2,788 research outputs found
Estimating graph parameters with random walks
An algorithm observes the trajectories of random walks over an unknown graph
, starting from the same vertex , as well as the degrees along the
trajectories. For all finite connected graphs, one can estimate the number of
edges up to a bounded factor in
steps, where
is the relaxation time of the lazy random walk on and
is the minimum degree in . Alternatively, can be estimated in
, where is
the number of vertices and is the uniform mixing time on
. The number of vertices can then be estimated up to a bounded factor in
an additional steps. Our
algorithms are based on counting the number of intersections of random walk
paths , i.e. the number of pairs such that . This
improves on previous estimates which only consider collisions (i.e., times
with ). We also show that the complexity of our algorithms is optimal,
even when restricting to graphs with a prescribed relaxation time. Finally, we
show that, given either or the mixing time of , we can compute the
"other parameter" with a self-stopping algorithm
Fast rate estimation of an unitary operation in SU(d)
We give an explicit procedure based on entangled input states for estimating
a operation with rate of convergence when sending
particles through the device. We prove that this rate is optimal. We also
evaluate the constant such that the asymptotic risk is . However
other strategies might yield a better const ant .Comment: 8 pages, 1 figure Rewritten version, accepted for publication in
Phys. Rev. A. The introduction is richer, the "tool section" on group
representations has been suppressed, and a section proving that the 1/N^2
rate is optimum has been adde
On the generalization of quantum state comparison
We investigate the unambiguous comparison of quantum states in a scenario
that is more general than the one that was originally suggested by Barnett et
al. First, we find the optimal solution for the comparison of two states taken
from a set of two pure states with arbitrary a priori probabilities. We show
that the optimal coherent measurement is always superior to the optimal
incoherent measurement. Second, we develop a strategy for the comparison of two
states from a set of N pure states, and find an optimal solution for some
parameter range when N=3. In both cases we use the reduction method for the
corresponding problem of mixed state discrimination, as introduced by Raynal et
al., which reduces the problem to the discrimination of two pure states only
for N=2. Finally, we provide a necessary and sufficient condition for
unambiguous comparison of mixed states to be possible.Comment: 8 pages, 4 figures, Proposition 1 corrected, appendix adde
Quantum mechanics explained
The physical motivation for the mathematical formalism of quantum mechanics
is made clear and compelling by starting from an obvious fact - essentially,
the stability of matter - and inquiring into its preconditions: what does it
take to make this fact possible?Comment: 29 pages, 5 figures. v2: revised in response to referee comment
Relativistic Doppler effect in quantum communication
When an electromagnetic signal propagates in vacuo, a polarization detector
cannot be rigorously perpendicular to the wave vector because of diffraction
effects. The vacuum behaves as a noisy channel, even if the detectors are
perfect. The ``noise'' can however be reduced and nearly cancelled by a
relative motion of the observer toward the source. The standard definition of a
reduced density matrix fails for photon polarization, because the
transversality condition behaves like a superselection rule. We can however
define an effective reduced density matrix which corresponds to a restricted
class of positive operator-valued measures. There are no pure photon qubits,
and no exactly orthogonal qubit states.Comment: 10 pages LaTe
Minimal optimal generalized quantum measurements
Optimal and finite positive operator valued measurements on a finite number
of identically prepared systems have been presented recently. With physical
realization in mind we propose here optimal and minimal generalized quantum
measurements for two-level systems.
We explicitly construct them up to N=7 and verify that they are minimal up to
N=5. We finally propose an expression which gives the size of the minimal
optimal measurements for arbitrary .Comment: 9 pages, Late
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