# Have "backwards-compatible" states on local QFT algebras been studied?

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There is a class of states on a local QFT algebra that are "backwards-compatible" in a natural sense I'll define below -- has this class been identified and studied? Any pointers would be hugely helpful in my research, and greatly appreciated!

Order the normal states of local von Neumann algebras by how "informative" they are, in terms of their support projections:

$\gamma \leq \omega \iff$ range$(s_\gamma) \subseteq$ range$(s_\omega)$.

Note 1: $s_\gamma, s_\omega$ are computed in the respective algebras on which $\gamma, \omega$ are states; the algebras may be distinct, but so long as they act on the same Hilbert space, the ranges can be meaningfully compared.

Note 2: if $\omega$ is a normal state on $\mathcal{S}$ and $\mathcal{R} \subseteq \mathcal{S}$, then $\omega$ is $\leq$ ("at least as informative as") its restriction $\omega \upharpoonright \mathcal{R}$; the inequality may or may not be strict.

Note 3: we assume each local algebra $\mathcal{R}$ is type III and so has no pure normal states; thus for each normal state $\omega$ on $\mathcal{R}$, there exist others that are strictly $< \omega$.

Call normal states $\omega, \omega'$ "compatible in $\mathcal{R}$" if there exists a normal state $\tau$ on $\mathcal{R}$ such that $\tau \leq \omega$ and $\tau \leq \omega'$. (By Note 1, $\omega, \omega'$ need not be states on $\mathcal{R}$.)

When $\mathcal{R} \subseteq \mathcal{S}$ and $\omega$ is a normal state on $\mathcal{S}$, call $\omega$ "backwards-compatible to $\mathcal{R}$" if every normal state $\gamma$ on $\mathcal{R}$ satisfying $\gamma \leq (\omega \upharpoonright \mathcal{R})$ is compatible with $\omega$ in $\mathcal{S}$.

Now let $\mathcal{O}$ be the set of open regions of some spacetime, and let $\{ \mathcal{R}(O) : O \in \mathcal{O} \}$ be our "typical net of local algebras."

Call a normal state $\omega$ on $\mathcal{R}(O_1)$ "backwards-compatible" for our net if it is backwards-compatible to $\mathcal{R}(O_2)$ for every $O_2 \subseteq O_1$ in $\mathcal{O}$.

For a spacetime region to be in a backwards-compatible state means something like this: any measurement designed in any sub-region to get strictly more information about that sub-region, would put it in a state compatible with what is known about the larger region. This seems like an excellent property for a state to have!

Has anyone identified or studied backwards-compatible states (presumably using a different term)?

Has anyone proved that nontrivial examples exist in typical AQFT models? I can prove, using the "split inclusion" property of such models, that for distinct $O_1 \subseteq O_2$, there exists a normal state $\omega$ on $\mathcal{R}(O_2)$, backwards-compatible to $\mathcal{R}(O_1)$, such that $\omega < (\omega \upharpoonright \mathcal{R}(O_1))$ (strictly). But it seems harder to find $\omega$ having these properties towards *all* proper sub-regions of $\mathcal{R}(O_2)$ simultaneously.

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