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Notice removed Authoritative reference needed by Niel de Beaudrap
Bounty Ended with Chris Ramsey's answer chosen by Niel de Beaudrap
Notice added Authoritative reference needed by Niel de Beaudrap
Bounty Started worth 100 reputation by Niel de Beaudrap
Revised to focus on the case I'm principally interested in, in particular to exclude the case of the transpose
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Niel de Beaudrap
Niel de Beaudrap
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Operator norm vs. completely Completely bounded norm for unital maps with completely positive sections

Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Operator norm vs. completely bounded norm for unital maps

Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Completely bounded norm for unital maps with completely positive sections

Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$?

Erratum (trace preserving --> unital)
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Niel de Beaudrap
Niel de Beaudrap
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Consider a completely bounded trace-preservingunital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Consider a completely bounded trace-preserving map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Fixed incorrect norm in equation (the result of a question revision)
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Niel de Beaudrap
Niel de Beaudrap
  • 1.4k
  • 11
  • 21

Consider a completely bounded trace-preserving map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert_1$$$$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_n(\mathbb C)$ and any $n > 1$$\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Consider a completely bounded trace-preserving map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert_1$$ for $\mathbf A = \mathbf M_n(\mathbb C)$ and any $n > 1$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

Consider a completely bounded trace-preserving map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Is the operator norm of $\Phi$ stable under tensor products with other spaces — i.e., is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert_{\mathrm{cb}} = \bigl\lVert \Phi \otimes \mathrm{id}_{\mathbf A} \bigr\rVert$$ for $\mathbf A = \mathbf M_h(\mathbb C)$? In particular, does it hold regardless of whether $\Phi$ is positive?

N.B. It is well-known that such a property does not hold if $\Phi$ is not unital, e.g. if we have $\Phi = C - D$ for two completely positive unital maps $C,D: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. However, what I am interested in is the retraction $\Phi$ of an injective completely positive unital map $C$. (Such a map $\Phi$ will be unital, but will not be positive unless $C$ is an isometry.)

(The dual question, of course, is whether the trace-norm is equal to the completely bounded trace norm for trace-preserving maps, again without assuming positivity.)

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Niel de Beaudrap
Niel de Beaudrap
  • 1.4k
  • 11
  • 21