Finding operator sum decomposition of $T(X)=AX+XA$

Question

Suppose $A$ is a positive definite $dtimes d$ 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?

rnva · Accepted Answer

The map $T(X)$ is not necessarily positive. Consider
$$A = begin{pmatrix}1&0 0&2end{pmatrix},quad X = begin{pmatrix}1&-1 -1&1end{pmatrix}.$$
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.

Finding operator sum decomposition of $T(X)=AX+XA$

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