Ancilla-Depth Phase Diagrams for Quantum Reference-Frame Comparison

This paper investigates the operational requirements for comparing two noisy quantum reference frames. Specifically, it asks how the dimension of an ancillary quantum memory (the 'ancilla depth') constrains the ability of a decision procedure to simulate one noisy channel using another. The study aims to quantify the gap between statistical simulability—where a source channel can mimic a target channel given sufficient memory—and physical convertibility, where one channel must be transformed into another by a single physical post-processing operation.

The author models noisy reference frames as finite-dimensional quantum channels. By focusing on invertible channels, the study uniquely identifies the statistical factor relating two channels as the composition of the target channel with the inverse of the source channel. The paper utilizes the hierarchy of r-positive maps to characterize the ancilla-restricted simulation. It derives exact phase boundaries for depolarizing channels, including cases with negative source parameters, and computes the 'physical conversion cost' (diamond-norm deficiency) to measure how far these statistical simulations remain from physical implementations.

The study establishes that for depolarizing channels, the ability to simulate a target channel from a source channel is governed by the ratio of their parameters relative to the ancilla dimension $r$. The author provides a closed-form phase diagram that maps these ratios to specific ancilla depths. A key result is the quantification of the 'hidden conversion gap': even when a statistical simulation is perfect for a given ancilla level, there often remains a strictly positive distance to the nearest physical post-processing channel. The paper also provides an activation law, showing how an unused 'spectator' ancilla can reduce the memory requirements for detecting these differences.