Topics in quantum information -- Continuous quantum measurements and quantum walks
Martin Varbanov · PhDT · 2015
The topics presented in this thesis have continuous quantum measurements and quantum walks at their core. The first topic being discussed centers around simulating a generalize measurement with a finite number of outcomes using a continuous measurement process with a continuous measurement history. We provide conditions under which it is possible to prove that such a process exists and that at long times it simulates faithfully the generalized measurement. We give the stochastic equations governing the feedback between the measurement history and the instantaneous weak measurements. The second topic examines a definition of hitting time for continuous-time quantum walks. A crucial component for such a definition is the use of weak measurements. Several methods using alternative but equivalent definitions of weak, continuous measurements are employed to derive a formula for the hitting time. The behavior of the thus defined hitting time is studied subsequently, in general and for specific graphs. The last topic explores continuous-time quantum walks on graphs with infinite tails. The equations for propagating and bound states are derived and the S-matrix is defined. Their properties, such as orthogonality of the propagating and bound states, unitarity of the S-matrix, are discussed. Formulas for the S-matrix under operations of cutting, adding or connecting tails are derived.