Theses

The technological application of quantum features relies upon the certification of the corresponding desirable properties of quantum states and processes. As our technical capabilities advance, so does our need for understanding complex properties that go beyond simple qubit entangled states and causal processes. This cumulative thesis addresses two of the current challenges in quantum state and process characterisation. First, we study the certification of high-dimensional entanglement that arises when systems are entangled in more than two degrees of freedom. Second, we study the certification of processes with indefinite causal order and the problem of channel discrimination. High-dimensional encoding of quantum information provides a promising method of transcending current limitations in quantum communication. The main goal of this part of this thesis was to develop practically implementable tools to certify the dimensionality of the entanglement taking into consideration the least possible amount of assumptions about the state, and when viable, no assumptions at all. In the first work, we developed an adaptive method that certifies the Schmidt number of a state using only two global product measurements. We put our method to test in an experiment that certified entanglement in 9 dimensions on a state encoded in the orbital angular momentum of two photons. When studying the most general transformations that act on a pair of quantum operations, an intriguing phenomenon emerges: some higher-order transformations may act on their input operations in an indefinite causal order. Such non-causal properties have found several theoretical advantages, from communication complexity to quantum computing. In the second part of the thesis, a formalism of certification of non-causal properties under different levels of assumptions was developed. This formalism can witness indefinite causal order in device-dependent, independent, and semi-device-independent experiments, while showing that a prominent higher-order operation, the quantum switch, can demonstrate stronger non-causal properties than what was previously known. For the other works in this part of the thesis, the focus was on the famous quantum information problem of channel discrimination, proving several novel instances of tasks for which sequential strategies outperform parallel ones, both for sets of unitary and general channels. We defined new classes of strategies for channel discrimination that employ indefinite causal order and proved them to be advantageous when compared to causally ordered ones. These strategies may themselves be interpreted as methods for certification of indefinite causal order.

Einstein-Podolsky-Rosen steering is a form of quantum correlation intermediate between entanglement and Bell nonlocality. Given that Alice and Bob share a quantum state, we say Alice can steer Bob when it is possible for her to condition Bob’s state by interacting only with her subsystems, in such a way that the resulting correlation between her inputs, her outcomes, and Bob’s conditioned states exhibits nonlocal properties. In this dissertation, we study the phenomenon of quantum steering from a fundamental perspective and as a semi-device-independent certification of entanglement. We formalize steering by making comparisons with other well studied quantum correlations and by investigating the hierarchy between them. We explore the close relationship between steering and joint measurability and present the phenomenon of one-way steering. The main focus of this work is the problem of determining the steerability of quantum states subjected to restricted measurement scenarios. In each scenario, we search for the family of measurements that reveals highest steerability of a given quantum state. We develop and apply a variety of methods for performing this task that are based on semidefinite programming. Analytical solutions for a particular family of measurements are achieved by exploring the theory of joint measurability and constructing steering inequalities. Finally, we study the relevance of general POVMs in exhibiting steering when compared to projective measurements.