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Bound entangled states are the states ρ which are (i) entangled (ii)
undistillable i.e. for which distillable entanglement is zero. To build
them, one needs pure entanglement, but once they are created, no pure
entanglement can be retrieved from them.
- There is no general rule how to decide if a given state is a bound
entangled one. To decide it one should show that it is entangled, and
that one cannot distill singlets from it. For both questions, no simple
algorithm has been found, however there are many different examples and
special constructions of bound entangled states.
- If a state has positive partial transposition (is PPT) and is
entangled, then it is also bound entangled.
- For a state to be bound entangled it is necessary to act on HA ⊗ HB
of dimension $dim ({H}_A) \times dim ({H}_B)
> 6$.
Examples
1. The first known example of bound entangled state given in
PawelHor1997 is a state acting on a tensor product of Hilbert spaces of
dimensions dim(HA) = 2
and dim(HB) = 4
respectively. It has the following form
\rho_p ={1\over 7p+1}\left[\begin{array}{cccccccc} p&
0&0&0& 0&p& 0&0 \\ 0& p&0&0&
0&0&p& 0 \\ 0& 0&p&0& 0&0&0& p\\
0& 0&0&p& 0&0&0& 0\\ 0&
0&0&0\over 2}&0&0\over 2}\\ p& 0&0&0&
0&p&0& 0\\ 0& p&0&0& 0&0&p& 0\\
0& 0&p&0\over 2}&0&0\over 2}\\ \end{array} \right]
%\end{eqnarray}
for $0 < p < 1$ and
the matrix is written in the
computational
basis. It is easy to check, that this matrix has
positive
partial transposition. Moreover it is not separable, as it does not
fulfill the
range
criterion of separability. Hence the above state is entangled.
2. Another example is the parametrised family of states in 3 ⊗ 3:
\sigma_{\alpha} = {2\over 7}|\psi_+\rangle\langle\psi_+| +
{\alpha\over 7}\sigma_+ + (I - P)
is bound entangled. where I
is the identity operator on the space Cd ⊗ Cd.
\bibitem{PawelHor1997} \bibitem{quant-ph/9808030}
Category:Entanglement
Last modified:
Monday, October 26, 2015 - 17:56
