2026/07/31 by Maxim V. Churilov
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.MP
12 pages, 3 figures; includes ancillary verification code and a deterministic regression certificate
arxiv created 2026/08/05 · arxiv updated 2026/08/06
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 Da and Gamma composed with Da on an exact interval determined by the smallest Choi eigenvalue. For depolarizing source and target channels Da and Db, including negative and singular source parameters, the phase boundary is Da \succeqr Db \Longleftrightarrow -1/(dr-1) ≤ b/a ≤ 1 for a≠ 0. We derive closed formulas for the restricted level-r deficiency and for the distance to every physical post-processing, δphys(Db\midDa)=(1-1/d2)dist(b,Ia), where Ia=conv\a,-a/(d2-1)\. The largest physical conversion cost hidden from all tests through level k is (d-k)/[d(d2-1)]. 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.