Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on ℂB ⊗ ℂC . For the sake of concision, I will write R = 1ℂA ⊗ 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 ℂA ⊗ ℂB. So I would like to know:
Is it also true that || trC( M(ρ) ) ||2 ≤ || trC(ρ) ||2 — where trC is the trace operator acting on ℂC, taken in tensor product with 1ℂA ⊗ 1ℂB ?