Vol.12
No.3&4, March 1, 2012
Research Articles:
Quantum McEliece
publickey cryptosystem
(pp01810203)
Hachiro
Fujita
The McEliece cryptosystem is one of the bestknown (classical)
publickey cryptosystems, which is based on algebraic coding theory. In
this paper, we present a quantum analogue of the classical McEliece
cryptosystem. Our quantum McEliece publickey cryptosystem is based on
the theory of stabilizer codes and has the key generation, encryption
and decryption algorithms similar to those in
the classical McEliece cryptosystem. We present an explicit construction
of the quantum McEliece publickey cryptosystem using
CalderbankShorSteane codes based on generalized ReedSolomon codes. We
examine the security of our quantum McEliece cryptosystem and compare it
with alternative systems.
Improved Data
PostProcessing in Quantum Key Distribution and Application to Loss
Thresholds in device independent QKD
(pp02030214)
Xiongfeng Ma
and Norbert Lutkenhaus
Security proofs of quantum key distribution (QKD) often require
postprocessing schemes to simplify the data structure, and hence the
security proof. We show a generic method to improve resulting secure key
rates by partially reversing the simplifying postprocessing for error
correction purposes. We apply our method to the security analysis of
deviceindependent QKD schemes and of detectiondeviceindependent QKD
schemes, where in both cases one is typically required to assign binary
values even to lost signals. In the deviceindependent case, the loss
tolerance threshold is cut down by our method from 92.4% to 90.9%. The
lowest tolerable transmittance of the detectiondeviceindependent
scheme can be improved from 78.0% to 65.9%
Onestep
implementation of the Fredkin gate via quantum Zeno dynamics
(pp02150230)
ZhiCheng
Shi, Yan Xia, Jie Song, and HeShan Song
We study onestep implementation of the Fredkin gate in a bimodal
cavity under both resonant and large detuning conditions based on
quantum Zeno dynamics, which reduces the complexity of experiment
operations. The influence of cavity decay and atomic spontaneous
emission is discussed by numerical calculation. The results demonstrate
that the fidelity and the success probability are robust against cavity
decay in both models and they are also insensitive to atomic spontaneous
emission in the large detuning model. In addition, the interaction time
is rather short in the resonant model compared to the large detuning
model.
Steadystate
correlations of two atoms interacting with a reservoir
(pp02310252)
Luis Octavio
Castanos
We consider two twolevel atoms fixed at different positions, driven by
a resonant monochromatic laser field, and interacting collectively with
the quantum electromagnetic field. A BornMarkovsecular master equation
is used to describe the dynamics of the two atoms and the steadystate
is obtained analytically for a configuration of the atoms. The
steadystate populations of the energy levels of the free atoms,
entanglement, quantum and geometric discords and degree of mixedness are
calculated analytically as a function of the laser field intensity and
the distance between the two atoms. It is found that there is a
possibility of considerable steadystate entanglement and left/right
quantum discord and that these can be controlled either by
increasing/decreasing the intensity of the laser field or by
increasing/decreasing the distance between atoms. It is shown that the
system of two atoms can be prepared in a separable mixed state with
nonzero quantum discord that turns into an $X$state for high laser
field intensities. The behavior and relationships between the different
correlations are studied and several limiting cases are investigated.
Probabilistic
secret sharing through noise quantum channel
(pp02530261)
Satyabrata
Adhikari, Indranil Chakrabarty, and Pankaj Agrawal
In a realistic situation, the secret sharing of classical or quantum
information will involve the transmission of this information through
noisy channels. We consider a three qubit pure state. This state becomes
a mixedstate when the qubits are distributed over noisy channels. We
focus on a specific noisy channel, the phasedamping channel. We propose
a protocol for secret sharing of classical information with this and
related noisy channels. This protocol can also be thought of as
cooperative superdense coding. We also discuss other noisy channels to
examine the possibility of secret sharing of classical information.
Decomposition of
orthogonal matrix and synthesis of twoqubit and threequbit orthogonal
gates
(pp02620270)
HaiRui Wei
and YaoMin Di
The decomposition of matrices associated to twoqubit and threequbit
orthogonal gates is studied, and based on the decomposition the
synthesis of these gates is investigated. The optimal synthesis of
general twoqubit orthogonal gate is obtained. For twoqubit unimodular
orthogonal gate, it requires at most 2 CNOT gates and 6 onequbit R_y
gates. For the general threequbit unimodular orthogonal gate, it
can be synthesized by 16 CNOT gates and 36 onequbit R_y and
R_z gates in the worst case.
