Recently, a number of new and surprising advances in the application of quantum mechanics to computer science are threatening to revolutionize the field of computer science. In this talk, we focus on one aspect of this revolution, the recent advances in quantum information theory.
The talk begins with the comparison of Shannon entropy
H(A) = - Sum( p_i lg p_i )
and von Neumann entropy
S(A) = - Trace( \rho lg \rho ),
where \rho denotes the density operator of quantum mechanics.
These entropies are compared with respect to three different
quantum mechanical systems of qubit pairs A, B, and C. The
first two A and B behave classically. The last C, an
entangled system, behaves non-classically. Both entropies agree
on A and B. But on C, Shannon entropy fails; and von
Neumann entropy clearly shows itself as the true entropy, i.e., as
an invariant of closed quantum mechanical systems.
Our observations in regard to von Neumann entropy are then used to reinterpret quantum measurement theory as a totally unitary theory without the so called "collapse" of the wave function. This interpretation is illustrated with an analysis of the Stern-Gerlach experiment.
If time permits, quantum error correcting codes will also be discussed.
The first three quarters of this talk are intended for those individuals of mathematical maturity who are not familiar with quantum mechanics.