3 corrected obvious typo

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 ℂB ⊗ ℂC throughout), B=C=2, and taking

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

for standard basis vectors ej for ℂ2. Note that P is the projector onto the antisymmetric subspace of ℂB ⊗ ℂC. The map M may then be re-presented as

M(ρ) = ½ ρ + ½ UρU*   where U = 1B ⊗ 1C − 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 a state an operator ρ for which the desired bound does not hold, by taking a tensor product of a state an operator with low 2-norm with one of high 2-norm, e.g.

ρ = 1Be1e1*.

We then have trC(ρ) = 1B , which has a 2-norm of $\sqrt 2$ ; and trC(UρU*) = 2 e1e1*, which has a 2-norm of $2$. By the convexity of the 2-norm, we may then show that || trC( M(ρ) ) ||2 > || trC(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 1B ⊗ 1 ℂC .

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

2 corrected minor error

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 ℂB ⊗ ℂC throughout), B=C=2, and taking

P = ¼ ½ ψψ*  where ψ = e1 ⊗ e2 − e2 ⊗ e1

for standard basis vectors ej for ℂ2. Note that P is the projector onto the antisymmetric subspace of ℂB ⊗ ℂC. The map M may then be re-presented as

M(ρ) = ½ ρ + ½ UρU*   where U = 1B ⊗ 1C − 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 a state ρ for which the desired bound does not hold, by taking a tensor product of a state with low 2-norm with one of high 2-norm, e.g.

ρ = 1Be1e1*.

We then have trC(ρ) = 1B , which has a 2-norm of $\sqrt 2$ ; and trC(UρU*) = 2 e1e1*, which has a 2-norm of $2$. By the convexity of the 2-norm, we may then show that || trC( M(ρ) ) ||2 > || trC(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 1B ⊗ 1C .

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

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 ℂB ⊗ ℂC throughout), B=C=2, and taking

P = ¼ ψψ*  where ψ = e1 ⊗ e2 − e2 ⊗ e1

for standard basis vectors ej for ℂ2. Note that P is the projector onto the antisymmetric subspace of ℂB ⊗ ℂC. The map M may then be re-presented as

M(ρ) = ½ ρ + ½ UρU*   where U = 1B ⊗ 1C − 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 a state ρ for which the desired bound does not hold, by taking a tensor product of a state with low 2-norm with one of high 2-norm, e.g.

ρ = 1Be1e1*.

We then have trC(ρ) = 1B , which has a 2-norm of $\sqrt 2$ ; and trC(UρU*) = 2 e1e1*, which has a 2-norm of $2$. By the convexity of the 2-norm, we may then show that || trC( M(ρ) ) ||2 > || trC(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 1B ⊗ 1C .

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