טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKvietny Lev
SubjectEntanglement in Quantum States
DepartmentDepartment of Physics
Supervisor Professor Emeritus Joseph Avron
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

Entanglement is a phenomenon in quantum mechanics in which quantum states of several systems are always described in relation one to the other, even if the systems are physically remote from each other. Such a correlation cannot be described in terms of classical physics probability, but rather by a wave function that describes the joint quantum state of all the systems. In this work we examined several existing methods for identifying entanglement and for defining entangled states. The work begins with ​​the Peres Criterion for Separability. This is a simple algebraic criterion, which can be executed on a matrix that describes a quantum state consisting of two subsystems. The criterion provides a necessary condition for separability, and for the special case of 2X2 system as well as for a 2X3 system, the Peres criterion is both necessary and sufficient. Bound-Entanglement states are then described - these are entangled states of several higher dimensional systems (e.g., two qutrits and above). The density matrices of these states, though entangled, remain positive under the partial transposition condition of Peres. One way to construct such states is by using UPB (Unextendible Product Basis). For a multipartite system UPB is an incomplete orthogonal product basis whose complementary subspace contains no product state. The uniform mixed state over the subspace complementary to any UPB is a Bound-Entangled state. The main part of the work deals with a numerical method for examing entanglement and separability. This method exploits the fact that each set of separable states is convex, and is based on an iterative scheme to find the closest separable state for any chosen density matrix, with ​​no limitations of this matrix’s dimensions. Further chapters of this work describe the implementation of the algorithm in MATLAB, its execution on matrices representing different two-qubit states, and the examples of different runs made, including comparisons of the outputs of the computer program against results of the execution of the Peres Criterion on these states. In conclusion, the program implementing the algorithm classifies correctly matrices that represent two-qubit states for entanglement and separability. One can adjust the program to support the studying of the composite states of higher dimensions, where Peres criterion is only a necessary condition (but not sufficient) for separability.