Exploring nonclassicality in quantum theory via convex optimization

The term nonclassicality refers to fundamental features of quantum theory that are incompatible with the predictions of classical physics. This thesis explores several manifestations of nonclassicality, including the negativity of the Glauber-Sudarshan function of light and spin systems, quantum entanglement, and quantum memory effects in dynamical processes. Since these phenomena are responsible for the operational advantages of modern quantum technologies, their detection and characterization is a key challenge in both theoretical and experimental quantum science.
A unifying theme throughout this work is the development of efficient, scalable methods to detect and quantify nonclassical features of quantum systems. Methodologically, the thesis is not only grounded on the formulation of nonclassicality in terms of abstract convex optimization problems, but also explores semidefinite programming relaxations, which provide tractable approximations for them.
In the first part, we introduce detection methods for nonclassicality in single quantum systems, such as light modes.
Here, we show how the theory of nonnegative polynomials can be used to optimally exploit data capturing the nonclassical nature of light.
Specifically, we show that the convex cone of nonnegative polynomials can reveal nonclassicality in data even when it is hidden from standard detection methods up to now. Further, inspired by the nonclassicality of quantum particles, we introduce Wigner representations in generalized probabilistic theories and provide conditions under which they are unique.
Turning to correlations between several systems, in the second part, we develop the polytope approximation technique for certifying separability and detecting entanglement in both bipartite and multipartite systems. This
leads to an algorithm which, for practical purposes, conclusively recognizes bipartite separability for small and medium-size dimensions. For multipartite systems, the approach allows characterizing a range of different separability classes for up to five qubits or three qutrits.
Finally, we systematically identify quantum states showing subtle forms of multipartite entanglement, such as strongly entangled three-qubit states which are separable in each bipartite split.
In the last part, we turn to nonclassical phenomena in quantum systems that evolve in time.
Specifically we investigate channel discrimination protocols under memory restrictions and the detection of quantum memory effects in quantum dynamical processes. By formulating these problems using the concept of convex optimization, we provide operational criteria to distinguish classical memory from genuine quantum memory. Practically, this allows us to systematically prove the presence of genuine quantum memory in spontaneous emission processes and to propose schemes for its experimental detection.