Physics 4999E Study Guide - Final Guide: Normal Operator, Closed Set, American Mathematical Society

• School: Western University
• Department: Physics
• Course: Physics 4999E
• Professor: john stockholm

Document Summary

Cs 766/qic 820 theory of quantum information (fall 2011) This lecture represents the second half of the discussion that we started in the previous lecture concerning basic mathematical concepts and tools used throughout the course. The spectral theorem, discussed in the previous lecture, is a valuable tool in quantum information theory. The fact that it is limited to normal operators can, however, restrict its applicability. The singular value theorem, which we will now discuss, is closely related to the spectral theo- rem, but holds for arbitrary operators even those of the form a l (x , y ) for different spaces. Like the spectral theorem, we will nd that the singular value decomposition is an indispensable tool in quantum information theory. Let us begin with a statement of the theorem. Let x and y be complex euclidean spaces, let a l (x , y ) be a nonzero operator, and let r = rank(a).