Optimal quantum correlations for (semi)-device-independent applications

Abstract

Quantum mechanics dramatically revolutionized our understanding of nature, challenged traditional information theory, and led to significant technological advancements. Two of the most counterintuitive features of quantum mechanics are quantum nonlocality and quantum contextuality. These features rule out hidden-variable models that attempt to explain quantum phenomena based on classical intuition. Quantum non-locality, first revealed by Bell's theorem in 1964, rules out local hidden-variable (LHV) models, originating from the famous Einstein-Podolsky-Rosen paradox. Meanwhile, quantum contextuality, originally pinpointed by the Kochen-Specker (KS) theorem in 1967, focuses on single-system correlations rather than between space-like separated systems, demonstrating quantum mechanics is incompatible with non-contextual hidden-variable (NCHV) models.

Quantum correlations are crucial in several applications, both in computing and communication tasks. They also enable the strongest form of cryptographic security through the device-independent (DI) paradigm, where devices are treated as black-boxes, and could even be supplied by potentially malicious adversaries. The semi-device-independent (SDI) paradigm offers a more practical solution by balancing security and feasibility, allowing for some assumptions about devices. Since different DI and SDI applications require specific properties, this thesis focuses on designing optimal quantum correlations for these applications.

DI randomness amplification (DIRA) is an important task in DI information processing, that converts a weak, biased seed into a fully random one. Quantum correlations on the boundary of the no-signaling set are necessary for this task. We introduce a class of almost-quantum Bell inequalities and develop a tilted version of the celebrated Hardy’s paradox showing corresponding quantum correlations enhance the robustness and efficiency in state-of-the-art DIRA protocols.

Another direction pursued in the thesis is the study of self-testing, a DI approach to certify the state and measurements of quantum systems in a unique manner. We investigated three families of quantum correlations, the tilted Hardy tests, which we prove self-test any two-qubit pure entangled state, except the maximally-entangled one, the $\alpha$-CHSH inequalities, where we prove robust self-test of the singlet and corresponding measurements, and weighted-Braunstein-Caves chained inequalities, which we prove delineate a boundary for the quantum correlation set, extending beyond the well-known Tsirelson-Landau-Masanes boundary.

A third main direction explored in the thesis is the efficient construction of novel state-dependent and state-independent contextuality proof. We provide an optimal and experiment-friendly toolbox for various contextuality-based applications, including demonstrating entanglement-assisted advantages in zero-error communication, separating binary-outcome no-signaling models from quantum theory, and achieving optimal randomness rate in the SDI paradigm. We also discuss and address issues arising from the noise and errors in real-world experimental tests of the KS theorem, and show that quantum mechanics also violates NCHV models under arbitrary relaxation of the so-called non-disturbance conditions.

Finally, a counter-intuitive phenomenon, termed ``bound randomness'', is introduced, revealing a deep resource-theoretical relationship between non-locality and DI-certified randomness. Analogous to Werner's famous result that entanglement is not sufficient for non-locality, ``bound randomness'' indicates non-locality is not always sufficient for DI randomness certification under standard spot-checking protocols. In conclusion, this thesis introduces key techniques in contextuality and non-locality for designing targeted correlations for various DI and SDI applications.