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Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, we have $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert \Phi \rVert$$ where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, for which $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert \Phi \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert \Phi \rVert ? \tag{2}\end{align} $$

(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$.

For such $\Psi$, consider a choice of retraction $\Phi$ and integer $n > 0$. Consider a Hermitian operator $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, and $\mathbf v, \mathbf w \in \mathbb{C}^{kn}$ be the eigenvectors of $(\Phi \otimes \mathrm{id}_n)(E)$ with the highest and lowest eigenvalues $\lambda_{\max}$ and $\lambda_{\min}$ respectively. Suppose that $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) = \lambda_{\max} - \lambda_{\min} > 2 \lVert \Phi \rVert ;$$ does it follow that $\mathbf v = (1 \otimes U)\, \mathbf w$ for some isometry $U \in \mathrm{U}_n(\mathbb C)$? 

For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, such that $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert \Phi \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert \Phi \rVert, \tag{2}\end{align} $$ but where the associated eigenvectors $\mathbf v$, $\mathbf w$ of $(\Phi \otimes \mathrm{id}_n)(E)$ do not merely differ by an isometry on the second tensor factor?

(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above. Note that I have made a significant revision to this question to exclude a construction which I had intended but failed to exclude, which follows as a corollary to the answer to that question.)

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$, we have $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert E \rVert$$ where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ for which $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert E \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert E \rVert ? \tag{2}\end{align} $$

(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, we have $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert \Phi \rVert$$ where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ with unit operator norm, for which $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert \Phi \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert \Phi \rVert ? \tag{2}\end{align} $$

(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$, we have $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert E \rVert$$ where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ in which $\Phi = \Psi^{-1}$ is unique, is there a choice of retraction $\Phi$ for which $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert E \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert E \rVert, \tag{2}\end{align} $$ for some choice of $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$?

(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$, we have $$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert E \rVert$$ where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ for which $$\begin{align} \lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert E \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert E \rVert ? \tag{2}\end{align} $$

(This question is a follow-up to a previous question, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)