Entangled Graphs Bipartite entanglement in multipartite states

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Entangled GraphsBipartite entanglement in
multipartite states
Research Center for Quantum Information

• Martin Plesch
• plesch_at_savba.sk

Collaborators Vladimír Buek, Mário Ziman
Supported by EQUIP, VEGA
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Outline

• Motivation
• Bipartite correlations in many-particle systems
• Entangled graphs
• Mixed states results
• Pure states results
• Conclusions

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Motivation

• Entanglement in big systems is a very complex
  phenomenon
• Bipartite entanglement is only a small fragment,
  but best theoretically understood
• Can give us a good overview about the whole
  system
• However, it cannot be shared freely CKW
  inequalitiesV. Coffman, J. Kundu, W. Wootters
  Phys.Rev. A61 (2000) 052306
• Tight limits are still not completely known

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Forerunners

• Entangled chains long chain of entangled
  qubitsW. Wootters, quant-ph/0001114 (2000)
• Entangled webs N qubits pairwise entangled M.
  Koashi, V. Buzek, N. Imoto Phys. Rev. A 62,
  050302(R)-14 (2000).
• Entangled molecules entanglement engineering on
  mixed statesW. Dur, Phys. Rev. A 63, 020303(R)
  (2001).
• Only condition on existence of entanglement
  imposed

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Entangled Graphs

• Particle (qubit) vertex
• Entanglement between 2 particles edge
• NO edge implies NO entanglement
• The graph is defined by the number of qubits N
  and a set of edges S
• particles i and j are
  entangled
• k S is the number of entangled pairs in the
  system

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Mixed States

• For a specified graph, we search for a mixed
  state, which would be characterized by that graph
• Suppose a state
• and a mixture

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Mixed States

• Reduced density operator for

• Reduced density operator for

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Pure States

• In general, the problem is more complicated
• Free parameters are limited in comparison to
  mixed state
• The same approach as for mixed states
  (combination of Bell states) does not work

• However, we are able to formulate a theorem

For each entangled graph there exists at least
one pure state
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Pure States

• The family of possible state vectors can be shown
  to be

• We use only a small, N2-dimensional part of the
  whole Hilbert space in comparison to the total
  dimension 2N

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Conclusions

• Could we control also the strength of
  entanglement?Yes, we are able to produce
  weighted graphs
• How effectively, what are the limitations?In our
  case rather strict,
• What about classical correlations?We can define
  Correlated graphs, with very interesting
  outcomes
• Recall the main theorem

For each entangled graph there exists at least
one pure state