Quantum physics has revolutionised our way of conceiving nature and is now bringing about a new technological revolution. The use of quantum information in technology promises to supersede the so-called classical devices used nowadays. Understanding what features are inherently non-classical is crucial for reaching better-than-classical performance. This thesis focuses on two nonclassical behaviours: quantum contextuality and Wigner negativity. To date, contextuality has mostly been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. In those scenarios, contextuality has been shown to be necessary and sufficient for advantages in some cases. On the other hand, negativity of the Wigner function is another unsettling non-classical feature of quantum states that originates from phase-space formulation in quantum optics. Wigner negativity is known to be a necessary resource for quantum speedup. We set out a robust framework for properly treating contextuality in continuous variables. We quantify contextuality in such scenarios by using tools from infinite-dimensional optimisation theory. Building upon this, we show that Wigner negativity is equivalent to contextuality in continuous variables with respect to Pauli measurements. We then introduce experimentally-friendly witnesses for Wigner negativity of multimode quantum states, based on fidelities with Fock states which again uses infinite-dimensional linear programming techniques. We further extend the range of previously known discrete-variable results linking contextuality and advantage into a new territory of discrete variable information retrieval.
Quantum physics has revolutionised our way of conceiving nature and is now bringing about a new technological revolution. The use of quantum information in technology promises to supersede the so-called classical devices used nowadays. Understanding what features are inherently non-classical is crucial for reaching better-than-classical performance. This thesis focuses on two nonclassical behaviours: quantum contextuality and Wigner negativity. To date, contextuality has mostly been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. In those scenarios, contextuality has been shown to be necessary and sufficient for advantages in some cases. On the other hand, negativity of the Wigner function is another unsettling non-classical feature of quantum states that originates from phase-space formulation in quantum optics. Wigner negativity is known to be a necessary resource for quantum speedup. We set out a robust framework for properly treating contextuality in continuous variables. We quantify contextuality in such scenarios by using tools from infinite-dimensional optimisation theory. Building upon this, we show that Wigner negativity is equivalent to contextuality in continuous variables with respect to Pauli measurements. We then introduce experimentally-friendly witnesses for Wigner negativity of multimode quantum states, based on fidelities with Fock states which again uses infinite-dimensional linear programming techniques. We further extend the range of previously known discrete-variable results linking contextuality and advantage into a new territory of discrete variable information retrieval.