Bipartite correlation in multipartite states Martin Plesch and Vladimr Buek Research Center for Quan

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Bipartite correlation in multipartite
statesMartin Plesch and Vladimír BuekResearch
Center for Quantum Information, Institute of
Physics, Slovak Academy of Sciences, Bratislava,
Slovakia
Abstract We investigate the properties of
multiparticle systems of qubits, concentrating on
the bipartite entanglement and classical
correlation. We introduce the terms Entangled
graph and Correlated graph, where qubits are
represented by vertices and pairwise correlation
and entanglement by edges. We examine for which
graphs one can find representatives in pure or
mixed states, where the entanglement or
correlation will be present only between a priori
selected pairs of qubits. In the case of weighted
graphs, we post a more strict condition we ask
for a specific concurrence between every pair of
qubits.

• Correlated graphs
• In this case we deal with two types of edges
  entanglement edges and classical correlation
  edges. As entanglement implies correlation, every
  pair of qubits can be either entangled
  (entanglement edge, full line), only classically
  correlated (correlation edge, dashed line) or
  completely separable (no edge, line).
• The graph is defined by two sets, the set of
  entangled pairs SE and the set of correlated
  pairs SC. Clearly, SE SC. The figure depicts
  all the correlation graphs for three particles.
  There are ten different correlation graphs in
  comparison to only four entangled graphs, and
  this ratio is growing enormously with N.

• 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 S
• particles i and j are
  entangled
• k S is the number of entangled pairs in the
  system
• Some examples of entangled graphs of five qubits

In the first case we want to recognize for which
graphs one can find a state, which could be
associated with that graph. For mixed states the
problem is rather easy, since it is enough to
prepare a mixture of states that exhibit
entanglement between single pairs. Such a mixture
will have exactly the desired properties
bipartite entanglement will be present only
between specific, well defined pairs. For pure
states the same idea does not work. However, with
the help of a little more complex state we are
able to formulate a theorem
For each entangled graph with non-weighted edges
there exists at least one pure state
First we will deal with mixed states. For this
case we can prove a general theorem
For each correlation graph there exists at least
one mixed state
Weighted entangled graphs In this case the edges
are weighted by a measure of entanglement,
concurrence. It is clear that not all graphs will
be realizable, since (at least) the CKW
inequalities and possibly also other restrictions
have to be fulfilled. However, when we confine
ourselves to week entanglement, where the maximal
concurrence is Cmax0.24/N, we can prove a theorem
For each weighted entangled graph, where
CijltCmax, there exists at least one pure state
For pure states the situation is much
complicated. For three qubits graphs c), d), e)
and f) do not exist, whereas the rest does. We
are not able to formulate a general theorem about
existence and non-existence of all the graphs.
However, we can at least decide about some
classes of graphs.
For each correlation graph with all pairs
entangled or correlated there exists at least one
pure state
There is a constructive proof of this theorem,
since the entangled graphs suggested above are
exactly of this type.
Pure states for unconnected graphs exist, iff
they exist for fragments of the graph
Unconnected graphs are defined as graphs with at
least two groups of vertices that are not
connected by any edge. Such a connected group of
vertices we call a fragment.
There exists a procedure to find parameters
gammaij of the state knowing all the
concurrencies. One starts with a specific state,
where all the concurrencies for all pairs will be
equal to Cmax (thus gammaijgamma_max). By
successive decrease of gammas, the concurrencies
will become closer to the desired values, but
always from the upper side. This means that after
each step all actual concurrencies will be
greater than or equal to the asked concurrencies.
However, when the gammas approach zero, so do the
concurrencies. So we have to cross after some
number of steps (finite for finite precision) the
desired state. In both cases of graphs with
non-weighted and weighted edged, we use only a
small, N2-dimensional part of the whole Hilbert
space in comparison to the total dimension 2N.
No pure states exist for correlation graphs with
an open edge
An open edge is an edge that connects two
vertices, of which at least one is not connected
with the rest of the system by any other edge.
The only exception is a pair of vertices
connected by an entanglement edge (a Bell pair).
This theorem explains why pure states for graphs
c), d), e) and f) according to the picture do not
exist. By numerical methods we have also proved
that for four qubits pure states exist for all
graphs except the ones with open edges. But it is
highly probable that in general there are more
restrictions, which become effective only in
higher dimensions.