Conditional preparation of maximal polarisation entanglement

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Conditional preparation of maximal polarisation
entanglement

• Event-ready creation of maximal entanglement in
  the photonic domain
• arXivquant-ph/0207117

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Summary

• Applications of entanglement
• Down conversion light sources
• Random vs. Event-Ready
• Possible solutions
• Proposed experimental set-up
• How it works
• Fidelity vs. Efficiency trade-off
• Concluding remarks

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Applications of entanglement

• Entanglement is an essential ingredient of
  quantum computing. We need maximal entanglement
  to implement the quantum information protocols
• teleportation,
• dense coding,
• cryptography.
• It contradicts Bells inequalities and hidden
  variable theories.

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Linear Optics Quantum Computing

• In Linear Optics Quantum Computing, one qubit is
  represented by a pair of photonic modes.
• The qubit states are represented by a photon in
  one of the two modes.
• Example two polarisation modes of one beam. A
  maximally entangled state of two qubits
  corresponds to a pair of photons, one in each of
  two beams, entangled in their polarisation states.

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Down conversion sources of entangled photon pairs
the so calledtype-II phase matching configuration

• P. G. Kwiat et al., New high-intensity source of
  polarization-entangled photons pairs, Phys. Rev.
  Letters 75, 4337 (1995)
• http//www.uebersetzung-reck.de/physics/entangled.
  htm

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Down conversion light sources a novel geometry

• P. G. Kwiat et al., Ultra-bright source of
  polarization-entangled photons, Phys. Rev. A 60,
  R737 (1999)

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Random vs. Event-Ready

• Since down-conversion is a spontaneous process,
  it occurs at random instants of time. A
  difficulty arises How to operate on an entangled
  pair until we are certain it is there?
• An event-ready source of entangled pairs would
  be one that provides a trigger signal to indicate
  a success of the preparation process.

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Possible solutions

• An event-ready source would be a good
  substitute for a deterministic one. But how to
  construct it?
• The possible answers
• quantum non-demolition detection (QND),
• post-selection (not universal),
• entanglement-swapping,
• general conditional-detection schemes.

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Post-selection

• In up-to-date experiments, the decision whether
  an entangled photon pair had been produced was
  deferred until the final detection of all the
  photons.
• This technique is known as post-selection.
• Implementing post-selection may require some
  additional experimental means, like
  single-photon-resolution detectors.

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Entanglement swapping

• In the procedure of entanglement-swapping, two
  crystals produce one photon pair each, and an
  appropriate measurement on two of the photons
  ascertains that the remaining two form an
  entangled pair.

M. Zukowski et al., Event-ready-detectors Bell
experiment via entanglement swapping, Phys. Rev.
Letters 71, 4287 (1993)
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Entanglement-swapping No-go

• But, as discovered later, it can not be detected
  whether each crystal produced one photon pair, or
  only one of the crystals produced both pairs. It
  was even shown that no conditional detection
  scheme utilising linear optics and
  down-conversion crystals can produce maximal
  entanglement from two down-converted pairs, i.e.
  conditioned on detection of two auxiliary
  detected photons.

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General conditional detection

• It was, however, not known whether maximal
  entanglement can be created by allowing for more
  than two auxiliary detected photons.

P. Kok, S. Braunstein, Limitations on the
creation of maximal entanglement, Phys. Rev. A
62, 064301 (2000)
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And the answer is

• Yes, conditional preparation of photon pairs in
  an entangled polarisation state is possible.
• The third-order of down-conversion is sufficient
  (three pairs of photons need to be produced, four
  of the produced photons are detected).
• A simple experimental set-up is possible
  consisting just of one down-conversion crystal,
  linear optics and detectors.

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We think

• Owing to the current progress in down-conversion
  sources, this fact may be of practical importance.

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Proposed experimental set-up

• PDC parametric down-conversion crystal
• a, b, spatial modes
• BS1, BS2 beam-splitters
• HWP half wave plate at 22.5
• PBS1, PBS2 polarisation analysers
• x, y polarisation modes

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How it works

• the output is conditioned on fourth-fold
  coincidence of detection
• the coincidence is not possible in the first
  order
• the second order vacuum term vanishes due to
  interference
• the third order yields the maximal entanglement

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Vanishing of the vacuum term

• All four photons must be reflected by the
  beam-splitters for the coincidence to take place.
• The second order down-conversion output is
• Only the middle term contributes to the
  coincidence in the upper arm. The lower arm is
  left in the state
• No coincidence is possible in the lower arm.

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Calculating the output state

• take the third order down-conversion term
• apply the linear optics transformation
• calculate a polynomial quotient

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Fidelity vs. Efficiency trade-off
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Fidelity vs. Efficiency trade-off
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Concluding remarks

• The output state can be easily calculated in
  terms of polynomials in the creation operators.
• In case of non-ideal detection (non-unit
  efficiency or lack of single-photon resolution),
  both the fidelity of the output state and the
  overall preparation efficiency are diminished.
• The transmittance/reflectivity of the
  beam-splitters can be adjusted to enhance
  fidelity at the cost of decreased efficiency.