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This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space representations. Please suggest the adequate tags if you think I'm missing some.

Let $V\subset\mathcal B(\mathcal H)$ be a subspace of Hermitian operators in finite-dimensional Hilbert space $\mathcal H$, containing $1$. Let $\{X_i,Y_j\}$, (with $1\leq i\leq r$ and $1\leq j\leq d$) be a basis of $V$ and let $\{\tilde X_i,\tilde Y_j\}$ be the dual basis w.r.t the Hilbert-Schmidt inner product \begin{align} \mathrm{tr}[X_i \tilde X_{i'}]=\delta_{ii'}, \quad \mathrm{tr}[Y_j \tilde Y_{j'}]=\delta_{jj'},\quad \mathrm{tr}[X_i \tilde Y_{j}]=0, \quad \mathrm{tr}[Y_i \tilde X_{j}]=0. \end{align} Note that defining $\alpha_i=\mathrm{tr}[\tilde X_i]$ and $\beta_j=\mathrm{tr}[\tilde Y_j]$ we have \begin{align} 1=\sum_{i=1}^r \alpha_i X_i+\sum_{j=1}^d \beta_j Y_j, \end{align} however not all $\alpha_i$'s are zero, meaning that $1\notin\mathrm{span}\{Y_j\}$.

Regard $\mathbb{R}^n\otimes\mathcal B(\mathcal H)$ as the algebra of $n$-tuples of elements from $\mathcal B(\mathcal H)$, as in operator systems theory. Let $\mathcal C\subset \mathbb{R}^r\otimes\mathcal B(\mathcal H)$ be the pointed cone defined by

\begin{align} \mathcal C=\left\{Z=\{Z_i\}\Big|\exists \{K_j\}\in\mathbb{R}^d\otimes\mathcal B(\mathcal H)\mathrm{~s.t.~} \sum_{i=1}^r X_i\otimes Z_i+\sum_{j=1}^d Y_j\otimes K_j\geq0\right\}. \end{align}

Question: Under what conditions does $\{\tilde X_i^\top\}\in\mathcal C$?

Note: If $1\in\mathrm{span}\{Y_j\}$ the answer is trivial: Always. This is however, not my case.

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