Liu Zixuan, Chiribella Giulio
QuIC, Ecole Polytechnique de Bruxelles, C.P. 165, Université Libre de Bruxelles, 1050, Brussels, Belgium.
QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China.
Nat Commun. 2025 Apr 7;16(1):3314. doi: 10.1038/s41467-025-58508-9.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson's bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order.
量子理论原则上与违反因果不等式的过程是兼容的，因果不等式是贝尔不等式的一种类似物，它限制了按确定因果顺序操作的各方所观察到的相关性。自从因果不等式被引入以来，确定它们的最大量子违背值，类似于贝尔不等式的Tsirelson界，一直是一个悬而未决的问题。在这里，我们提供了一种通用方法来界定任意因果不等式的违背情况，确定了任意局部实验以及具有不确定因果顺序的任意量子过程所能实现的相关性的极限。我们证明，最大违背值通常小于相应相关性的代数最大值，并为一类因果不等式确定了类似Tsirelson的界，其中包括一些最典型的例子。我们的结果促使人们寻找表征具有不确定因果顺序的量子相关性集合边界的物理原理。
