用于量子参考系比较的辅助深度相图
Ancilla-Depth Phase Diagrams for Quantum Reference-Frame Comparison
AI总结：
研究量子参考系比较中辅助存储器维度的作用，通过推导公式等方法，分析了信道因子的正性、相位边界等，量化了辅助受限统计模拟与单量子信道实现的区别。
AI中文摘要：
将两个有噪声的量子参考系作为统计实验进行比较，取决于决策过程可用的辅助存储器维度。对于有限维且 A 可逆的信道 A 和 B，我们表明由 r 维辅助器辅助的所有测量的精确模拟等同于唯一因子 Gamma = BA^{-1} 的 r 正性。该层次结构可由物理信道对实现。对于去极化源信道和目标信道 D_a 和 D_b，包括负的和奇异的源参数，相位边界为 $\mathcal{D}_a \succeq_r \mathcal{D}_b \Longleftrightarrow -1/(dr - 1) \leq b/a \leq 1$（当 $a\neq 0$ 时）。我们推导了受限 r 级缺陷和到每个物理后处理距离的封闭公式。通过 k 级隐藏于所有测试的最大物理转换成本为 $(d - k)/[d(d^2 - 1)]$。未受影响的 m 级旁观者会将第一个检测外部级别从 k + 1 变为 $\lfloor k/m\rfloor + 1$。转置 - 去极化构造表明这种分离不限于去极化因子。结果量化了辅助受限统计模拟与单量子信道实现之间的区别。
英文摘要：
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, $δ_{\mathrm{phys}}(\mathcal{D}_b\mid\mathcal{D}_a)=(1-1/d^2)\operatorname{dist}(b,I_a)$, 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.
