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Suppose $A$ is a positive definite $d\times d$ matrix and $T$ is a positive map over such matrices defined as follows

$$T(X)=AX+XA$$

I'm wondering if if it possible to get a decomposition of this operator, ie, set of $V_i's$ such that

$$T(X)=\sum_i V_i X V_i^T$$

and if so, how do I go about it?

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  • $\begingroup$ What constant k? $\endgroup$
    – joseph h
    Commented Oct 10, 2020 at 5:30

1 Answer 1

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The map $T(X)$ is not necessarily positive. Consider

$$A = \begin{pmatrix}1&0\\ 0&2\end{pmatrix},\quad X = \begin{pmatrix}1&-1\\ -1&1\end{pmatrix}.$$

The matrix $AX +XA$ is not positive semidefinite. On the other hand, if $X$ is positive semidefinite, so is $V_iXV_i^T$ (or $V_iXV_i^\dagger$ when considering complex-valued vectors) for any $V_i$.

Hence, an operator sum decomposition of the form you ask is impossible.

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  • $\begingroup$ I realize you said positive definite, not semidefinite, but the argument is the same. Simply set $X_{22} = 1 + \varepsilon$ for $\varepsilon > 0$. $\endgroup$
    – rnva
    Commented Oct 10, 2020 at 16:33

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