The Heisenberg–Robertson uncertainty relation bounds the product of the variances in the two possible measurement outcomes in terms of the expectation of the commutator of the observables. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two noncompatible observables. Here, we give two stronger uncertainty relations, relating to the sum of variances with respect to density matrix, whose lower bounds are guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system; moreover, two stronger uncertainty relations in terms of the product of the variances of two observables are established. Also, several stronger uncertainty relations for three observables are established, relating to the sum and product of variances with respect to density matrix, respectively.

中文翻译：

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混合状态的不确定性关系更强

海森堡-罗伯逊不确定性关系根据可观测物换向器的期望，限制了两种可能的测量结果中方差的乘积。值得注意的是，它没有捕获不兼容可观测对象的概念，因为它可能是微不足道的，即，即使对于两个不兼容的可观测对象，下界也可以为空。在这里，我们给出了两个更强的不确定性关系，它们与密度矩阵的方差之和有关，只要两个可观测值在系统状态不兼容，就可以保证其下限是不平凡的。此外，根据两个可观测值的方差的乘积，建立了两个更强的不确定性关系。此外，针对三个可观察物建立了几个更强的不确定性关系，