# Problem 2: Undistillability implies ppt?

# Problem

A state on a bipartite quantum system is called distillable, if from sufficiently many pairs prepared in that state one can obtain a close approximation of a maximally entangled singlet state, using only local quantum operations and classical communication (LOCC). It is well-known that states with positive partial transpose (PPT) are not distillable. The problem is to decide the converse.

# Background

This problem has been evident ever since it was shown in [HHH1] that entangled PPT states are undistillable. The two properties, PPT on the one hand and being undistillable on the other, are mathematically as different as they can be. Whereas the latter is a variational problem on an unbounded number of tensor products of density matrices, the first is a simple eigenvalue problem:

- A bipartite density operator [math]\rho[/math] is said to be
**PPT**if its partial transpose [math]\rho^{\text{T}_\text{A}}[/math], defined with respect to some product basis via [math]\langle ij |\rho^{\text{T}_\text{A}}| kl \rangle = \langle kj | \rho | il \rangle[/math], is positive semi-definite, i. e., has only non-negative eigenvalues. - A bipartite state characterized by a density matrix [math]\rho[/math] is
**distillable**if there is a number n, such that [math]\rho^{\otimes n}[/math] can locally be projected onto an entangled two qubit state. That is, there are two dimensional projectors Q and P acting on the n-fold tensor product corresponding to Alice respectively Bob, such that [math]\left( (P\otimes Q) \rho^{\otimes n} (P\otimes Q) \right)^{\text{T}_\text{A}}[/math] has at least one negative eigenvalue. If n copies of [math]\rho[/math] have such an entangled two qubit subspace, then the state is called n-**distillable**. There is yet no example of a state, which is distillable but not 1-distillable.

Using the above criterion of distillability, which was proven by the Horodeckis in [HH], the problem can be reformulated as [DSST]:

Given a completely positive map S such that T S is 2-positive (i. e. [math]id_2 \otimes T S[/math] is positive), where [math]T[/math] denotes the transpose map. Decide whether [math]T S \otimes T S[/math] is necessarily 2-positive.

# Partial Results

- For special cases like states on Hilbert spaces of dimension [math]2 \times m[/math] or Gaussian states it was proven in [DCLB], [HHH2] respectively [GDCZ], that every such state having a non-positive partial transpose (NPPT) is distillable.
- It was proven in [HH], that every NPPT state can be mapped onto an NPPT Werner state by means of LOCC operations. Hence, the matter can be decided considering the one-parameter family of Werner states only: if there exist any undistillable NPPT states, then there are undistillable entangled Werner states.
- In [DCLB], [DSST] numerical evidence has been presented, that there may be undistillable NPPT states. Moreover, it was proven analytically in [DSST], that for every fixed finite n there is an interval of n-undistillable entangled Werner states. However, the parameter interval for which this statement has been proven, goes to zero for [math]n\rightarrow\infty[/math].
- It was proven in [EVWW] that if one enlarges the class of allowed operations from LOCC to PPT preserving maps, then every NPPT state becomes 1-distillable. For a proof using entanglemet witnesses and the discussion of the tripartite case see [KLC]. Note that every PPT-preserving map can stochastically be implemented as LOCC operation with an additional PPT entangled state as a resource [CDKL].
- If an additional entangled PPT state [math]\sigma[/math] makes an NPPT state [math]\rho[/math],
which is not 1-distillable itself, become 1-distillable, then we say that
[math]\sigma[/math]
**activates**the distillability of [math]\rho[/math]. It has been proven in [VW] that there are PPT states [math]\sigma[/math], which are capable of activating every NPPT state. Moreover, the required amount of entanglement (measured in terms of any entanglement measure, which is continuous at the separable boundary) has been shown to be infinitesimally small [VW]. In [KLC] a formalism was introduced that connects entanglement witnesses and the activation properties of a state. Here it was shown that there exist three-partite NPPT states with the property that two copies can neither be distilled, nor activated. - It was shown in [W] that for any choice of integers [math]d\geq 3[/math] and [math]n\geq 1[/math], there exists a [math]d^2 \otimes d^2[/math] bipartite mixed quantum state that is distillable but not n-distillable.
- It was shown in [C] that the distillability problem can be reformulated as a special instance of the seperability problem or as a Schmidt number problem. Furthermore a numerical method for detecting distillability is presented, which can be used to estimate the volume of distillable states. A set of one parameter states is presented which is conjectured to exhibit all forms of distillability.
- The authors of [VD] use Robust Semidefinte Programs and Entanglement Witness to study the distillablity of Werner states. They confirm the undistillablity conjecture for two copies and dimension [math]d=3[/math].

# Solution

# Literature

[CDKL] | J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, »Entangling Operations and Their Implementation Using a Small Amount of Entanglement«, Phys. Rev. Lett. 86, 544 (2001) and quant-ph/0007057 (2000). |

[DCLB] | W. Dür, J. I. Cirac, M. Lewenstein, and D. Bruß, »Distillability and transposition in bipartite systems«, Phys. Rev. A 61, 062313 (2000) and quant-ph/9910022 (1999). |

[DSST] | D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, »Evidence for bound entangled states with negative partial transpose«, Phys. Rev. A 61, 062312 (2000) and quant-ph/9910026 (1999). |

[EVWW] | T. Eggeling, K. G. H. Vollbrecht, R. F. Werner, and M. M. Wolf, »Distillability via Protocols Respecting the Positivity of Partial Transpose«, Phys. Rev. Lett. 87, 257902 (2001) and quant-ph/0104095 (2001). |

[GDCZ] | G. Giedke, L.-M. Duan, J. I. Cirac, and P. Zoller, »Distillability criterion for all bipartite Gaussian states«, Quant. Inf. Comp. 1(3), 79 (2001) and quant-ph/0104072 (2001). |

[HH] | M. Horodecki and P. Horodecki, »Reduction criterion of seperability and limits for a class of protocols of entanglement distillation«, Phys. Rev. A 59, 4206-4216 (1999) and quant-ph/9708015 (1997). |

[HHH1] | M. Horodecki, P. Horodecki, and R. Horodecki, »Mixed-State Entanglement and Distillation: Is there a 'Bound' Entanglement in Nature?«, Phys. Rev. Let. 80, 5239-5242 (1998) and quant-ph/9801069 (1998). |

[HHH2] | M. Horodecki, P. Horodecki, and R. Horodecki, »Inseparable Two Spin-1/2 Density Matrices Can Be Distilled to a Singlet Form«, Phys. Rev. Lett. 78, 574 (1997). |

[KLC] | B. Kraus, M. Lewenstein, and J. I. Cirac, »Characterization of distillable and activatable states using entanglement witnesses«, Phys. Rev. A 65, 042327 (2002) and quant-ph/0110174 (2001). |

[VW] | K. G. H. Vollbrecht and M. M. Wolf, »Activating Distillation with an Infinitesimal Amount of Bound Entanglement«, Phys. Rev. Lett. 88, 247901 (2002) and quant-ph/0201103 (2002). |

[W] | J. Watrous,»Many copies may be required for entanglement distillation«, http://xxx.lanl.gov/abs/quant-ph/0312123 quant-ph/0312123] (2004). |

[C] | L. Clarisse,»The distillability problem revisited«, quant-ph/0510035 (2006). |

[VD] | Reinaldo O. Vianna, Andrew C. Doherty,»Study of the Distillability of Werner States Using Entanglement Witnesses and Robust Semidefinite Programs «, quant-ph/0608095 (2006). |