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



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> M(ρ) = (1 − R) ρ (1 − R) + R ρ R .
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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 ?



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



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> P = ½ **ψ****ψ\***  where **ψ** = **e**1 ⊗ **e**2 − **e**2 ⊗ **e**1
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for standard basis vectors **e**j for ℂ2. Note that P is the projector onto the antisymmetric subspace of ℂB ⊗ ℂC. The map M may then be re-presented as



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> M(ρ) = ½ ρ + ½ UρU\*   where U = 1ℂB ⊗ 1ℂC − 2P.
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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 = 1ℂB ⊗ **e**1**e**1\*.
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We then have trC(ρ) = 1ℂB , which has a 2-norm of $\sqrt 2$ ; and trC(UρU\*) = 2 **e**1**e**1\*, 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 ℂB ⊗ 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.


