Proposition: Static state data do not determine the operational realization

Even an exact labelled bipartite output state, and hence all its static reduced spectra and mutual-information values, can have realizations both with and without a signalling transmission. Such state data alone do not specify an intervention-based event order.

Proof. Consider the labelled Bell output

One realization distributes this state to two laboratories with no communication. For any trace-preserving local intervention ${\mathcal{I}}_A$,

Indeed, pairing with a $B$-observable and using ${\mathcal{I}}_A^{\ast }(I_A)=I_A$ proves the equality. There is no unconditional signalling from that local intervention.

Another realization prepares the Bell pair at a source, retains the $A$-register, and transmits its $B$-register to a receiver. The baseline labelled output is exactly the same ${\rho }_{AB}$. If the source is allowed to replace its outgoing register by $|0⟩⟨0|$ or $|1⟩⟨1|$, the receiver can distinguish the alternatives perfectly.

The two realizations have the same static output datum but different operational access and transmission structure. Precisely those missing declarations are not encoded by the state or its spectra. The first equality is the standard distinction between local correlations and signalling Beckman.