Entanglement and coherence are fundamental properties of quantum systems, promising to power near-future quantum technologies, such as quantum computation, quantum communication, and quantum metrology. Yet, their quantification, rather than mere detection, generally requires reconstructing the spectrum of quantum states, i.e., experimentally challenging measurement sets that increase exponentially with the system size. Here, we demonstrate quantitative bounds to operationally useful entanglement and coherence that are universally valid, analytically computable, and experimentally friendly. Specifically, our main theoretical results are lower and upper bounds to the coherent information and the relative entropy of coherence in terms of local and global purities of quantum states. To validate our proposal, we experimentally implement two purity detection methods in an optical system: shadow estimation with random measurements and collective measurements on pairs of state copies. The experiment shows that both the coherent information and the relative entropy of coherence of pure and mixed unknown quantum states can be bounded by purity functions. Our research offers an efficient means of verifying large-scale quantum information processing.

纠缠和相干性是量子系统的基本属性，有望为量子计算、量子通信和量子计量等近期量子技术提供动力。然而，它们的量化，而不仅仅是检测，通常需要重建量子态的光谱，即，具有实验挑战性的测量集，这些测量集随着系统尺寸呈指数级增长。在这里，我们证明了可操作的有用纠缠和相干性的定量界限，这些界限普遍有效、分析可计算且实验友好。具体来说，我们的主要理论结果是相干信息的下界和上限以及量子态局部和全局纯度方面相干性的相对熵。为了验证我们的建议，我们在光学系统中实验性地实现了两种纯度检测方法：随机测量的阴影估计和状态副本对的集体测量。实验表明，纯未知量子态和混合未知量子态的相干信息和相干性相对熵都可以受到纯度函数的限制。我们的研究提供了验证大规模量子信息处理的有效方法。