Superadditivity of Quantum Relative Entropy for General States

The property of superadditivity of the quantum relative entropy states that, in a bipartite system H
_AB
= H
_A
⊗ H
_B
, for every density operator ρ
_AB
, one has D(ρ
_AB
||σ
_A
⊗ σ
_B
)≥ D(ρ
_A
||σ
_A
) + D(ρB||σB). In this paper, we provide an extension of this inequality for arbitrary density operators σ
_AB
. More specifically, we prove that α(σ
_AB
)· D(ρ
_AB
||σ
_AB
)≥D(ρ
_A
||σ
_A
)+ D(ρ
_B
||σ
_B
) holds for all bipartite states ρ
_AB
and σ
_AB
, where α(σ
_AB
) = 1 + 2||σ
_A
^-1/2
⊗ σ
_AB
σ
_A
^-1/2
⊗ σ
_B
^-1/2
- ||
_AB
||
_∞
.