New Methods In The Study of Partial Tracing Random States and the Disordered Majumdar Ghosh Model

Abstract

newline Abstract newlineIn this thesis three closely related works have been done. We develop a novel method newlinefor the partial tracing operation, which is not only faster but also the most accurate newlinemethod reported till now to the best of our knowledge. The partial trace operation newlineis an important mathematical operation used in a plethora of applications like calculating entanglement measures, calculating subsystem reduced density matrices etc newlineWe then investigate the two-qubit reduced density matrix using the partial trace newlinemethod developed, for the superposed state of an entangled 2N qubit Bell state and newline2N qubit completely random pure state. We then embark into the entanglement newlinestudy of the disordered Majumdar-Ghosh model and its interesting ramifications. newlineIn the first two chapters the essential prerequisites are motivated which is very newlineimportant in understanding the main results of this thesis in Chapters 3, 4, 5 which newlinealso acts as a brief review of the literature of this field. In the first chapter, we have newlinediscussed the development of the concept of entanglement and its measures in detail. newlineWe then proceed further into the development of quantum spin chain systems, and newlinehow entanglement helps us detect a possible quantum phase transitions in a spin newlinechain system. Vast majority of entanglement measures are calculated using the newlinepartial trace operation, and calculating partial trace becomes computationally very newlineintensive with increasing number of qubits as the Hilbert space dimension increases newlineexponentially. Therefore in the third chapter of this thesis, we discuss about our newlinenew method of partial tracing which is based on set theory, to be specific, the power newlineset method and it is much more efficient than the presently known methods. The newlineproposed method of partial tracing overcomes all the limitations of the other well newlineknown methods such as being computationally intensive and being limited to lower newlinedimensional Hilbert spaces. We give a detailed theoretical description of our method newlineand also provide an explicit example