Quantum tomography: asymptotic theory and statistical methodology

Recent experimental progress in the preparation and control of quantum systems has brought to light the importance of Quantum State Tomography (QST) in validating the results. In this thesis we investigate several aspects of QST, whose central problem is to devise estimation schemes for the recovery of an unknown state, given an ensemble of n independent identically prepared systems. The key issues in tackling QST for large dimensional systems is the construction of physically relevant low dimensional state models, and the design of appropriate measurements. Inspired by compressed sensing tomography, in chapters 4, 5 we consider the statistical problem of estimating low rank states (r ≪ d) in the set-up of Multiple Ions Tomography (MIT), where r and d are the rank and the dimension of the state respectively. We investigate how the estimation error behaves with a reduction in the number of measurement settings, compared to ‘full’ QST in two setups - Pauli and random bases measurement designs. We study the estimation errors in this ‘incomplete’ measurement setup in terms of a concentration of the Fisher information matrix. For the random bases design we demonstrate that O(r logd) settings suffice for the mean square error w.r.t the Frobenius norm to achieve the optimal O(1/n) rate of estimation. When the error functions are locally quadratic, like the Frobenius norm, then the expected error (or risk) of standard procedures achieves this optimal rate. However, for fidelity based errors such as the Bures distance we show that no ‘compressive’ recovery exists for states close to the boundary, and it is known that even with conventional ‘full’ tomography schemes the risk scales as O(1/√n) for such states and error functions. For qubit states this boundary is the surface of the Bloch sphere. Several estimators have been proposed to improve this scaling with ‘adaptive’ tomography. In chapter 6 we analyse this problem from the perspective of the maximum Bures risk over all qubit states. We propose two adaptive estimation strategies, one based on local measurements and another based on collective measurements utilising the results of quantum local asymptotic normality. We demonstrate a scaling of O(1/n) for the maximum Bures risk with both estimation strategies, and also discuss the construction of a minimax optimal estimator. In chapter 7 we return to the MIT setup and systematically compare several tomographic estimators in an extensive simulation study. We present and analyse results from this study, investigating the performance of the estimators across various states, measurement designs and error functions. Along with commonly used estimators like maximum likelihood, we propose and evaluate a few new ones. We finally introduce two web-based applications designed as tools for performing QST simulations online

Statistically efficient tomography of low rank states with incomplete measurements

The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random selection of settings, and that the number of settings can be significantly reduced with only a negligible increase in error. We present an argument to explain these findings based on a concentration inequality for the Fisher information matrix. In the more general setup of random basis measurements we use this argument to show that for certain rank r states it suffices to measure in $O(r\mathrm{log}d)$ bases to achieve the average Fisher information over all bases. We present numerical evidence for random states of up to eight atoms, which suggests that a similar behaviour holds in the case of Pauli bases measurements, for randomly chosen states. The relation to similar problems in compressed sensing is also discussed

A comparative study of estimation methods in quantum tomography

As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators. The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of [36] which point to strong optimality properties, and high computational efficiency of the projected linear estimators. To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online

Quantum tomography: asymptotic theory and statistical methodology

Recent experimental progress in the preparation and control of quantum systems has brought to light the importance of Quantum State Tomography (QST) in validating the results. In this thesis we investigate several aspects of QST, whose central problem is to devise estimation schemes for the recovery of an unknown state, given an ensemble of n independent identically prepared systems. The key issues in tackling QST for large dimensional systems is the construction of physically relevant low dimensional state models, and the design of appropriate measurements. Inspired by compressed sensing tomography, in chapters 4, 5 we consider the statistical problem of estimating low rank states (r ≪ d) in the set-up of Multiple Ions Tomography (MIT), where r and d are the rank and the dimension of the state respectively. We investigate how the estimation error behaves with a reduction in the number of measurement settings, compared to ‘full’ QST in two setups - Pauli and random bases measurement designs. We study the estimation errors in this ‘incomplete’ measurement setup in terms of a concentration of the Fisher information matrix. For the random bases design we demonstrate that O(r logd) settings suffice for the mean square error w.r.t the Frobenius norm to achieve the optimal O(1/n) rate of estimation. When the error functions are locally quadratic, like the Frobenius norm, then the expected error (or risk) of standard procedures achieves this optimal rate. However, for fidelity based errors such as the Bures distance we show that no ‘compressive’ recovery exists for states close to the boundary, and it is known that even with conventional ‘full’ tomography schemes the risk scales as O(1/√n) for such states and error functions. For qubit states this boundary is the surface of the Bloch sphere. Several estimators have been proposed to improve this scaling with ‘adaptive’ tomography. In chapter 6 we analyse this problem from the perspective of the maximum Bures risk over all qubit states. We propose two adaptive estimation strategies, one based on local measurements and another based on collective measurements utilising the results of quantum local asymptotic normality. We demonstrate a scaling of O(1/n) for the maximum Bures risk with both estimation strategies, and also discuss the construction of a minimax optimal estimator. In chapter 7 we return to the MIT setup and systematically compare several tomographic estimators in an extensive simulation study. We present and analyse results from this study, investigating the performance of the estimators across various states, measurement designs and error functions. Along with commonly used estimators like maximum likelihood, we propose and evaluate a few new ones. We finally introduce two web-based applications designed as tools for performing QST simulations online

