Continuity of quantum mutual information
Robert Alicki, M. Fannes · arXiv (Cornell University) · 2003
We prove continuity of quantum mutual information S(ρ 12 |ρ 2) with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party. This can, e.g., be used to prove the continuity of squashed entanglement. A, generally mixed, state of a bipartite system is given by a density matrix ρ 12 on a Hilbert space H 12 = H 1 ⊗ H 2. We shall, in order to avoid technical complications, restrict our attention to finite dimensional systems and not distinguish between the density matrix ρ 12 and its associated expectation functional a ↦ → ρ 12 (a): = Tr ρ 12 a, a a linear operator on H 12. The restrictions of ρ 12 to the subsystems 1 and 2 are denoted by ρ 1 and ρ 2, e.g. ρ 1 (a): = ρ 1 (a ⊗ ) = Tr ρ 12 a ⊗ , a a linear operator on H 1. The von Neumann entropy S(ρ) of a state ρ is the quantity Tr η(ρ) with η(x): = −x log x for 0 < x ≤ 1 and η(0) = 0. The mutual information S(ρ 12 | ρ 2) of ρ 12 with respect to the second system is the quantity S(ρ 12 | ρ 2): = S(ρ 12) − S(ρ 2),