Locally
unextendible nonmaximally entangled basis
(pp02710282)
Indranil
Chakrabarty, Pankaj Agrawal, and Arun K. Pati
We introduce the concept of the locally unextendible nonmaximally
entangled basis (LUNMEB) in H^d \bigotimes H^d. It is shown that
such a basis consists of d orthogonal vectors for a nonmaximally
entangled state. However, there can be a maximum of (d1)^2
orthogonal vectors for nonmaximally entangled state if it is maximally
entangled in (d1) dimensional subspace. Such a basis plays an
important role in determining the number of classical bits that one can
send in a superdense coding protocol using a nonmaximally entangled
state as a resource. By constructing appropriate POVM operators, we find
that the number of classical bits one can transmit using a nonmaximally
entangled state as a resource is (1+p_0\frac{d}{d1})\log d,
where p_0 is the smallest Schmidt coefficient. However, when the
state is maximally entangled in its subspace then one can send up to
2\log (d1) bits. We also find that for d= 3, former may be
more suitable for the superdense coding.
An exact tensor network for the
3SAT problem
(pp02830292)
Artur GarciaSaez
and Jose I. Latorre
We construct a tensor network that delivers an unnormalized quantum
state whose coefficients are the solutions to a given instance of 3SAT,
an NPcomplete problem. The tensor network contraction that corresponds
to the norm of the state counts the number of solutions to the instance.
It follows that exact contractions of this tensor network are in the
\#Pcomplete computational complexity class, thus believed to be a hard
task. Furthermore, we show that for a 3SAT instance with $n$ bits, it is
enough to perform a polynomial number of contractions of the tensor
network structure associated to the computation of local observables to
obtain one of the explicit solutions to the problem, if any. Physical
realization of a state described by a generic tensor network is
equivalent to finding the satisfying assignment of a 3SAT instance and,
consequently, this experimental task is expected to be hard.
Perfect state
transfer on quotient graphs
(pp02930313)
Rachel
Bachman, Eric Fredette, Jessica Fuller, Michael Landry, Michael Opperman,
Christino Tamon, and Andrew Tollefson
We prove new results on perfect state transfer of quantum walks on
quotient graphs. Since a graph G has perfect state transfer if
and only if its quotient G/\pi, under any equitable partition
\pi, has perfect state transfer, we exhibit graphs with perfect
state transfer between two vertices but which lack automorphism swapping
them. This answers a question of Godsil (Discrete Mathematics
312(1):129147, 2011). We also show that the Cartesian product of
quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the
quotient graph \Box_{k} G_{k}/\pi, for some equitable partition
\pi. This provides an algebraic description of a construction due
to Feder (Physical Review Letters 97, 180502, 2006) which
is based on manyboson quantum walk.
Limit theorems
for the discretetime quantum walk on a graph with joined half lines
(pp03140333)
Kota Chisaki,
Norio Konno, and Etsuo Segawa
We consider a discretetime quantum walk W_{t,\kappa} at time
t on a graph with joined half lines J_\kappa, which is
composed of \kappa half lines with the same origin. Our analysis
is based on a reduction of the walk on a half line. The idea plays an
important role to analyze the walks on some class of graphs with
symmetric initial states. In this paper, we introduce a quantum walk
with an enlarged basis and show that W_{t,\kappa} can be reduced
to the walk on a half line even if the initial state is asymmetric.
For W_{t,\kappa}, we obtain two types of limit theorems. The
first one is an asymptotic behavior of W_{t,\kappa} which
corresponds to localization. For some conditions, we find that the
asymptotic behavior oscillates. The second one is the weak convergence
theorem for W_{t,\kappa}. On each half line, W_{t,\kappa}
converges to a density function like the case of the onedimensional
lattice with a scaling order of t. The results contain the cases
of quantum walks starting from the general initial state on a half line
with the general coin and homogeneous trees with the Grover coin.
Realization of an economical phasecovariant
telecloning in separate cavities
(pp03340345)
BaoLong Fang, Tao Wu, and Liu Ye
We present a scheme to realize an economical $1 \to 2$ phasecovariant
telecloning in separate cavities. In the scheme, an entangled state
between a photon pulse and the tapped atoms will be prepared through
cavityassisted interaction. Next, the Bell state measurement is execute
on the photon pulse and a trapped atom or another photon pulse on which
the quantum information is encoded. In this way, the system can provide
symmetric (asymmetric) economical $1 \to 2$ phasecovariant telecloning.
On the geometry of tensor network states
(pp03460354)
Joseph M. Landsburg, Yang
Qi, and Ke Ye
We answer a question of L. Grasedyck that arose in quantum information
theory, showing that the limit of tensors in a space of tensor network
states need not be a tensor network state. We also give geometric
descriptions of spaces of tensor networks states corresponding to trees
and loops. Grasedyck's question has a surprising connection to the area
of Geometric Complexity Theory, in that the result is equivalent to the
statement that the boundary of the MulmuleySohoni type variety
associated to matrix multiplication is strictly larger than the
projections of matrix multiplication (and reexpressions of matrix
multiplication and its projections after changes of bases). Tensor
Network States are also related to graphical models in algebraic
statistics.
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