Vol.5 No.1
January 01,
2005
Research and
Review Articles:
New encoding schemes for quantum
authentication (pp001012)
P. GarciaFernandez, E.
FernandezMartinez, E. Perez and D.J. Santos
We study the potential of general quantum operations,
TracePreserving CompletelyPositive Maps (TPCPs), as encoding and
decoding mechanisms in quantum authentication protocols. The study shows
that these general operations do not offer significant advantage over
unitary encodings. We also propose a practical authentication protocol
based on the use of two successive unitary encodings.
Qubit channels
which require four inputs to achieve capacity:
Implications for additivity conjectures (pp013031)
M. Hayashi, H. Imai, K. Matsumoto, M.B.
Ruskai and T. Shimono
An example is given of a qubit quantum channel which
requires four inputs to maximize the Holevo capacity. The example is one
of a family of channels which are related to 3state channels. The
capacity of the product channel is studied and numerical evidence
presented which strongly suggests additivity. The numerical evidence
also supports a conjecture about the concavity of output entropy as a
function of entanglement parameters. However, an example is presented
which shows that for some channels this conjecture does not hold for all
input states. A numerical algorithm for finding the capacity and optimal
inputs is presented and its relation to a relative entropy optimization
discussed.
Asymmetric phase
covariant ddimensional cloning (pp032039)
L.P.
Lamoureux and N.J. Cerf
We consider cloning transformations of ddimensional
states of the form $e^{i\phi_0}0> + e^{i\phi_1}1> +...+ e^{i\phi_{d1}}d1>$
that are covariant with respect to rotations of the phases $\phi_i$'s.
The optimal cloning maps are easily obtained using a welldefined
general characterization of statedependent $1 \rightarrow 2$ cloning
transformations in arbitrary dimensions. Our results apply to symmetric
as well as asymmetric cloners, so that the balance between the fidelity
of the two clones can be analyzed.
Some attacks on
quantumbased cryptographic protocols (pp040047)
HK Lo and TM Ko
Quantumbased cryptographic protocols are often said to
enjoy security guaranteed by the fundamental laws of physics. However,
even carefully designed quantumbased cryptographic schemes may be
susceptible to subtle attacks that are outside the original design. As
an example, we give attacks against a recently proposed ``secure
communication using mesoscopic coherent states'', which employs
mesoscopic states, rather than singlephoton states. Our attacks can be
used either as a knownplaintext attack or in the case where the
plaintext has not been randomized. One of our attacks requires
beamsplitters and the replacement of a lossy channel by a lossless one.
It is successful provided that the original loss in the channel is so
big that Eve can obtain $2^k$ copies of what Bob receives, where $k$ is
the length of the seed key preshared by Alice and Bob. In addition,
substantial improvements over such an exhaustive key search attack can
be made, whenever a key is reused. Furthermore, we remark that, under
the same assumption of a known or nonrandom plaintext, Grover's
exhaustive key search attack can be applied directly to "secure
communication using mesoscopic coherent states", whenever the channel
loss is more than 50 percent. Therefore, as far as informationtheoretic
security is concerned, optically amplified signals necessarily degrade
the security of the proposed scheme, when the plaintext is known or
nonrandom. Our attacks apply even if the mesoscopic scheme is used only
for key generation with a subsequent use of the key for onetimepad
encryption. Studying those attacks can help us to better define the risk
models and parameter spaces in which quantumbased cryptographic schemes
can operate securely. Finally, we remark that our attacks do not affect
standard protocols such as BennettBrassard BB84 protocol or Bennett B92
protocol, which rely on singlephoton signals.
Quantum circuits
for incompletely specified twoqubit operators (pp048056)
V.V. Shende and I.L. Markov
While the question ``how many CNOT gates are needed to
simulate an arbitrary twoqubit operator'' has been conclusively
answered  three are necessary and sufficient  previous work on this
topic assumes that one wants to simulate a given unitary operator up to
global phase. However, in many practical cases additional degrees of
freedom are allowed. For example, if the computation is to be followed
by a given projective measurement, many dissimilar operators achieve the
same output distributions on all input states. Alternatively, if it is
known that the input state is $\ket{0}$, the action of the given
operator on all orthogonal states is immaterial. In such cases, we say
that the unitary operator is incompletely specified; in this work, we
take up the practical challenge of satisfying a given specification with
the smallest possible circuit. In particular, we identify cases in which
such operators can be implemented using fewer quantum gates than are
required for generic completely specified operators.
Notes on
superoperator norms induced by Schatten norms (pp057067)
J. Watrous
Let $\Phi$ be a superoperator, i.e., a linear mapping of
the form $\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})$
for finite dimensional Hilbert spaces $\mathcal{F}$ and $\mathcal{G}$.
This paper considers basic properties of the superoperator norms
defined by $\\Phi\_{q\rightarrow p}= \sup\{\\Phi(X)\_p/\X\_q\,:\,X\not=0\}$,
induced by Schatten norms for $1\leq p,q\leq\infty$. These
superoperator norms arise in various contexts in the study of quantum
information. In this paper it is proved that if $\Phi$ is completely
positive, the value of the supremum in the definition of $\\Phi\_{q\rightarrow
p}$ is achieved by a positive semidefinite operator $X$, answering a
question recently posed by King and Ruskai~\cite{KingR04}. However, for
any choice of $p\in [1,\infty]$, there exists a superoperator $\Phi$
that is the {\em difference} of two completely positive,
tracepreserving superoperators such that all Hermitian $X$ fail to
achieve the supremum in the definition of $\\Phi\_{1\rightarrow p}$.
Also considered are the properties of the above norms for
superoperators tensored with the identity superoperator. In
particular, it is proved that for all $p\geq 2$, $q\leq 2$, and
arbitrary $\Phi$, the norm $\\Phi \_{q\rightarrow p}$ is stable under
tensoring $\Phi$ with the identity superoperator, meaning that $\\Phi
\_{q\rightarrow p} = \\Phi \otimes I\_{q\rightarrow p}$. For $1\leq p
< 2$, the norm $\\Phi\_{1\rightarrow p}$ may fail to be stable with
respect to tensoring $\Phi$ with the identity superoperator as just
described, but $\\Phi\otimes I\_{1\rightarrow p}$ is stable in this
sense for $I$ the identity superoperator on $\mathrm{L}(\mathcal{H})$
for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes
and simplifies a proof due to Kitaev \cite{Kitaev97} that established
this fact for the case $p=1$.
An information
theoretical model for quantum secret sharing schemes (pp068079)
H. Imai, J. MuellerQuade,
A.C. A. Nascimento, P. Tuyls and A. Winter
Similarly to earlier models for quantum error correcting
codes, we introduce a quantum information theoretical model for quantum
secret sharing schemes. This model provides new insights into the theory
of quantum secret sharing. By using our model, among other results, we
give a shorter proof of Gottesman's theorem that the size of the shares
in a quantum secret sharing scheme must be as large as the secret
itself. Also, we introduced approximate quantum secret sharing schemes
and showed robustness of quantum secret sharing schemes by extending
Gottesman's theorem to the approximate case.
Equiangular
spherical codes in quantum cryptography (pp080091)
J M Renes
Quantum key distribution protocols based on equiangular
spherical codes are introduced and their behavior under the
intercept/resend attack investigated. Such protocols offer a greater
range of secure noise tolerance and speed options than protocols based
on their cousins, the mutuallyunbiased bases, while also enabling the
determination of the channel noise rate without the need to sacrifice
key bits. For fixed number of signal states in a given dimension, the
spherical code protocols offer Alice and Bob more noise tolerance at the
price of slower key generation rates.
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