A short term comparative evaluation of the efficacy of diode laser with desensitizing toothpastes and mouthwashes in the treatment of dentinal hypersensitivity

Treatment of dentinal hypersensitivity (DH) has always been challenging with a wide variety of therapeutic options, in-office and home care. The study objective was to compare the clinical efficacy of diode laser [DL] with four commercially available des

A clinical study of the effect of calcium sodium phosphosilicate on dentin hypersensitivity

Objective: Dentinal hypersensitivity is a commonly encountered problem with varied treatment options for its management. A large number of home use products have been tested and used for the management of dentinal hypersensitivity. This 8 week clinical trial investigates the temporal efficacy of commercially available calcium sodium phosphosilicate containing toothpaste in comparison to a potassium nitrate containing toothpaste. Methods: A total 20 subjects between the ages of 18 to 65 years were screened for a visual analogue score (VAS) for sensitivity of 5 or more by testing with a cold stimulus and randomly divided into test and positive control groups. Baseline sensitivity VAS scores to air evaporative stimulus were recorded for minimum two teeth. The subjects were prescribed respective dentifrices and revaluated for sensitivity scores at 2, 4 and 8 weeks. Results:The study demonstrated reduction in symptoms for all treatment groups from baseline to 2, 4 and 8 weeks. The calcium sodium phosphosilicate group showed a higher degree of effectiveness at reducing hypersensitivity to air evaporative stimulus at 2 weeks, than commercially available potassium nitrate. However, there was no significant difference in scores of subjects using the calcium sodium phosphosilicate toothpaste as compared to potassium nitrate at 4 weeks and 8 weeks. Conclusion: Calcium sodium phosphosilicate showed greater reduction in sensitivity compared to potassium nitrate at an earlier stage which is of high clinical value. However, based on the findings of the present study long term effects of calcium sodium phosphosilicate seem to be less promising than previously claimed

Minimax estimation of qubit states with Bures risk

The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/n. However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as 1/√n for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states. We propose qubit estimation strategies based on separate adaptive measurements, and collective measurements, that achieve 1/n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us to derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate O(n-1 log n) under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the 'standard' rate n −1

Salivary and gingival crevicular fluid histatin in periodontal health and disease

Objectives: Histatin, with its anti bacterial, anti protease, and wound closure stimulating property might influence the pathogenesis of periodontal disease. This study assessed the presence of histatin in gingival crevicular fluid (GCF); the levels of salivary and GCF histatin in periodontal disease. Material and methods: It was a cross sectional study that included systemically healthy forty five subjects (22 males and 23 females) between the age group of 20 to 45 years. Based on Gingival Index (Loe and Silness ,1963) and Russell's Periodontal Index they were grouped as 15 healthy (Group 1), 15 gingivitis (Group 2), and 15 periodontitis (Group 3) subjects. Whole pooled unstimulated saliva was collected by asking the patient to spit in a sterile container and GCF samples were collected using a micropipette from all the subjects. Histatin levels were assessed using Enzyme Linked Immunosorbent Assay (ELISA). The intergroup comparison was done by ANOVA and Mann Whitney U Test was done for pair wise comparison. Results: The results of this study show that histatin is present in saliva and gingival crevicular fluid. When the salivary histatin levels were compared it was found that the levels of histatin increase from health to periodontitis but the levels of histatin in the gingival crevicular fluid and saliva had no correlation with severity of periodontal disease as there was no statistically significant difference between the three groups. Conclusions: It can be concluded that histatin cannot be used as a potential marker of periodontal disease

Removal of Crystal Violet dye from aqueous solution using water hyacinth: Equilibrium, kinetics and thermodynamics study

Effluent water from dyeing industries has now for long been a taxing issue. Of the various dyes which are extremely toxic, Crystal Violet which is used in the dyeing industry is known for its mutagenic and mitotic poisoning nature. Water hyacinth (Eichhornia crassipes) is a perennial aquatic plant notorious for its rapid invasive growth on the surface of water bodies causing ill-effects on the biodiversity. The potential of powdered roots of water hyacinth was studied for decolorization of Crystal Violet dye. Influence of parameters such as initial pH (2.0-10.0), initial dye concentration (100-500 ppm), biosorbent dosage (0.5-5 g/l), contact time (10-240 min) and temperature (300-323 K) were examined. Maximum removal of dye was observed at pH 7.8. The obtained data were fit into different kinetic models and the biosorption was found to follow pseudo second order kinetic model. The Langmuir monolayer biosorption capacity of water hyacinth was estimated as 322.58 mg/g. The study has demonstrated water hyacinth as a potential low cost biosorbent for effective removal of Crystal Violet dye from aqueous solution