Abstract: The aim of this talk is to outline the results obtained by me (during my PhD studies) and my collaborators.

In quantum information theory, there is a rich collection of analytic tools to study tensor product of Hilbert spaces. Geometric quantization attaches Hilbert spaces to symplectic manifolds. The principal objective of the first set of study, supervised by T Barron, is to investigate how the geometry of the manifold influences the quantum information theoretic aspects of the Hilbert space and vice versa. For instance, in one of the works, we presented an asymptotic result for the average entropy over all the pure states on the Hilbert space , where is a Hermitian ample line bundle over a compact complex manifold . In another work, we associated states with subsets of a product of two integral Kahler manifolds using a quantum line bundle in a particular manner. We proved that the state associated this way is separable when the subset is a finite union of products.

In the second part of the talk, we present a work on quantum circuit synthesis, joint with A.R. Kalra, D. Valluri, S. Winnick and J. Yard. In classical computing, we choose a small set of special gates (known as a universal gate set) and make circuits using these gates to generate any classical gates (Boolean functions). However, the quantum version of circuit synthesis is a bit more complicated. In this talk, we will introduce what it means to be a universal gate set in quantum computing and see some examples. Finally, we shall present our construction of an exact circuit synthesis algorithm of unitaries in the groups and over the multi-qutrit Clifford+T universal gate set with the of help of ancilla.

Abstract: Monotone functions over the real numbers are very well-behaved compared to general measurable functions. Consequently, a wide variety of techniques and applications are in place for working with them. In this talk, we explore the notion of an ordered core, which allows us to define core decreasing functions and generalize monotone functions to general measure spaces without reliance on a strict ordering among elements.

We will begin by introducing a definition of monotone functions compatible with the ordered core. This allows us to extend the down space construction, a variant of the Köthe dual restricted to positive decreasing functions, to all measure spaces. We will look at their associate spaces and their relationship with a suitable version of the least decreasing majorant construction in this more general setting. We will discuss the interpolation structure of these spaces and find strong similarities to the real line case; the down spaces corresponding to L1 and L∞ form an exact Calderón couple and as a consequence, we can describe all their exact interpolation spaces in terms of the K−functional.

We illustrate the versatility and adaptability of this generalized perspective on decreasing functions by proving a new characterization for the boundedness of an abstract Hardy operator between L^1 to L^q with general measures.