10 Maximum-Likelihood Estimationin Experimental Quantum Physics

Abstract

The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system -a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. 10.1 Introduction Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system. However, in experimental physics one often faces a different question: " Given the outcomes of a particular set of measurements, what quantum state do they imply? " Such inverse problems may arise for instance in the stage of setting up and calibrating laboratory sources of quantum states, in the analysis of decoherence and other deteriorating effects of the environ-ment, or in some special tasks in quantum information processing such as eavesdropping on a quantum channel in quantum cryptography. Quantum state reconstruction is a highly nontrivial problem. The quan-tum state of a system, however simple, cannot be determined by a single measurement. Repeated identical measurements performed on multiple iden-tical copies of a quantum state generally will not yield the complete infor-mation about the ensemble in question. Such a set of measurements will, G. Badurek et al., Maximum-Likelihood Estimation in Experimental Quantum Physics, Lect.