Cite this problem as **Problem 20**.

**Problem**

Are ppt-operations suffcient to ensure asymptotically reversibly interconversion of all, i.e. pure and mixed, bipartite entangled states?

What is the smallest non-trivial class of operations that permits asymptotically reversibly interconversion of all, i.e. pure and mixed, bipartite entangled states?

Background

The concept of entanglement as a resource for quantum information processes motivates the study of its transformation properties under restricted classes of allowed operations, such as local operations and classical communication (LOCC). The key questions in such a resource theory are

- Does a given state contain the resource, i.e. is it entangled?
- What is the set of states we can transform a given state into under the restricted set of operations, e.g. LOCC?

A similar resource theory is given by the second law of thermodynamics where we know that one equilibrium state can be transformed adiabatically into another one if and only if the entropy increases in the process.

It is known that for a finite number of identically prepared quantum system entanglement under LOCC is in general irreversible for both pure and mixed states. However in the case of infinitely many identical copies of a pure state bipartite entanglement can be interconverted reversibly [1]. For mixed states, however, this asymptotic reversibility under LOCC operations is lost [2],[3].

There are more general sets of operations for which entanglement manipulation might become reversible again. One such example is the set of positive partial transpose preserving operations (ppt-operations) [4], which are all those completely positive maps that map the set of ppt-states into itself. Asymptotic reversibility under such a class of operations would lead to a unique entanglement measure and impose a unique ordering on entangled states thereby playing a role similar to entropy in thermodynamics.

**Partial Results**

- Under ppt-operations there are some mixed states, the totally anti-symmetric Werner states , that can be reversible converted into pure singlet states in the asymptotic limit [5].
- In [6] it was shown that under certain conditions and for a set of operations (denoted Hyper-set in [6]) that is smaller than ppt-operations and strictly larger than LOCC asymptotic irreversibility persists.
- In [7] it is shown that multipartite entangled pure states are not reversible interconvertible under ppt-operations.

**Solution**

In [8] it is shown that multipartite entangled states can be reversible interconverted by asymptotically non-entangling operations and the unique entanglement measure in this context is given by the regularized relative entropy of entanglement given by:

where is the usual qunatum relative entropy defined as:

More precisely they showed that for two multipartite states and there exists a sequence of asymptotically non-entangling operations such that:

if, and only if

.

For further details for example the precise definition of asymptotically non entangling operations and a proof of the theorem see [8]. These results establish a reversible theory in the multipartite case, whether this set of operations is minmal for bipartite entanlged states or whether ppt-operations are sufficient in the bipartite setting remain open aspects of the problem.

**References**

[1] C.H. Bennett, H.J. Bernstein, S. Popescu and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046 (1996) and quant-ph/9511030 (1995).

[2] G. Vidal and J.I. Cirac, Irreversibility in asymptotic manipulations of entanglement, Phys. Rev. Lett. 86, 5803 (2002) and quant-ph/0102036 (2001)

[3] <tr><td align=right valign=top>[HSS02] </td><td> M. Horodecki, A. Sen, and U. Sen, Rates of asymptotic entanglement transformations for bipartite mixed states: maximally entangled states are not special, Phys. Rev. A 67, 062314 (2003) and quant-ph/0207031 (2002).

[4] E.M. Rains, A semidednite program for distillable entanglement, IEEE T. Inform. Theory 47, 2921 (2001) and quant-ph/0008047 (2000).

[5] K. Audenaert, M.B. Plenio and J. Eisert, Entanglement cost under positive-partial-transpose-preserving operations, Phys. Rev. Lett. 90, 027901 (2003)

[6] M. Horodecki, J.Oppenheim and R. Horodecki, Are the laws of entanglement theory thermodynamical?, Phys. Rev. Lett. 89, 240403 (2002) and quantph/0207177 (2002).

[7] Ishizaka and M. B. Plenio, Multiparticle entanglement under asymptotic positive-partial-transpose-preserving operations, Phys. Rev. A 72, 042325 (2005) and quant-ph/0503025 (2005).

[8] F.Brandão and M.B. Plenio, A Reversible Theory of Entanglement and its Relation to the Second Law, CMP 295(3) (2010) and arXiv:0710.5827