Bound entangled states

'''Bound entangled states''' are the states $\backslash rho$ 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 $\{H\}\_A\; \backslash otimes\; \{H\}\_B$ of dimension $dim\; (\{H\}\_A)\; \backslash times\; dim\; (\{H\}\_B)\; >\; 6$. == 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 $dim\; (\{H\}\_A)\; =\; 2$ and $dim\; (\{H\}\_B)\; =\; 4$ 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\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 and the matrix is written in the [[computational basis]]. It is easy to check, that this matrix has [[positive partial transpose|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 $3\backslash otimes\; 3$: \sigma_{\alpha} = {2\over 7}|\psi_+\rangle\langle\psi_+| + {\alpha\over 7}\sigma_+ + {{5 - \alpha}\over 7} \sigma_- \quad for \quad 2\leq \alpha \leq 5, where \sigma_+ = {1\over 3}(|0\rangle |1\rangle\langle 0|\langle 1| + |1\rangle |2\rangle\langle 1|\langle 2| + |2\rangle |0\rangle \langle 2|\langle 0|) \sigma_- = {1\over 3}(|1\rangle |0\rangle \langle 1|\langle 0| + |2\rangle |1\rangle \langle 2|\langle 1| + |0\rangle |2\rangle \langle 0|\langle 2|) States of that family are [[separable]] for $\backslash alpha\; \backslash in\; [2,3]$, bound entangled for $\backslash alpha\; \backslash in\; (3,4]$ and free entangled for $\backslash alpha\; \backslash in\; (4,5]$. 3. One of the simplest constructions of familes of bound entangled states is from quant-ph/9808030 and is based on the so called [[unextendible product basis|unextendible product bases]] (UPB) Namely, given a projector onto UPB: P =\sum_{i=1}^k |v_i\rangle\langle v_i| the state \rho = {1\over {d^2 - k}}(I - P) is bound entangled. where $I$ is the identity operator on the space $C^d\backslash otimes\; C^d$. \bibitem{PawelHor1997} \bibitem{quant-ph/9808030} [[Category:Entanglement]]