- Definition Properties
- Logarithmic negativity Properties
- References
In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability.[1] It has shown to be an entanglement monotone [2][3] and hence a proper measure of entanglement.
Definition
The negativity of a subsystem 
can be defined in terms of a density matrix 
as:
where:
is the partial transpose of with respect to subsystem 
is the trace norm or the sum of the singular values of the operator 
.
An alternative and equivalent definition is the absolute sum of the negative eigenvalues of :
where 
are all of the eigenvalues.
Properties
- Is a convex function of :
- Is an entanglement monotone:
where 
is an arbitrary LOCC operation over
Logarithmic negativity
The logarithmic negativity is an entanglement measure which is easily computable and an upper bound to the distillable entanglement.[4]
It is defined as
where 
is the partial transpose operation and 
denotes the trace norm.
It relates to the negativity as follows:[1]
Properties
The logarithmic negativity
- can be zero even if the state is entangled (if the state is PPT entangled).
- does not reduce to the entropy of entanglement on pure states like most other entanglement measures.
- is additive on tensor products:
- is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces
(typically with increasing dimension) we can have a sequence of quantum states 
which converges to 
(typically with increasing 
) in the trace distance, but the sequence 
does not converge to 
. - is an upper bound to the distillable entanglement
References
- This page uses material from Quantwiki licensed under GNU Free Documentation License 1.2
2. ^{{cite thesis|author=J. Eisert|title=Entanglement in quantum information theory|year=2001|publisher=University of Potsdam|arxiv=quant-ph/0610253||bibcode=2006PhDT........59E}}
3. ^{{cite journal|author1=G. Vidal |author2=R. F. Werner |title=A computable measure of entanglement|journal=Phys. Rev.|year=2002|volume=A 65|series=032314|doi= 10.1103/PhysRevA.65.032314|accessdate=|arxiv = quant-ph/0102117 |bibcode = 2002PhRvA..65c2314V }}
4. ^{{cite journal|author=M. B. Plenio|title=The logarithmic negativity: A full entanglement monotone that is not convex|journal=Phys. Rev. Lett.|year=2005|volume=95|series=090503|doi=10.1103/PhysRevLett.95.090503|accessdate=|arxiv = quant-ph/0505071 |bibcode = 2005PhRvL..95i0503P|pmid=16197196|page=090503}}
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