Static assumptions

For every finite labeled qubit set $S$, consider

where $\mathcal{D}({\mathcal{H}}_S)$ denotes density operators. Require:

Diagonal anchoring.

If $\rho$ is diagonal, $F_S(\rho )=\mathrm{diag}\rho$.

Marginal compatibility.

For $T\subseteq S$,

$${\mathrm{marg}}_T{F}_S(\rho )=F_T({\rho }_T).$$

Continuity.

Every $F_S$ is continuous.

Positivity, when imposed.

Every coordinate is nonnegative.

Set $F_{\varnothing }(1)=1$. Compatibility then implies normalization. Without positivity, call the family signed.

Algebraic partial-trace compatibility does not describe how a first measurement prepares a state for a second measurement.