Bounds on operator 2-norms on partial traces of linearly related operators

Question

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 ?

Answer 1

Okay, it turns out in retrospect that the problem is trivial. The answer is "no": such a bound does not hold in general.

A simple counterexample is yielded by taking A=1 (so that we effectively deal with ℂ^B ⊗ ℂ^C throughout), B=C=2, and taking

P = ½ ψψ*  where ψ = e₁ ⊗ e₂ − e₂ ⊗ e₁

for standard basis vectors e_j for ℂ². Note that P is the projector onto the antisymmetric subspace of ℂ^B ⊗ ℂ^C. The map M may then be re-presented as

M(ρ) = ½ ρ + ½ UρU*   where U = 1_ℂ^B ⊗ 1_ℂ^C − 2P.

The operator U is unitary, and has the effect of 'swapping' the two spaces B and C; that is, for all tensor products α ⊗ β , we have U(α ⊗ β) = β ⊗ α . We may then construct an operator ρ for which the desired bound does not hold, by taking a tensor product of an operator with low 2-norm with one of high 2-norm, e.g.

ρ = 1_ℂ^B ⊗ e₁e₁*.

We then have tr_C(ρ) = 1_ℂ^B , which has a 2-norm of $\sqrt 2$ ; and tr_C(UρU*) = 2 e₁e₁*, which has a 2-norm of $2$. By the convexity of the 2-norm, we may then show that || tr_C( M(ρ) ) ||₂ > || tr_C(ρ) ||₂ for this choice of P and ρ. A similar construction can be made for any B=C>1, and letting P be the projector onto the antisymmetric space of ℂ^B ⊗ ℂ^C .

I'm interested now in what upper bounds may be obtained for || tr_C( M(ρ) ) ||₂ − || tr_C(ρ) ||₂ , or related quantities, in the case that P is a rank-1 projector on ℂ^B ⊗ ℂ^C . If anyone can show such an interesting such bound, I may 'accept' it; but for the meantime, this answers my original question.