In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.
In this thesis we study the problem of continuous measurement on an open quantum system.The starting point of our analysis is the time-evolution of an open quantum system where the evolution of the environment is not explicitly described, i.e. traced out.More explicitly we restrict to time-evolutions having the structure of a semigroup or evolution system.That is, for time-points 0 ≤ r ≤ s ≤ t there exists a two-parameter family of completely-positive maps E * (t, s), such that E * (t, s)E * (s, r) = E * (t, r) and the systems state evolved according to ρ(tPhysically this means that in our Ansatz the interaction between system and environment is already fixed and can only be weak.The main goal of this thesis is now to describe, mathematically rigorous, all continuous measurements compatible with the given evolution.A measurement is compatible with the given time-evolution, if the evolution does not change when measurement results are ignored.In other words the measurement is indirect and does not further disturb the systems evolution.Our results can be interpreted as a rigorous construction of continuous matrix product states, and are related to the Hudson-Parthasarathy quantum stochastic calculus.The analysis in this thesis separates into two parts.In the first we study the class of quantum systems having an evolution system structure as described above.Precisely we study the theory of minimal solutions to a Cauchy equation with a generator of Gorini-Kossakowski-Sudarshan-Lindblad type.We do this for the case of time-dependent and unbounded generators.This analysis culminates in the description of a counterexample due to Holevo of a semigroup which is not the minimal solution to such a standard Lindblad equation.In the second part we construct a description of all measurements compatible with a given evolution and thus achieve our main goal.The analysis starts with the discrete-time version of the problem, i.e.where the time-points r, s, t in the above description of the evolution are chosen from a finite set.This discrete-time problem is completely solved by Stinespring dilation theory.We then construct our solution of the continuous-time case as a refinement limit over arbitrary discretizations of the evolution.Hence, our construction gives a continuous-time analogue of the Stinespring dilation for semigroups and evolution systems.It can be interpreted as a complete quantum description of the information the system radiates into the environment, i.e. it yields a quantum state of the environment.We complement the description of the state with a description of the observables for continuous measurements and the calculation expectation values.
In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.