Certifying quantum states and their properties through correlations, communication and tomography

Quantum systems offer significant advantages in information processing and communication, but leveraging these benefits requires reliable methods for extracting relevant information. Due to the probabilistic nature of quantum mechanics, the no-cloning theorem, and the existence of incompatible observables, measuring quantum states is an inherently indirect and often costly process. As engineered quantum systems scale beyond the laboratory, the ability to efficiently certify quantum states and their properties becomes indispensable.
This thesis examines quantum state certification from three perspectives: correlations, communication, and tomography. Correlations play a key role in security protocols, while communication scenarios highlight potential applications of quantum systems in information transmission. Beyond practical applications, they can also reveal fundamental capabilities and limitations of quantum information. Tomography, on the other hand, provides techniques for reconstructing unknown quantum states and is essential not only for validating quantum devices, but also for extracting results from simulations and computations. Our main contributions include:
• Establishing a connection between sum-of-squares uncertainty relations and graph-theoretic quantities.
• Constructing a complete set of criteria for certifying entanglement dimension with an untrusted adversary.
• Proving an equivalence between classical and quantum communication with entangled parties.
• Identifying a class of communication games for which having an entanglement dimension larger than the communication dimension can be useful.
• Providing rigorous statistical guarantees for quantum state tomography experiments applicable to any experimental setup.
• Connecting the optimal measurement choices for partial state tomography to graph covering problems.
• Proposing a protocol for characterizing one- and two-body observables in native fermionic simulators.