Entanglement

Entanglement is the property of quantum mechanics that there can be stronger correlations between subsystems than what is possible in any classical theory. It is important for many phenomena, and for such novel ideas as quantum teleportation, quantum cryptography and some quantum computer algorithms it is used as a resource.

In that respect it is important to quantify entanglement. Our project is to gain a better understanding of entanglement by understanding the geometry of entanglement.

Relevant articles

Recent articles

Simplifying monotonicity conditions for entanglement; Michal Horodecki.
Previously Guifre Vidal has shown some necessary and sufficient inequalities that convex measures of entanglement has to satisfy to be monotone under Local Operations and Classical Communication. In this paper M. Horodecki use this result to derive two equalities that are necessary and sufficient conditions for monotonicity under LOCC. Again for convex measures, such as the minimal geometric distance to the set of separable states.
The new conditions looks to be easier to use in proving the monotonicity under LOCC for a given measure, and they are also much stronger since they are equalities. With the only inequality hidden in the convexity of the measure.

Review works

Measures of entanglement in quantum mechanics; Geir Myhr.
A master thesis from Jan Myrheim's student Geir Myhr on entanglement measures

Bipartite entanglement - criteria, measures, applications; Ingmar Weber.
This is a 34 page manuscript, not an article, on bipartite entanglement. Topics such as distillation, negativity, teleportation and dense coding.

Relativity of Pure States Entanglement; Zyczkowski and Bengtsson.
Analysis of various measures of entanglement, with geometric interpretations.

Characterizing entanglement; Dagmar Bruss.
A good overviem of entanglement criteria, entanglement types and entanglement measures.

Separability criteria

Separability of mixed states: Necessary and sufficient conditions; Horodeckis
This is the article where they first prove that for the 2x2 and 2x3 case, ppt is sufficient for separability.

Separability criterion for density matrices; Asher Peres.
This is the articles where Peres finds his criterion, that PPT is necessary for separability.

Geometric distance as entanglement measure

On the geometry of entanglement; Verstraete et al.
Article with an algorithm for finding the closest PPT state.

Optimal entanglement witnesses for qubits and qutrits; Bertlmann et al.
The optimal entanglement witness is the same as the closest separable state in the Hilbert-Schmidt metric.

Entanglement measures and the Hilbert-Schmidt distance; Masanao Ozawa.
This article shows that we do not know if the entanglement measure of the shortest distance to a separable state in the Hilbert-Schmidt metric, is non-increasing under LOCC.

Monotone metrics on matrix spaces; Denes Petz.
Some mathematics on monotone metrics, including Hilbert-Schmidt, that might be useful.

Measures of entanglement

Concentrating partial entanglement by local operations; Bennet et al.
In this article they show that one can take an ensemble of non-maximally entangled pairs of spin half particles and distill out a lesser amount of pairs of bells state pairs.

Computable measure of entanglement; Vidal and Werner.
Here they introduce negativity as a measure of entanglement and show it does not increase under local manipulations of the system.

Mixed state entanglement and distillation: Is there a "bound" entanglement in nature? ; Horodeckis.
This is the article where the concept of bound entanglement is introduced, with a direct example of a ppt entangled state.

Monotonicity under LOCC

Affine maps of density matrices; Thomas F. Jordan
A short article on complete positivity and unitary evolution.

Assorted

A classical analogue of entanglement; Collins and Popescu
An article in which they find what sort of non-allowed devices are needed for having quantum correlations in a classical system.