# Integrated Information in Relational Quantum Dynamics (RQD)
## 6 Preferred Observers: The Max-\(\Phi\) Principle  
We have shown that no observer (CPTP map) can increase integrated information (Theorem 1). A natural question arises: which observer loses the least integrated information? In other words, suppose we have a system in state \(\rho\), and we are allowed to \"observe\" it in some fashion, perhaps by making a measurement or by coarse-graining its degrees of freedom. Different observation schemes will preserve different fractions of the holistic correlations present in \(\rho\). Is there a principled way to choose the optimal observation that retains as much of \(\Phi(\rho)\) as possible? Remarkably, the answer is yes. We call this the Max-\(\Phi\) observer principle: among any reasonable family of observation channels, one can find a channel \(F^{*}\) that maximizes the integrated information of the observed state \(F(\rho)\). Moreover, \(F^{*}\) can be interpreted as selecting the \"best\" basis or representation, in which to obtain classical information about \(\rho\) without destroying its holistic structure. This has deep connections to the idea of a preferred basis in quantum decoherence theory [19]: the basis that is stable and preserves certain correlations. Here, we derive such a basis from first principles, using only the monotonicity of \(\Phi\) and the compactness of the channel space. **Theorem 4** (Optimal Observer Channel): _Let \(\mathcal{F}\) be any compact set of CPTP maps (quantum channels) that one considers possible observers on the state \(\rho\). For example, \(\mathcal{F}\) could be the set of all local projective measurements of each qubit (with the observer recording classical outcomes), or the set of all partial trace operations onto some subsystem, or more generally any parametrized family of channels. Then there exists at least one channel \(F^{*}\in\mathcal{F}\) that maximizes the integrated information of the output:_  
\[F^{*}\ =\ \arg\max_{F\in\mathcal{F}}\ \Phi\big{(}F(\rho)\big{)}\,. \tag{24}\]Existence is guaranteed; there may in principle be more than one maximizer, but we can pick one. In other words, there is an optimal observer \(F^{*}\) that preserves \(\rho\)'s integrated information better than any other in \(\mathcal{F}\). Furthermore,  
\[\Phi(F^{*}(\rho))=\max_{F\in\mathcal{F}}\Phi(F(\rho))\leq\Phi(\rho), \tag{25}\]  
with equality only if the optimal channel effectively does nothing to disturb the relevant correlations. For example, \(F^{*}\) could be the identity channel or an isometry embedding into a larger space. Proof. : Because \(\Phi(F(\rho))\) is a function with real value in the set of channels and \(\mathcal{F}\) is compact, the maximum value is attained (by the extreme value theorem). We only need to argue for the continuity of \(F\mapsto\Phi(F(\rho))\) in a suitable topology in the space of CPTP maps. If we parametrize channels by their finite-dimensional Kraus operators, small changes in those operators yield small changes in the output state (in trace norm, say), which in turn yield small changes in \(\Phi\) (by Lipschitz continuity of \(\Phi\)). Thus, \(\Phi(F(\rho))\) is a continuous function in the compact set \(\mathcal{F}\), and therefore there is a maximum. The inequality \(\Phi(F(\rho))\leq\Phi(\rho)\) for all \(F\) is just Theorem 1, so the maximal value is bounded by \(\Phi(\rho)\). The equality \(\Phi(F(\rho))=\Phi(\rho)\) would require that  
\[D^{Q}_{\mathrm{S}}(F(\rho)\|F(\sigma))=D^{Q}_{\mathrm{S}}(\rho\|\sigma) \tag{26}\]  
for some product state \(\sigma\) that achieves \(\Phi(\rho)\). In practice, this implies that \(F\) does not erase any of the distinguishing information between \(\rho\) and \(\sigma\)--essentially, \(F\) is invertible on the support of \(\rho,\sigma\)--which typically means that \(F\) is an embedding or trivial operation. Theorem 4 is simple but profound. It tells us that, given any constraint on how we observe the system, we can find a \"best\" way to observe it, if our criterion is to preserve integrated information.