Optimal error regions for quantum state estimation

Abstract

The strategies of bottom-up design of inorganic structures from biological templates enable cheap, eco-friendly and efficient fabrication of nano-structured materials. Here, template assembly of silica nanostructures were achieved using different protein hydrogels. Ovalbumin and fibrinogen gels were prepared by heat treatment at different pHs and protein concentrations. These hydrogels have been morphologically (SEM) and mechanically (rheology) well characterized. Next, a silica precursor is added, the condensation reaction is initiated and finally the protein hydrogel template is removed by calcination. A variety of 3D nanostructures ranging from highly porosity structures to spherical particles have been identified and characterized. Furthermore, it was observed that the fractal dimension of silica structures follow the same pattern than their corresponding templates. Consequently, the bio-scaffolding method proposed here helps the bottom-up assembly of silica precursors in nanostructures with defined three dimensional dimensions and provides a versatile route for the design of new architectures under green conditions. " 2013 The Royal Society of Chemistry.",,,,,,"10.1039/c3ra42204f",,,"http://hdl.handle.net/20.500.12104/42047","http://www.scopus.com/inward/record.url?eid=2-s2.0-84887362158&partnerID=40&md5=0254b95010772e10498523acc3901edf",,,,,,"46",,"RSC Advances",,"24256

24265",,"3",,"Scopus

WOS",,,,,,,,,,,,"Identifying emerging trends of protein hydrogels for biological scaffolding",,"Article" "45173","123456789/35008",,"Shang, J., Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore; Ng, H.K., Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore, Applied Physics Lab, DSO National Laboratories, 20 Science Park Drive, Singapore 118230, Singapore, Yale-NUS College, 6 College Avenue East, Singapore 138614, Singapore; Sehrawat, A., Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore, Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico; Li, X., Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore; Englert, B.-G., Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore, Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore",,"Shang, J.

Ng, H.K.

Sehrawat, A.

Li, X.

Englert, B.-G.",,"2013",,"An estimator is a state that represents one's best guess of the actual state of the quantum system for the given data. Such estimators are points in the state space. To be statistically meaningful, they have to be endowed with error regions, the generalization of error bars beyond one dimension. As opposed to standard ad hoc constructions of error regions, we introduce the maximum-likelihood region - the region of largest likelihood among all regions of the same size - as the natural counterpart of the popular maximum-likelihood estimator. Here, the size of a region is its prior probability. A related concept is the smallest credible region - the smallest region with pre-chosen posterior probability. In both cases, the optimal error region has constant likelihood on its boundary. This surprisingly simple characterization permits concise reporting of the error regions, even in high-dimensional problems. For illustration, we identify optimal error regions for single-qubit and two-qubit states from computer-generated data that simulate incomplete tomography with few measured copies. " IOP Publishing and Deutsche Physikalische Gesellschaft.