Quantum information processing

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Quantum information processing

• Myungshik Kim
• Queens University, Belfast

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What is quantum information processing?

• A research in quantum information processing is
  to understand how quantum mechanics can improve
  acquisition, transmission and processing of
  information.
• Review article Bennett DiVincenzo, Nature 404,
  247 (2000)
• Who may be involved?
• Computer scientists What are difficult problems
  ?
• Mathematicians Coding and information theory
• Electrical engineers electronic quant info
  processor
• Chemists Chemical quantum devices
• Physicists Developer of quantum theory

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Quantum coherence
PBS

• Example 1
• By focusing a ?/2 pulse on a two level atom
• Example 2
• Schroedinger cat states
• ? denotes the amplitude of a coherent state
• By nonlinear Kerr interaction

Kerr medium
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Why quantum coherence?

• Consider the density operator
• Recall Youngs double slit interference
• Allows massive parallelism in quantum computation

Classical
Quantum
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What is entanglement ?

• Two important ingredients
• Random
• Deterministic

Type II nonlinear crystal
Energy momentum conservation
One photon is ? polarised the other ? polarised
? polarised
? or ?
? or ?
? polarised
HongMandel, PRL 59, 2044 (1987) Kwiat et al.,
PRL 75, 4337 (1995)
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• There are four basis states
• They are so called the Bell states
• Braunstein et al. PRL 68, 3259 (1992)
• States maximally violating Bells inequality

Bell
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• How about
  ?
• Is it entangled? Yes, as far as a?0 b?1.
• How do we define a degree of entanglement?
• For a pure state, the von Neumann entropy is a
  useful tool.
• Find ? for two particles
• Find the reduced density operator for one
  particle
• Substitute ?1 into the definition of the von
  Neumann entropy to find out the degree of
  entanglement
• S -Tr ?1 ln
  ?1.

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How to manufacture an arbitrary entangled state?

• White Kwiat (PRL 83, 3103 (1999))
• a) Twin photons are emitted
• at two identical down-conversion
• crystals. The crystals are oriented
• so that the optic axis of the first
• (second) lies in the vertical
• (horizontal) plane
• b) Adjustable quarter- half-wave
• plates and polarising beam splitters
• allow polarisation analysis
• in any basis.

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• Experimentally reconstructed
• density matrices of states that are
• a)
• b)
• c)

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• Is a mixed state always classical?
• Eh? What is a mixed state?
• Why mixed states? As soon as a quantum state is
  embedded in an environment, the pure state
  becomes mixed.

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A mixed state is not always classical.

• Consider
• For ab1/?2 the density operator describes a
  pure state and for b0 a classical mixture.
• There should be some critical point when the
  system loses quantum nature (entanglement).
• We need to define a measure of entanglement for a
  mixed state.

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Multi-particle entanglement BouwmeesterZeiling
er, PRL 82, 1345 (1999)

• Three-particle entanglement
• so called GHZ(Greenberger-Horne-Zeilinger) state
• Generation
• Use type II interaction
• Two pairs are generated
• GHZ for four simultaneous clicks
• How do we define a degree
• of entanglement? Any use?
• Sackett Wineland,
• Experimental entanglement of four paritcles,
• Nature 404, 256 (2000)

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Higher-dimensional entanglement

• Quantum information processing for qubits.
• Quantum effects for a d-dimensional system
  qudits
• Most quantum systems require the use of an
  infinite dimensional vector space.
• Possibility to test experimentally.
• Laser
• Provide a test ground for paradoxes in quantum
  mechanics.
• Photonic state Defined in phase space-continuous
  variables
• Braunstein error correction for continuous
  quantum variables, PRL 80, 4084 (1998)
• Braunstein, quantum error correction for
  communication with linear optics, Nature 394, 47
  (1998)
• BraunsteinLloyd, quantum computation over
  continuous variables, PRL 82, 1784 (1999)
• BotoBraunstein, quantum interferometric optical
  lithography, PRL 85, 2733 (2000)