2,385 research outputs found
Discriminating quantum-optical beam-splitter channels with number-diagonal signal states: Applications to quantum reading and target detection
We consider the problem of distinguishing, with minimum probability of error,
two optical beam-splitter channels with unequal complex-valued reflectivities
using general quantum probe states entangled over M signal and M' idler mode
pairs of which the signal modes are bounced off the beam splitter while the
idler modes are retained losslessly. We obtain a lower bound on the output
state fidelity valid for any pure input state. We define number-diagonal signal
(NDS) states to be input states whose density operator in the signal modes is
diagonal in the multimode number basis. For such input states, we derive series
formulas for the optimal error probability, the output state fidelity, and the
Chernoff-type upper bounds on the error probability. For the special cases of
quantum reading of a classical digital memory and target detection (for which
the reflectivities are real valued), we show that for a given input signal
photon probability distribution, the fidelity is minimized by the NDS states
with that distribution and that for a given average total signal energy N_s,
the fidelity is minimized by any multimode Fock state with N_s total signal
photons. For reading of an ideal memory, it is shown that Fock state inputs
minimize the Chernoff bound. For target detection under high-loss conditions, a
no-go result showing the lack of appreciable quantum advantage over coherent
state transmitters is derived. A comparison of the error probability
performance for quantum reading of number state and two-mode squeezed vacuum
state (or EPR state) transmitters relative to coherent state transmitters is
presented for various values of the reflectances. While the nonclassical states
in general perform better than the coherent state, the quantitative performance
gains differ depending on the values of the reflectances.Comment: 12 pages, 7 figures. This closely approximates the published version.
The major change from v2 is that Section IV has been re-organized, with a
no-go result for target detection under high loss conditions highlighted. The
last sentence of the abstract has been deleted to conform to the arXiv word
limit. Please see the PDF for the full abstrac
Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
A strong direct product theorem says that if we want to compute k independent
instances of a function, using less than k times the resources needed for one
instance, then our overall success probability will be exponentially small in
k. We establish such theorems for the classical as well as quantum query
complexity of the OR function. This implies slightly weaker direct product
results for all total functions. We prove a similar result for quantum
communication protocols computing k instances of the Disjointness function.
Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for
sorting N items on a quantum computer, which is optimal up to polylog factors.
They also give several tight time-space and communication-space tradeoffs for
the problems of Boolean matrix-vector multiplication and matrix multiplication.Comment: 22 pages LaTeX. 2nd version: some parts rewritten, results are
essentially the same. A shorter version will appear in IEEE FOCS 0
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
Chernoff's density is log-concave
We show that the density of ,
sometimes known as Chernoff's density, is log-concave. We conjecture that
Chernoff's density is strongly log-concave or "super-Gaussian", and provide
evidence in support of the conjecture.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ483 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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