Heisenberg-limited measurement protocols can be used to gain an
increase in measurement precision over classical protocols. Such
measurements can be implemented using, e.g., optical Mach-Zehnder
interferometers and Ramsey spectroscopes. We address the formal
equivalence between the Mach-Zehnder interferometer, the Ramsey
spectroscope, and the discrete Fourier transform. Based on this
equivalence we introduce the quantum ``Rosetta stone''.
Large photon-number path entanglement is an important resource for
Heisenberg-limited measurement protocols. We present a general
constructive protocol to create any large photon number
path-entangled state based on the conditional detection of single
photons. The influence of imperfect detectors is considered and an
asymptotic scaling law is derived.
Another important application of projective measurements in linear
optics is the interferometer that signals the presence of a single
photon in a particular input state without destroying it. This quantum nondemolition
device can be used to create practical, efficient, quantum repeaters,
employing double-photon guns, for long-distance optical quantum
communication. The guns create polarization-entangled photon pairs on
demand. One such source might be a semiconducter quantum dot, which
has the distinct advantage over parametric down-conversion that the
probability of creating a photon pair is close to one, while the
probability of creating multiple pairs vanishes. The swapping and
purifying components are implemented by polarizing beam splitters and
probabilistic optical CNOT gates.
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