二分可观测量的分类I：稳定与一致稳定可观测量

中文摘要 AI 辅助

当两个量子系统被独立制备且各自被推向理想化极限时，二分可观测量的期望值可能取决于先对哪个系统进行理想化。我们将不出现这种情况的可观测量称为稳定可观测量。我们证明：若先对探测系统进行理想化，则可观测量稳定当且仅当对一个系统的每一种理想化制备，都能通过实际制备的有限混合以任意精度、在所有探测上一致地模拟；对于稳定可观测量，两种理想化制备具有唯一的对称独立乘积。在一致稳定性下，n个自适应探测在固定边际下可区分的制备数量受限于n的多项式，且响应可通过有限次错误学习；否则，在某一边际下，n个探测可区分2ⁿ种制备。交换测试、Hong–Ou–Mandel符合及离散相等测试是一致稳定的，其界限与维度无关，而对无上界的离散谱的序比较是不稳定的。这些证明基于一个经典事实：单一双重极限条件以不同名称出现在拓扑学（Grothendieck）、泛函分析（Krivine–Maurey）、代数学（Arens）、概率论（Simons）、组合学（Pták）中，在模型论中则表现为Shelah稳定性；该独立乘积对应Harrington引理。模型论为这些条件提供了共同语言，我们将其作为罗塞塔石碑，将每个物理问题与其数学表述配对。

英文摘要

When two quantum systems are prepared independently and each is driven to an idealized limit, the expectation of a bipartite observable may depend on which system is idealized first. We call an observable stable when it does not. We show that an observable is stable if and only if every idealized preparation of one system can be simulated to arbitrary accuracy, uniformly over all probes, by finite mixtures of actual preparations, provided that the probing system is idealized first; on stable observables, two idealized preparations have a unique, symmetric independent product. Under a uniform form of stability, the number of preparations that $n$ adaptive probes can tell apart at a fixed margin is bounded by a polynomial in $n$, and responses can be learned with a bounded number of mistakes; otherwise, at some margin, $n$ probes tell apart $2^n$ preparations. The swap test, Hong--Ou--Mandel coincidences and discrete equality tests are uniformly stable, with bounds independent of the dimension, while order comparisons of discrete spectra unbounded above are unstable. The proofs rest on the classical fact that a single double limit condition appears, under different names, in topology (Grothendieck), functional analysis (Krivine--Maurey), algebra (Arens), probability (Simons) and combinatorics (Pták), and in model theory as Shelah's stability; the independent product corresponds to Harrington's lemma. Model theory supplies a common language for these conditions; we use it as a Rosetta stone, pairing each physical question with its mathematical formulation.

发表机构

• The University of Texas at San Antonio（圣安东尼奥德克萨斯大学）

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