Analytic Properties of Quantum States on Manifolds

Manimugdha Saikia, The University of Western Ontario

Abstract

The principal objective of this study is to investigate how the Kahler geometry of a classical phase space influences the quantum information aspects of the quantum Hilbert space obtained from geometric quantization and vice versa. 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 states associated this way are separable when the subset is a finite union of products. We presented an asymptotic result for the average entropy over all the pure states on the Hilbert space H⁰(M₁,L₁^⊗N) ⊗ H⁰(M₂,L₂^⊗N), where H⁰(M_j,L_j^⊗N) is the space of holomorphic sections of the N-th tensor powers of hermitian ample line bundle L_j on compact complex manifolds M_j. The coefficients of this asymptotic expression capture certain topological and geometric properties of the manifold.

In another project related to quantum computing, we constructed an exact synthesis algorithm for quantum gates in the groups U_3ⁿ(Z[1/3,e^2πi/3]) and U_3ⁿ(Z[1/3,e^2πi/3]) over the multi-qutrit Clifford+T gate set with the help of ancilla.