Question

27. Some More Facts about Density Operators: Let ? be a density operator acting in an N-dimensional complex vector space. (a) Show that ?-?2 if and only if ? is a pure state. (b) show that Trlo l 1, with equality if and only if p is a pure state. Thus, Tr ?2 is one convenient m easure of the purity or impurity of the state. (Borrowing from the interpretation of mixed states for spin-systems, this quantity is sometimes referred to as the polarization of the state). As a trace, this purity measure is independent of the choice of basis. (c) If is any unitary operator, and ? is any density operator, show that ??Ut is also a density matrix HINT: Show that the transformed operator ??Ut is Hermitian, positive semidefinite, and unit-trace. (d) Show that Trl2, where the equality holds if and only if p corresponds to the completely mized state, i.e., ?-.. characteristic of maximal ignorance as to the pure state. Another frequently encountered measure of the purity (or rather, non-purity) of the state is the von Neumann entropy defined by where S is measured in natural units, or nats, if the logarithms are chosen to be natural logarithm, and S is measured in binary digits, or bits, if the logarithms are taken in base-2. (it can also be put into SI units by using the natural logarithm, by multiplying the whole thing by Boltzmann's constant). In the resent context, we can think of it as a measure of the uncertainty as to which pure state the system can be located, or equivalently, the best-case uncertainty as to the results of performing a measurement over of a CSCO. As a trace, the entropy must also independent of the choice of basis.

Answer 1

"given that Definition of density operator: - e^= sigma_n