Additivity of Entanglement of Formation

Cite this problem as Problem 7.

Problem

The entanglement of formation [1] is one of the standard measures of entanglement. It is defined, for any density operator $\rho$ on a bipartite system, as

$E_F(\rho)=\text{inf} \left\{ \sum_ir_i S(\rho_i|A)\,\Big|\, \sum_ir_i \rho_i = \rho \right\},$

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 $\rho_i$ are density operators and the $r_i$ 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

$E_F(\rho) \le E_F(\rho^{(1)})+ E_F(\rho^{(2)})$ .

The problem is to show that equality always holds here.

Background

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.

Partial Results

The additivity of entanglement of formation could be proven for several examples of states by Vidal et al. [2].

This problem has been shown to be equivalent to problem 10: Additivity of classical capacity and related problems

Solution

See problem 10: Additivity of classical capacity and related problems

References

[1] 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).

[2] G. Vidal, W. Dür, and J. I. Cirac, Entanglement cost of mixed states, Phys. Rev. Lett. 89, 027901 (2002) and quant-ph/0112131 (2001).