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Reversible entanglement manipulation |
| contact: |
M. B. Plenio |
date: |
08 Feb 2005 |
last progress: |
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solved by: |
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The concept of entanglement as a resource motivates the study of
its transformation properties under certain classes of operations
such as local operations and classical communication (LOCC).
For a finite number of identically prepared quantum systems the
manipulation of entanglement under LOCC is generally irreversible,
both for pure and mixed states. In the asymptotic limit of
infinitely many identical copies of a pure state, in contrast,
pure bi-partite entanglement can be interconverted reversibly
[BBPS96]. For mixed states, however, this asymptotic
reversibility under LOCC operations is lost [VC02,HSS02].
However, 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) [Ra00] which are all those
completely positive maps that map the set of ppt-states into
itself. It has been shown that under ppt-operations there are some
mixed states that can be reversible converted into pure singlet
states in the asymptotic limit [APE03]. This has
been proven for the totally anti-symmetric Werner state and weak
numerical evidence suggests that this is true for all Werner
states [Pl]. On the other hand in [HOH02]
it was shown that under certain conditions and for a set of
operations (denoted Hyper-set in [HOH02]) that is
smaller than ppt-operations and strictly larger than LOCC
asymptotic irreversibility persists.
Asymptotic reversibility under 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.
The following are open questions:
- Are ppt-operations sufficient to ensure asymptotically
reversibly interconversion of all, i.e. pure and mixed,
bi-partite entangled states [BFC]?
- What is the smallest non-trivial class of operations that permits
asymptotically reversibly interconversion of all, i.e. pure and
mixed, bi-partite entangled states [Bet]?
| [BBPS96] | 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). |
| [VC02] | G. Vidal and J.I. Cirac, »Irreversibility in
asymptotic manipulations of entanglement«, Phys. Rev. Lett. 86, 5803
(2002) and quant-ph/0102036
(2001). |
| [HSS02] | 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). |
| [Ra00] | E.M. Rains, »A semidefinite program for distillable
entanglement«, IEEE T. Inform. Theory 47, 2921 (2001) and
quant-ph/0008047 (2000). |
| [APE03] | K. Audenaert, M.B. Plenio and J. Eisert,
»Entanglement cost under positive-partial-transpose-preserving
operations«, Phys. Rev. Lett. 90, 027901 (2003) and
quant-ph/0207146 (2002). |
| [HOH02] | M. Horodecki, J.Oppenheim and R. Horodecki, »Are
the laws of entanglement theory thermodynamical?«, Phys. Rev. Lett.
89, 240403 (2002) and
quant-ph/0207177 (2002). |
| [Pl] | M.B. Plenio, unpublished |
| [BFC] | This was boldly conjectured by the author and is in certain
circles known as the Big-Fat-Conjecture. |
| [Bet] | The existence of such a class is the subject of a
bet between Michal Horodecki and Reinhard Werner. |
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