More precisely, there is a smooth function $Q_{\boldsymbol{M}}$ on $\boldsymbol{M}$, constructed locally from the metric of $\boldsymbol{M}$ (so that the family $(Q_{\boldsymbol{M}})$ is local covariant), such that $$\begin{aligned}
 \label{TminusQ} 
\langle {\sf T}_{\boldsymbol{M}ab}(x) \rangle_{\omega_{\boldsymbol{M}}}  & =
 \langle \tilde{\sf T}_{\boldsymbol{M}ab}(x) \rangle_{\omega_{\boldsymbol{M}}}
 - Q_{\boldsymbol{M}}(x) g_{ab}(x)\,,\end{aligned}$$ where $g_{ab}$ denotes the spacetime metric of $\boldsymbol{M}$, defines an expectation value of the stress energy tensor which is divergence-free. Moreover, with this definition, the conditions [**LCSE-1...4**]{} hold when interpreted as valid for expectation values of states fulfilling the Hadamard condition [@WaldQFTCST; @BFV] (possibly after suitable symmetrization in order to obtain [**LCSE-4**]{}). This may actually be seen as one of the main motivations for introducing the Hadamard condition in [@Wald78], and in the same paper, Wald proposed conditions on the expectation values of the stress energy tensor which are variants of our [**LCSE-1...4**]{} above. Wald [@Wald78; @WaldQFTCST] also proved that, if there are two differing definitions for the expectation values of the stress energy tensor complying with the conditions, then the difference is given by a state-independent, local covariant family $(C_{\boldsymbol{Ma}b})$ of smooth, symmetric, divergence-free tensor fields, in complete analogy to our discussion above. Actually, the formulation of the local covariance condition for the expectation values of the 