QIC Abstracts

 Vol.7 No.8 November 1, 2007

Research Articles: 
Multi-partite quantum cryptographic protocols with noisy GHZ states (pp689-715) PDF
          K. Chen and H.-K. Lo
We propose a wide class of distillation schemes for multi-partite entangled states that are CSS-states. Our proposal provides not only superior efficiency, but also new insights on the connection between CSS-states and bipartite graph states. We then apply our distillation schemes to the tri-partite case for three cryptographic tasks---namely, (a) conference key agreement, (b) quantum sharing of classical secrets and (c) third-man cryptography. Moreover, we construct ``prepare-and-measure'' protocols for the above three cryptographic tasks which can be implemented with the generation of only a single entangled pair at a time. This gives significant simplification over previous experimental implementations which require two entangled pairs generated simultaneously. We also study the yields of those protocols and the threshold values of the fidelity above which the protocols can function securely. Rather surprisingly, our protocols will function securely even when the initial state does not violate the standard Bell-inequalities for GHZ states.

Asymmetric quantum teleclonging of multiqubit states (pp716-729) PDF
          L. Chen and Y.-X. Chen
A scheme of 1$\rightarrow$2 optimal universal asymmetric quantum telecloning for pure multiqubit states is proposed. We first investigate the telecloning of arbitrary 2-qubit states and then extend it to the case of multiqubit system. We discuss the scheme in terms of the quantum channels and fidelities of clones, as well as the entanglement of states in the telecloning.

Quantumness, generalized spherical 2-design, and symmetric informationally complete POVM (pp730-737) PDF
          I.H. Kim
Fuchs and Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural generalization of spherical $t$-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to $d^2$, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.

Universal Mixing of Quantum Walk on Graphs (pp738-751) PDF
          W. Carlson, A. Ford, E. Harris, J. Rosen, C. Tamon, and K. Wrobel   
We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph $G$ is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on $G$ ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is {\em uniform} mixing if it visits the uniform distribution. Our results include the following:
1) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no {\em unweighted} complete multipartite graphs are uniform mixing (except for the four-cycle $K_{2,2}$).
2) For all $n \ge 1$, the weighted claw $K_{1,n}$ is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted $K_{1,n}$ is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes.
3) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs.
4) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case.
Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.

For distinguishing conjugate Hidden subgroups, the pretty good measurement is as good as it gets (pp752-765) PDF
          C. Moore and A. Russell
Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group $D_n$ in the case where the hidden subgroup is chosen uniformly from the $n$ involutions. We show that, for any group and any subgroup $H$, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of $H$. We go on to show that when $H$ forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest.

Fully multi-qubit entangled states (pp766-774) PDF
          J.-M. Cai, Z.-W. Zhou, and G.-C. Guo
We investigate the properties of different levels of entanglement in graph states which correspond to connected graphs. Combining the operational definition of graph states and the postulates of entanglement measures, we prove that in connected graph states of $N$ qubits there is no genuine $k$-qubit entanglement, $2\leq k\leq N-1$, among every $k$ qubits. These results about connected graph states naturally lead to the definition of fully multi-qubit entangled states. We also find that the connected graph states of four qubits is one but not the only one class of fully four-qubit entangled states.

Macroscopic displaced thermal field as the entanglement catalyst (pp775-781) PDF
          S.-B. Zheng
We show that entanglement of multiple atoms can arise via resonant interaction with a displaced thermal field with a macroscopic photon-number. The cavity field acts as the catalyst, which is disentangled with the atomic system after the operation. Remarkably, the entanglement speed does not decrease as the average photon-number of the mixed thermal state increases. The atoms may evolve to a highly entangled state even when the photon-number of the cavity mode approaches infinity.

Unambiguous unitary quantum channels (pp782-798) PDF
          S.-J. Wu and X.-M. Chen
Unambiguous unitary maps and unambiguous unitary quantum channels are introduced and some of their properties are derived. These properties ensures certain simple form for the measurements involved in realizing an unambiguous unitary quantum channel. Error correction and unambiguous error correction with nonzero probability are discussed in terms of unambiguous unitary quantum channels. We not only re-derive the well-known condition for a set of errors to
be correctable with certainty, but also obtain a necessary and sufficient condition for the errors caused by a noisy channel to be correctable with any nonzero probability. Dense coding with a partially entangled state can also be viewed as an unambiguous unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained.

Book Review: 
On “An introduction to quantum computing (authored by P. Kaye, R. Laflamme and M. Mosca) (pp799-800) PDF
          G.J. Milburn

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