QIC Abstracts

 Vol.3 No.2  March 1, 2003
Asymptotic entanglement capacity of the Ising and anisotropic Heisenberg interactions (pp97-105)
        A.M. Childs, D.W. Leung, F. Verstrarte, and G. Vidal
We calculate the asymptotic entanglement capacity of the Ising interaction \sigma_z\otimes\sigma_z, the anisotropic Heisenberg interaction \sigma_x\otimes\sigma_x + \sigma_y\otimes\sigma_y, and more generally, any two-qubit Hamiltonian with canonical form K = \mu_x  \sigma_x\otimes \sigma_x + \mu_y  \sigma_y \otimes \sigma_y. We also describe an entanglement assisted classical communication protocol using the Hamiltonian K with rate equal to the asymptotic entanglement capacity.

Entanglement and nonlocality for a mixture of a pair-coherent state (pp106-115)
        S. Mancini and P. Tombesi
We consider a bipartite continuous variables quantum mixture coming from phase randomization of a pair-coherent state. We study the nonclassical properties of such a mixture. In particular, we quantify its degree of entanglement, then we show possible violations of Bell's inequalities. We also consider the use of this mixture in quantum teleportation. Finally, we compare this mixture with that obtained from a pair-coherent state by single photon loss.

Generation and degree of entanglement in a relativistic formulation (pp115-120)
        J. Pachos and E. Solano
The generation of entangled states and their degree of entanglement are studied in a relativistic formulation for the case of two interacting spin-1/2 charged particles. In the realm of quantum electrodynamics, we revisit the interaction that produces entanglement between the spin components of covariant Dirac spinors describing the two particles. In this way, we derive the relativistic version of the spin-spin interaction, widely used in the nonrelativistic regime. Following this consistent approach, the relativistic invariance of the generated entanglement is discussed.

An analysis of reading out the state of a charge quantum bit (pp121-138)
        H-S. Goan 
We provide a unified picture for the master equation approach and the quantum trajectory approach to a measurement problem of a two-state quantum system (a qubit), an electron coherently tunneling between two coupled quantum dots (CQD's) measured by a low transparency point contact (PC) detector. We show that the master equation of ``partially'' reduced density matrix can be derived from the quantum trajectory equation (stochastic master equation) by simply taking a ``partial'' average over the all possible outcomes of the measurement. If a full ensemble average is taken, the traditional (unconditional) master equation of reduced density matrix is then obtained. This unified picture, in terms of averaging over (tracing out) different amount of detection records (detector states), for these seemingly different approaches reported in the literature is particularly easy to understand using our formalism. To further demonstrate this connection, we analyze an important ensemble quantity for an initial qubit state readout experiment, P(N,t), the probability distribution of finding N electron that have tunneled through the PC barrier(s) in time t. The simulation results of P(N,t) using 10000 quantum trajectories and corresponding measurement records are, as expected, in very good agreement with those obtained from the Fourier analysis of the ``partially'' reduced density matrix. However, the quantum trajectory approach provides more information and more physical insights into the ensemble and time averaged quantity P(N,t). Each quantum trajectory resembles a single history of the qubit state in a single run of the continuous measurement experiment. We finally discuss, in this approach, the possibility of reading out the state of the qubit system in a single-shot experiment.

Optimal realizations of controlled unitary gates (pp139-155)
        G. Song and A. Klappenecker 
The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases.

Bell inequality for quNits with binary measurements (pp157-164)
        H. Bechmann-Pasquinucci and N. Gisin
We present a generalized Bell inequality for two entangled quNits. On one quNit the choice is between two standard von Neumann measurements, whereas for the other quNit there are N^2 different binary measurements. These binary measurements are related to the intermediate states known from eavesdropping in quantum cryptography. The maximum violation by \sqrt{N} is reached for the maximally entangled state. Moreover, for N=2 it coincides with the familiar CHSH-inequality.

Quantum lower bound for recursive Fourier sampling (pp165-174)
        S. Aaronson
We revisit the oft-neglected `recursive Fourier sampling' (RFS) problem, introduced by Bernstein and Vazirani to prove an oracle separation between BPP and BQP. We show that the known quantum algorithm for RFS is essentially optimal, despite its seemingly wasteful need to uncompute information. This implies that, to place \mathsf{BQP} outside of PH[\log] relative to an oracle, one would need to go outside the RFS framework. Our proof argues that, given any variant of RFS, either the adversary method of Ambainis yields a good quantum lower bound, or else there is an efficient classical algorithm. This technique may be of independent interest.

Circuit for Shor's algorithm using 2n+3 qubits (pp175-185)
        S. Beauregard 
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored.

Maximal p-norms of entanglement breaking channels (pp186-190)
        C. King
It shown that when one of the components of a product channel is entanglement breaking, the output state with maximal p-norm is always a product state. This result complements Shor's theorem that both minimal entropy and Holevo capacity are additive for entanglement breaking channels. It is also shown how Shor's results can be recovered from the p-norm results by considering their behavior for p close to one.

Book Review:
On “Statistical Structure of Quantum Theory” by A.S. Holevo (pp191-192)
        C. Fuchs

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