A physical system is determined by a finite set of initial conditions and laws represented by equations. The system is computable if we can solve the equations in all instances using a ``finite body of mathematical knowledge". In this case, if the laws of the system can be coded into a computer program, then given the system's initial conditions of the system, one can compute the system's evolution. This scenario is tacitly taken for granted. But is this reasonable? The answer is negative, and a straightforward example is when the initial conditions or equations use irrational numbers, like Chaitin's Omega Number: no program can deal with such numbers because of their ``infinity''. Are there incomputable physical systems? This question has been theoretically studied in the last 30--40 years. This article presents a class of quantum protocols producing quantum random bits. Theoretically, we prove that every infinite sequence generated by these quantum protocols is strongly incomputable -- no algorithm computing any bit of such a sequence can be proved correct. This theoretical result is not only more robust than the ones in the literature: experimental results support and complement it.

翻译：物理系统由一组有限的初始条件和以方程形式表示的定律决定。若我们能用“有限的数学知识体系”求解所有情况下的方程，则该系统是可计算的。此时，若系统定律可编码为计算机程序，则给定系统初始条件后，即可计算其演化过程。这种场景被默认为理所当然。但这是否合理？答案是否定的，一个直观的例子是当初始条件或方程涉及无理数时——比如蔡廷的Ω数：没有任何程序能处理这类数因其“无限性”。是否存在不可计算的物理系统？这一问题在过去三四十年间得到了理论研究。本文提出了一类能产生量子随机比特的量子协议。理论上，我们证明这些量子协议生成的每个无穷序列都是强不可计算的——没有任何算法能通过正确性证明来计算此类序列中的任意比特位。这一理论结果不仅比文献中的现有结论更具鲁棒性，实验数据也为其提供了支持与补充。