|
|
 |
Additivity of Entanglement of Formation |
The entanglement of formation [BD96] is one of the
standard measures of entanglement. It is defined, for any density operator
\rho on a bipartite system, as
E F(\rho)=
inf { \sumi
ri S(\rhoi|A)
\Big| \sumi
ri \rhoi = \rho },
where S(.) denotes the von Neumann entropy and \rho|A denotes the
restriction of a density operator \rho to the "Alice''
subsystem (partial trace over the other subsystem), the
\rhoi are density operators and the
ri are positive, adding up to one. Since S is
concave, the infimum is attained at a convex decomposition of \rho
into pure states, and the definition is often given as this
restricted infimum.
Consider now a pair \rho(i), i=1,2 of
bipartite density operators, and their tensor product
\rho=\rho(1)\otimes\rho(2),
which lives on a tensor product of four Hilbert spaces, but can be
considered as a bipartite state when the two Alice subsepaces and
the two Bob subspaces are grouped together. Then it is easy to
show (by plugging the tensor product of the optimal decompositions
of the factors into the variational expression and using the
additivity of the entropy) that
E F(\rho)
<= E F(\rho(1))
+ E F(\rho(2)).
The problem is to show that equality always holds here.
This inequality is crucial to settle the interpretation of
E F as a resource quantity. The typical kind of
tensor products appearing in the theory are pairs created by
(maybe different) sources of entangled states, and kept for later
use.
The additivity of entanglement of formation could be proven for several
examples of states by Vidal et al. [VDC02].
This problem has been shown to be equivalent to
problem 10.
| [BD96] | C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters,
»Mixed-state entanglement and quantum error correction«,
Phys. Rev. A 54, 3824 (1996) and
quant-ph/9604024 (1996). |
| [VDC02] | G. Vidal, W. Dür, and J. I. Cirac, »Entanglement
cost of mixed states«, Phys. Rev. Lett. 89, 027901 (2002)and
quant-ph/0112131 (2001). |
[LaTeX -> HTML by ltoh]
7
