# Bound entangled states

**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
*H**A*⊗*H**B*of dimension*d**i**m*(*H**A*) ×*d**i**m*(*H**B*) > 6.

### Examples

1. The first known example of bound entangled state given in 1 is a state acting on a tensor product of Hilbert spaces of dimensions *d**i**m*(*H**A*) = 2 and *d**i**m*(*H**B*) = 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& {{1+ p}\over 2}&0&0&{\sqrt{1- p^2}\over 2}\\ p& 0&0&0& 0&p&0& 0\\ 0& p&0&0& 0&0&p& 0\\ 0& 0&p&0& {\sqrt{1- p^2}\over 2}&0&0& {{1+ p}\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 *C**d* ⊗ *C**d*.

\bibitem{PawelHor1997} \bibitem{quant-ph/9808030}

- PawelHor1997↩