Vol.4 No.5 September 08, 2004
Research and Review Articles:
Security of quantum key distribution with imperfect devices  (pp325-360) 
        D. Gottesman, H.-K. Lo, N. L\"utkenhaus, and J. Preskill
We prove the security of the Bennett-Brassard (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 basis-dependent flaws, as is typical in practical implementations of the protocol. We derive a general lower bound on the asymptotic key generation rate for weakly basis-dependent eavesdropping attacks, and also estimate the rate in some special cases: sources that emit weak coherent states with random phases, detectors with basis-dependent efficiency, and misaligned sources and detectors.

Optimal realizations of simplified Toffoli gates  (pp361-372) 
        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} controlled-not gates, and subject to that, a sharp lower bound of {\em four} single-qubit 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  (pp373-382) 
        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 multi-partite entangled states allowing for a simple entanglement criterion  (pp383-395) 
        H. Aschauer, J. Calsamiglia, M. Hein, and H.~J. Briegel
We present local invariants of multi-partite pure or mixed states, which can be easily calculated and have a straight-forward 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  (pp396-400) 
        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 three-qubit 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 Cosine-Sine Decomposition (CSD), a well-known 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 $\mathfrak{a}=\mbox{span}_{\mathbb{R}} \{ \chi_1^n ( \ket{j}\bra{N-j-1} -\ket{N-j-1}\bra{j}) \chi_1^n \}$ 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  (pp401-408) 
        T.A. Brun
In this paper I show that any $m$th-degree 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.

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