Robust self-testing of (almost) all pure two-qubit states
Tim Coopmans

Abstract:
In a nonlocal scenario, physically isolated players each have a device that inputs and outputs classical information. Certain correlations between the joint input and output of the devices almost uniquely identify the quantum state that they share. This phenomenon is known as self-testing and has applications in quantum cryptography with untrusted devices. It was for example shown that for every pure two-qubit state, there exists a two-player Bell experiment whose correlations, or rather the Bell value that is computed from the correlations, can be used here to self-test that state; the Bell value that is used stems from a family of Bell inequalities called tilted CHSH inequalities.  A special case is the regular CHSH inequality, which is used to self-test the singlet, a maximally entangled state of two qubits. For practical applications, estimation errors and the presence of external noise require self-testing statements to be robust to errors.
In this thesis, we extend previous work on self-testing of the singlet with the CHSH game. First, we use tilted CHSH inequalities to improve the robustness of previously found self-testing statements for (almost) all pure partially entangled states. Our result consists of the explicit construction of local quantum channels for the two players, from which we derive operator inequalities that we verify numerically. Using a recently developed method, the improved bounds can be inferred. Furthermore, we construct a state that violates the CHSH inequality but for which there exist no local quantum channels that achieve greater fidelity than a trivial lower bound (i.e. achieve fidelity with the singlet greater than what is achievable using a separable state). This result implies that CHSH violation is not sufficient for the two players to `extract' a singlet from their actual state by just local operations. Future research could focus on extending our results to different self-testable states such as the GHZ-states and on self-testing in the scenario where only one of the two players has a potentially untrusted device (quantum steering).