Entangled Graphs

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Entangled Graphs
Research Center for Quantum Information

• Martin Plesch
• plesch_at_savba.sk
• www.quniverse.sk/plesch

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

• Entanglementis a very complex phenomenon in big
  systems

• Interesting feature Limited sharing CKW
  inequalitiesV. Coffman, J. Kundu, W. Wootters
  Phys.Rev. A61 (2000) 052306

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Classical Correlations

• Correlation in quantum systems has two principal
  origins
• Correlation induced by entanglement
• Correlation due to statistical mixing
• More-partite entanglement fragments into
  bipartite correlation
• Problem with a suitable measure, that could be
  compared to concurrence
• Basic question which bipartite entanglement and
  classical correlation configurations are allowed?

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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).
• No conditions on separability
• Classical correlation were not considered at all

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Entangled Graphs

• Particle (qubit) vertex
• Entanglement between 2 particles edge, weighted
  by concurrence
• NO edge implies NO entanglement
• The graph is defined by the number of qubits N
  and a set of concurrencies Cij

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Weighted Graphs for Pure States

• Edges in graphs are weighted by concurrence
• Definitely not all graphs have representatives
  (CKW inequalities and more)
• If we post a strict condition for maximal
  concurrence , we can
  show that

M. Plesch, J. Novotný, Z. Dzuráková, V. Buek,
Controlling of bipartite entanglement in
many-partite states, quant-ph/0311069
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Weighted Graphs for Pure States

• There exists a procedure to find for given
  Cij
• We start with
• Step by step we lower gammas to update the
  concurrence
• In every step, every concurrence is greater than
  or equal to the desired concurrence (we approach
  the desired state from to top in the viewpoint of
  concurrence)
• After every step, gammas are smaller than before
  the sequence is convergent

• Again, we use only the N2-dimensional part of the
  Hilbert space

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Classical Correlations

• Given a state of N qubits, a pair of them is
  correlated, iff its density matrix is correlated
• A density matrix uncorrelation condition
• There are three types of states of two qubits
• Entangled pair full line
• Correlated, but not entangled pair dashed line
• Not correlated, factorized pair no line
• No measure is assigned to the edges problems
  with classical correlations

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Graphs with Classical Correlations

• The graph is given by
• The set of entangled pairs SE
• The set of correlated pairs SC
• For the definition of the state vector one needs
  to specify
• The number of qubits correlated with the ith
  qubit mi
• The total number of correlated pairs M

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Mixed States

• We can utilize the true classical correlation
  coming out of the classical uncertainty of the
  state
• The state space is big
• On the other side, uncorrelation condition is, in
  comparison to entanglement, very tight
• Still, we are able to prove, that

For each correlation graph there exists at least
one mixed state
Martin Plesch a Vladimír Buek, Entangled graphs
II Classical correlations in multi-qubit
entangled systems, quant-ph/0306008, Phys. Rev. A
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Pure States

• We know that not all graphs with double
  constraints can be realized by pure states
• The not-realizable graphs have a common property
  an open edge

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Pure States

• In cases with more particles we are able to state
  only some general assertions

Pure states for unconnected correlation graphs
exist, if they exist for fragments of the graph
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Pure States
No pure states exist for correlation graphs with
an open edge
For each correlation graph, where every pair of
qubits is entangled or correlated, there exists
at least one pure state
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Operation Tomography
Research Center for Quantum Information

• Open problem talk

Collaborators Vladimír Buek, Mário Ziman
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Black box Problem

• Having a black box (with no memory) processing
  one qubit in a time, how can we determine its
  parameters?

• How many different states do we need for a
  complete guess?

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Complete Estimation

• For a complete estimation one needs four
  different states, which are linearly independent.

?
?
?
?

• What to do, if we do not have them?

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Incomplete Estimation

• One needs to create a general and reasonable rule
  that would stand for the missing information
• The basic question is, what guess should one make
  if NO information is available
• For symmetry reasons only two candidates are
  relevant
• Identity
• Contraction to the complete mixture
• Identity is not suitable, as it can not be an
  average operation (it is an extremal operation,
  as any unitary operation)

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Known states

• What to do, if we have a complete information
  about incoming and outgoing states, but there are
  not enough different ones?
• Take such an operation, that is
• As close to contraction to the complete mixture
  as possible
• Fits to the known data
• The problem complete positivity
• Additional criteria
• Perpendicular states remain perpendicular
• Is an analytical solution possible in general
  also for two and three different states? the
  small open question

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Unknown states

• What to do, if we have an incomplete information
  about outgoing states (for instance, we have a
  limited resource of a few states, that can be
  prepared and send through our device)?
• Idea state tomography and then operation
  tomography
• Data from state tomography can be inconsistent
• Definitely not the best way to do it
• This is the BIG open question