Ancilla-Depth Phase Diagrams for Quantum Reference-Frame Comparison

Comparing two noisy quantum reference frames as statistical experiments depends on the dimension of the ancillary memory available to the decision procedure.

For finite-dimensional channels A and B with invertible A, we show that exact simulation of all measurements assisted by an r-dimensional ancilla is equivalent to r-positivity of the unique factor Gamma = BA^{-1}.

The hierarchy can be realized by physical channel pairs: every unital, trace-preserving map that is k-positive but not (k+1)-positive embeds as the factor between the channels D_a and Gamma composed with D_a on an exact interval determined by the smallest Choi eigenvalue.

For depolarizing source and target channels D_a and D_b, including negative and singular source parameters, the phase boundary is $\mathcal{D}_a \succeq_r \mathcal{D}_b \Longleftrightarrow -1/(dr-1) \leq b/a \leq 1$ for $a\neq 0$.

We derive closed formulas for the restricted level-r deficiency and for the distance to every physical post-processing, $\delta_{\mathrm{phys}}(\mathcal{D}_b\mid\mathcal{D}_a)=(1-1/d^2)\operatorname{dist}(b,I_a)$, where $I_a=\operatorname{conv}\{a,-a/(d^2-1)\}$.

The largest physical conversion cost hidden from all tests through level k is $(d-k)/[d(d^2-1)]$.

An untouched m-level spectator changes the first detecting external level from k+1 to $\lfloor k/m\rfloor+1$.

A transpose--depolarizing construction shows that the separation is not confined to depolarizing factors.

The results quantify the distinction between ancilla-restricted statistical simulation and implementation by a single quantum channel.