量子理论最大程度地违反了进动协议的 Tsirelson 不等式
Tsirelson's Inequality for the Precession Protocol is Maximally Violated by Quantum Theory
AI总结：
针对进动协议中 Tsirelson 不等式的最大违反问题，本文以理论无关方式给出仅依赖测量结果极值的一般界，并构造证明量子理论总能饱和该界。
摘要：
进动协议涉及测量 $P_3$，即一个均匀进动的可观测量（例如谐振子的位置，或经历空间旋转的坐标）在三个等间隔时刻之一为正的概率。Tsirelson 不等式指出，在经典理论中 $P_3 \leq 2/3$，但在量子理论中却被某些态违反。在这篇 Letter 中，我们处理关于该不等式的一些开放问题：量子理论中可能出现的对 Tsirelson 不等式的最大违反是多少？其他理论是否可能做得更好？通过以独立于理论的方式考察具有有限多个结果的系统的进动协议，我们导出了最大可能违反的一个一般界。任何理论，只要其期望值是可观测量的线性函数——包括经典理论、量子理论以及所有广义概率理论——都必须满足这个独立于理论的界；该界仅依赖于最小的正测量结果和负测量结果。给定任意两个这样的值，我们通过构造证明，量子理论总是使该界达到饱和。一些值得注意的例子包括自旋 $3/2$ 粒子的角动量，以及在进动协议中表现优于量子谐振子的一族可观测量。最后，我们还将所得结果与最近提出的受约束条件概率概念联系起来。
英文摘要：
The precession protocol involves measuring $P_3$, the probability that a uniformly precessing observable (like the position of a harmonic oscillator or a coordinate undergoing spatial rotation) is positive at one of three equally spaced times. Tsirelson's inequality, which states that $P_3 \leq 2/3$ in classical theory, is violated in quantum theory by certain states. In this Letter, we address some open questions about the inequality: What is the maximum violation of Tsirelson's inequality possible in quantum theory? Might other theories do better? By considering the precession protocol in a theory-independent manner for systems with finitely many outcomes, we derive a general bound for the maximum possible violation. This theory-independent bound must be satisfied by any theory whose expectation values are linear functions of observables -- which includes classical, quantum, and all general probabilistic theories -- and depends only on the minimum positive and negative measurement outcomes. Given any such two values, we prove by construction that quantum theory always saturates this bound. Some notable examples include the angular momentum of a spin-$3/2$ particle and a family of observables that outperform the quantum harmonic oscillator in the precession protocol. Finally, we also relate our findings to the recently introduced notion of constrained conditional probabilities.
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