On the quantum theory of sequential measurements

Willis E. Lamb

John van de Wetering

We study general physical systems with a notion of sequential measurement that satisfies a few simple and intuitive conditions. We show that, apart from two exceptional cases, the systems satisfying these conditions are exactly the real, complex or quaternionic quantum systems. Moreover, only the complex quantum systems remain if systems must be allowed to be composed together in a locally tomographic manner. In other words, quantum theory is the unique locally tomographic theory allowing sequential measurement.

Tuomas Tammero

This thesis studies the field of quantum measurements and more precisely sequential measurements. The aim is to provide answers to questions like when is performing a sequential or a repeated measurement meaningful, or even possible. This is important in order to understand what type of information can be acquired when performing measurements on quantum states. In basic quantum theory observables are identified with self-adjoint operators. We introduce the concept of a positive operator valued measure (POVM) to gain better understanding on observables. The downside to POVMs is that they don't produce a new state after the measurement, only measurement statistics. In order to perform multiple measurements on a quantum state, the concepts of measurement model and instrument are defined in this thesis. Using instruments to perform measurements is essential in order to describe a state after the initial measurement is performed. This way it's possible to make measurements sequentially. After defining the necessary tools for sequential measurements and a few different properties related to them, some applicable situations for these kind of measurements are shown. This way sequential measurements end up being somewhat meaningful in order to gain information from quantum systems.

Alberto Barchielli

Jingliang Gao

We present two quantum union bounds for sequential projective measurements. These bounds estimate the disturbance accumulation and probability of outcomes when the measurements are performed sequentially. These results are based on a trigonometric representation of quantum states and should have wide application in quantum information theory for information-processing tasks such as communication and state discrimination, and perhaps even in the analysis of quantum algorithms.

W. H. Furry