Debabrata Ghoshal

Weak measurements, weak values, and entanglement

As a resource, quantum entanglement provides enormous power to quantum information processing and quantum communication. We focus on new properties of entanglement as revealed by quantum weak measurements. Weak measurements are performed between a pre-selected and post-selected states, one, or both or which are entangled.

A statistical and comparative study of quantum walks under weak measurements and weak values regimes

Quantum walks have been studied under several regimes. Motivated by experimental results on quantum weak measurements and weak values as well as by the need to develop new insights for quantum algorithm development, we are extending our knowledge by studying the behavior of quantum walks under the regime of quantum weak measurements and weak values of pre- and postselected measurements (QWWM hereinafter). In particular, we investigate the limiting position probability distribution and several statistical measures (such as standard deviation) of a QWWM on an infinite line, and compare such results with corresponding classical and quantum walks position probability distributions and statistical measures, stressing the differences provided by weak measurements and weak values with respect to results computed by using canonical observables. We start by producing a concise introduction to quantum weak values and quantum weak measurements. We then introduce definitions as well as both analytical and numerical results for a QWWM under Hadamard evolution and extend our analysis to quantum evolution ruled by general unitary operators. Moreover, we propose a definition and focus on the properties of mixing time of QWWM on an infinite line, followed by a comparison of known corresponding results for classical and quantum walks mixing times. We finish this paper by presenting a plausible experimental implementation of a QWWM.

NP problems, post-selection and weak measurements

If operations in a quantum computer were conditioned on the results of a subsequent post-selection measurement, then NP-complete problems could be solved in polynomial time. Using the natural connection between post-selection and NP, we show that this result is un-physical by considering constraints on new kinds of measurements which depend on the future post-selection in a non-trivial way. We review practical quantum information advantages of post-selection.

Finite temperature quantum entanglement

Some of our previous research showed some interesting results regarding the effect of non-zero temperature on a specified quantum computation. For example, our analysis revealed that more Grover iterations are required to amplify the amplitude of the solution in a quantum search problem when the system is found at some finite temperature. We want to further study the effects of temperature on quantum entanglement using a finite temperature field theoretical description. Such a framework could prove to be useful for the understanding of computational dynamics inside a quantum computer. Other issues that we will address in our discussion include analytical descriptions of the effects of the temperature in the Von Newman entropy and others as a measure of entanglement.

Quantum computation in the analysis of Hyperspectral data

Recent research on the topic of quantum computation provides us with some quantum algorithms with higher efficiency and speedup compared to their classical counterparts. In this paper, it is our intent to provide the results of our investigation of several applications of such quantum algorithms - especially the Grovers Search algorithm - in the analysis of Hyperspectral Data. We found many parallels with Grovers method in existing data processing work that make use of classical spectral matching algorithms. Our efforts also included the study of several methods dealing with hyperspectral image analysis work where classical computation methods involving large data sets could be replaced with quantum computation methods. The crux of the problem in computation involving a hyperspectral image data cube is to convert the large amount of data in high dimensional space to real information. Currently, using the classical model, different time consuming methods and steps are necessary to analyze these data including: Animation, Minimum Noise Fraction Transform, Pixel Purity Index algorithm, N-dimensional scatter plot, Identification of Endmember spectra - are such steps. If a quantum model of computation involving hyperspectral image data can be developed and formalized - it is highly likely that information retrieval from hyperspectral image data cubes would be a much easier process and the final information content would be much more meaningful and timely. In this case, dimensionality would not be a curse, but a blessing.

Quantum walk search factors in the regime of weak measurement

Our previous work brought some interesting results of the discrete Quantum Walks in the regime of Weak Measurement (QWWM or QWWV). Using the knowledge of such results of QWWM, we are now exploring the search algorithms and investigating the factors associated with such walk. The study of such factors like dimensionality, connectivity of the dataset and the strength of disorder or percolation are already studied by others in the context of general quantum walks. It is our interest to show the similarities and/or differences of such factors of general quantum walks with QWWV. The subject of decoherence in quantum walks is another challenging research topic at present. We are also exploring the topic of decoherence in QWWM or QWWV.

Finite Temperature Quantum Algorithm And Majorization

It is often believed that quantum entanglement plays an important role in the speed-up of quantum algorithms. In addition, a few research groups have found that Majorization behavior may also play an important role in some quantum algorithms. In some of our previous work we showed that for a simple spin 1/2 system, consisting of two or three qubits, the value of a Groverian entanglement (a rather useful measure of entanglement) varies inversely with the temperature. In practical terms this means that more iterations of the Grovers algorithm may be needed when a quantum computer is working at finite temperature. That is, the performance of a quantum algorithm suffers due to temperature-dependent changes on the density matrix of the system. Most recently, we have been interested in the behavior of Majorization for the same types of quantum system, and we are trying to determine the relationship between Groverian entanglement and Majorization at finite temperature. As Majorization entails the probability distribution arising out of the evolving quantum state from the probabilities of the final outcomes, our study will reveal how Majorization affects the evolution of Grovers algorithm at finite temperature.