Consider an arbitary positive semidefinite operator ρ, acting on ℂ^A ⊗ ℂ^B ⊗ ℂ^C, for A,B,C finite. Also, let P be an orthogonal projector on ℂ^B ⊗ ℂ^C . For the sake of concision, I will write R = 1_ℂ^A ⊗ P ; this of course is also an orthogonal projector. Consider the completely positive transformation

M(ρ) = (1 − R) ρ (1 − R) + R ρ R .

As R is an orthogonal projector, it is easy to show that || M(ρ) ||₂ ≤ || ρ ||₂ . This is because we may represent ρ as matrix in a basis consisting of the eigenvectors of R; if we divide ρ into block according to rows/columns representing vectors in the image or the kernel of R, the effect of the map M is to set the non-diagonal blocks to zero.

I am interested in how the map M may similarly affect the operator 2 norm of reduced operators on ℂ^A ⊗ ℂ^B. So I would like to know:

Is it also true that || tr_C( M(ρ) ) ||₂ ≤ || tr_C(ρ) ||₂  —  where tr_C is the trace operator acting on ℂ^C, taken in tensor product with 1_ℂ^A ⊗ 1_ℂ^B ?