlinear EnCE maps are
investigated. We also prove subadditivity of the measures in the form of
logarithmic fidelity.
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