טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBisker Gili
SubjectThe World of Two Qubits
DepartmentDepartment of Physics
Supervisor Professor Emeritus Joseph Avron
Full Thesis textFull thesis text - English Version


Abstract

The world of states, witnesses and Bell inequalities of two qubits can be visualized geometrically. Since it is represented by 4X4 Hermitian matrices, it is 16 dimensional. Thus, visualization in three dimensions requires introducing an appropriate equivalence relation. The SLOCC equivalence relation is often used for states and here we show that entanglement witnesses and CHSH Bell inequalities can be incorporated in this description as well.

                       

The geometric description is faithful to the duality between separable states and witnesses, and allows one to give elementary and elegant proofs of non-elementary results.


This visualization allows us to give a ``proof by inspection'' that for two qubits, the Peres test is `if and only if'. We show that the CHSH Bell inequalities, where both Alice and Bob have two experiments to choose from, can be visualized as circles in the figure. This allows us to solve geometrically the optimization problem of the CHSH inequality violation. Finally, we give numerical evidence that, remarkably, allowing Alice and Bob to use three rather than two measurements, does not help them to distinguish any new entangled SLOCC equivalence class beyond the CHSH class.



The concept of entanglement witnesses has a practical meaning as well. If one wants to prove the existence of entanglement, two approaches can be used. The first one is to find the state explicitly and use the Peres separability test mentioned above. The second is to measure the expectation value of a corresponding entanglement witness. The latter is more economical, since the number of measurements needed is smaller, whereas in the first approach no less then $16$ measurements are needed. We present a set of product projective measurements, optimal for proving entanglement. Our goal is to minimize the number of measurements needed in order to find the expectation value of an entanglement witness.