Vol.4 No.5
September 08,
2004
Research and
Review Articles: Security
of quantum key distribution with imperfect devices (pp325360)
D. Gottesman, H.K. Lo, N.
L\"utkenhaus, and J. Preskill
We prove the security of the BennettBrassard (BB84) quantum key
distribution protocol in the case where the source and detector are
under the limited control of an adversary. Our proof applies when both
the source and the detector have small basisdependent flaws, as is
typical in practical implementations of the protocol. We derive a
general lower bound on the asymptotic key generation rate for weakly
basisdependent eavesdropping attacks, and also estimate the rate in
some special cases: sources that emit weak coherent states with random
phases, detectors with basisdependent efficiency, and misaligned
sources and detectors.
Optimal
realizations of simplified Toffoli gates (pp361372)
G. Song and A. Klappenecker
A simplified Toffoli gate coincides with the Toffoli gate except that
the result is allowed to differ on one computational basis state by a
phase factor. We prove that the simplified Toffoli gate implementation
by Margolus is optimal, in the sense that it attains a lower bound of {\em
three} controllednot gates, and subject to that, a sharp lower bound of
{\em four} singlequbit gates. We also discuss optimal implementations
of other simplified Toffoli gates, and explain why the phase factor $1$
invariably occurs in such implementations.
Erasure
versus teleportation scheme of optical CNOT gate (pp373382)
A. Wójcik and A. Grudka
We clarify the connections between the erasure scheme of probabilistic
CNOT gate implementation recently proposed by Pittman, Jacobs and
Franson [Phys. Rev. A 64, 062311 (2001)] and quantum teleportation.
Local invariants for
multipartite entangled states allowing for a simple entanglement
criterion
(pp383395)
H. Aschauer, J. Calsamiglia,
M. Hein, and H.~J. Briegel
We present local invariants of multipartite pure or
mixed states, which can be easily calculated and have a straightforward
physical meaning. As an application, we derive a new entanglement
criterion for arbitrary mixed states of $n$ parties. The new criterion
is weaker than the partial transposition criterion but offers advantages
for the study of multipartite systems. A straightforward generalization
of these invariants allows for the construction of a complete set of
observable polynomial invariants.
Note on the Khaneja
Glaser decomposition (pp396400)
S.S. Bullock
Recently, Vatan and Williams utilize a matrix
decomposition of $SU(2^n)$ introduced by Khaneja and Glaser to produce
{\tt CNOT}efficient circuits for arbitrary threequbit unitary
evolutions. In this note, we place the Khaneja Glaser Decomposition ({\tt
KGD}) in context as a $SU(2^n)=KAK$ decomposition by proving that its
Cartan involution is type {\bf AIII}, given $n \geq 3$. The standard
type {\bf AIII} involution produces the CosineSine Decomposition (CSD),
a wellknown decomposition in numerical linear algebra which may be
computed using mature, stable algorithms. In the course of our proof
that the new decomposition is type {\bf AIII}, we further establish the
following. Khaneja and Glaser allow for a particular degree of freedom,
namely the choice of a commutative algebra $\mathfrak{a}$, in their
construction. Let $\chi_1^n$ be a {\tt SWAP} gate applied on qubits $1$,
$n$. Then $\chi_1^n v \chi_1^n=k_1\; a \; k_2$ is a KGD for $\mathfrak{a}=\mbox{span}_{\mathbb{R}}
\{ \chi_1^n ( \ket{j}\bra{Nj1} \ket{Nj1}\bra{j}) \chi_1^n \}$ if
and only if $v=(\chi_1^n k_1 \chi_1^n) (\chi_1^n a \chi_1^n)(\chi_1^n
k_2 \chi_1^n)$ is a CSD.
Measuring polynomial functions of states (pp401408)
T.A. Brun
In this paper I show that any $m$thdegree polynomial
function of the elements of the density matrix $\rho$ can be determined
by finding the expectation value of an observable on $m$ copies of $\rho$,
without performing state tomography. Since a circuit exists which can
approximate the measurement of any observable, in principle one can find
a circuit which will estimate any such polynomial function by averaging
over many runs. I construct some simple examples and compare these
results to existing procedures.
back to QIC online Front page
