Vol.13
No.7&8, July 1, 2013
Research Articles:
Multiaccess
quantum communication and product higher rank numerical range
(pp05410566)
Maciej
Demianowicz, Pawel Horodecki, and Karol Zyczkowski
In the present paper we initiate the study of the product higher
rank numerical range. The latter, being a variant of the higher rank
numerical range [M.D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77
(2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for
studying a construction of quantum error correction codes for multiple
access channels. We review properties of this set and relate it to other
numerical ranges, which were recently introduced in the literature.
Further, the concept is applied to the construction of codes for
biunitary twoaccess channels with a hermitian noise model.
Analytical techniques for both outerbounding the product higher rank
numerical range and determining its exact shape are developed for this
case. Finally, the reverse problem of constructing a noise model for a
given product range is considered.
Nobroadcasting
of nonsignalling boxes via operations which transform local boxes into
local ones (pp05670582)
P. Joshi,
Andrzej Grudka, Karol Horodecki, Michal Horodecki, Pawel Horodecki, and
Ryszard Horodecki
We deal with families of probability distributions satisfying nonsignalling
condition, called nonsignalling boxes and consider a class of
operations that transform local boxes into local ones (the one that
admit LHV model). We prove that any operation from this class cannot
broadcast a bipartite nonlocal box with 2 binary inputs and outputs. We
consider a function called antiRobustness which can not decrease under
these operations. The proof reduces to showing that antiRobustness
would decrease after broadcasting.
Quantum
fingerprints that keep secrets (pp05830606)
Dmitry
Gavinsky and Tsuyoshi Ito
We introduce a new type of cryptographic primitive that we call a \e{hiding
fingerprinting scheme}.
A (quantum) fingerprinting scheme maps a binary string of length $n$ to
$d$ (qu)bits, typically $d\ll n$, such that given any string $y$ and a
fingerprint of $x$, one can decide with high accuracy whether $x=y$. It
can be seen that a classical fingerprint of $x$ that guarantees error
$\le\eps$ necessarily reveals \asOm{\Min{n,\log(1/\eps)}} bits of
information about $x$. We call a scheme \e{hiding} if it reveals \aso{\Min{n,\log(1/\eps)}}
bits; accordingly, no classical scheme is hiding. }{We construct quantum
hiding fingerprinting schemes. Our schemes are computationally efficient
and their hiding properties are shown to be optimal.
Fast and
efficient exact synthesis of singlequbit unitaries generated by
Clifford and T gates (pp06070630)
Vadym
Kliuchnikov, Dmitri Maslov, and Michele Mosca
In this paper, we show the equivalence of the set of unitaries
computable by the circuits over the Clifford and T library and the set
of unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$, in the
singlequbit case. We report an efficient synthesis algorithm, with an
exact optimality guarantee on the number of Hadamard gates used. We
conjecture that the equivalence of the sets of unitaries implementable
by circuits over the Clifford and T library and unitaries over the ring
$\mathbb{Z}[\frac{1}{\sqrt{2}},i]$ holds in the $n$qubit case.
Efficient quantum
circuits for binary elliptic curve arithmetic: reducing $T$gate
complexity (pp06310644)
Brittanney
Amento, Martin Rotteler, and Rainer Steinwandt
Elliptic curves over finite fields ${\mathbb F}_{2^n}$ play a
prominent role in modern cryptography. Published quantum algorithms
dealing with such curves build on a short Weierstrass form in
combination with affine or projective coordinates. In this paper we show
that changing the curve representation allows a substantial reduction in
the number of $T$gates needed to implement the curve arithmetic. As a
tool, we present a quantum circuit for computing multiplicative inverses
in $\mathbb F_{2^n}$ in depth $\bigO(n\log_2 n)$ using a polynomial
basis representation, which may be of independent interest.
Most robust and
fragile twoqubit entangled states under deploarizing channels (pp06450660)
ChaoQian
Pang, FuLin Zhang, Yue Jiang, MaiLin Liang, and JingLing Chen
For a twoqubit system under local depolarizing channels, the most
robust and most fragile states are derived for a given concurrence or
negativity. For the onesided channel, the pure states are proved to be
the most robust ones, with the aid of the evolution equation for
entanglement given by Konrad \emph{et al.} [Nat. Phys. 4, 99 (2008)].
Based on a generalization of the evolution equation for entanglement, we
classify the ansatz states in our investigation by the amount of
robustness, and consequently derive the most fragile states. For the
twosided channel, the pure states are the most robust for a fixed
concurrence. Under the uniform channel, the most fragile states have the
minimal negativity when the concurrence is given in the region
$[1/2,1]$. For a given negativity, the most robust states are the ones
with the maximal concurrence, and the most fragile ones are the pure
states with minimum of concurrence. When the entanglement approaches
zero, the most fragile states under general nonuniform channels tend to
the ones in the uniform channel. Influences on robustness by
entanglement, degree of mixture, and asymmetry between the two qubits
are discussed through numerical calculations. It turns out that
the concurrence and negativity are major factors for the robustness.
When they are fixed, the impact of the mixedness becomes obvious. In the
nonuniform channels, the most fragile states are closely correlated with
the asymmetry, while the most robust ones with the degree of mixture.
Limit theorems
for the interference terms of discretetime quantum walks on the line (pp06610671)
Takuya Machida
The probability distributions of discretetime quantum walks have
been often investigated, and many interesting properties of them have
been discovered. The probability that the walker can be find at a
position is defined by diagonal elements of the density matrix. On the
other hand, although offdiagonal parts of the density matrices have an
important role to quantify quantumness, they have not received attention
in quantum walks. We focus on the offdiagonal parts of the density
matrices for discretetime quantum walks on the line and derive limit
theorems for them.
Galois automorphisms of a
symmetric measurement (pp06720720)
D.M. Appleby,
Hulya YadsanAppleby, and Gerhard Zauner
Symmetric Informationally Complete Positive Operator Valued Measures
(usually referred to as SICPOVMs or simply as SICs) have been
constructed in every dimension $\le 67$. However, a proof that they
exist in every finite dimension has yet to be constructed. In this paper
we examine the Galois group of SICs covariant with respect to the WeylHeisenberg
group (or WH SICs as we refer to them). The great majority (though not
all) of the known examples are of this type. Scott and Grassl have noted
that every known exact WH SIC is expressible in radicals (except for
dimension $3$ which is exceptional in this and several other respects),
which means that the corresponding Galois group is solvable. They have
also calculated the Galois group for most known exact examples. The
purpose of this paper is to take the analysis of Scott and Grassl
further. We first prove a number of theorems regarding the structure of
the Galois group and the relation between it and the extended Clifford
group. We then examine the Galois group for the known exact fiducials
and on the basis of this we propose a list of nine conjectures
concerning its structure. These conjectures represent a considerable
strengthening of the theorems we have actually been able to prove.
Finally we generalize the concept of an antiunitary to the concept of a
$g$unitary, and show that every WH SIC fiducial is an eigenvector of a
family of $g$unitaries (apart from dimension 3).
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