shown in Chapter 17, the trace of any matrix or operator is independent of the choice of the base representation {|xi}}, or is invariant by unitary base transformation (see Eq. (17.33)). A useful property is that tr(p2) < 1 applies for any density operator p, with equality only for the case of pure states (p = which is left as an exercise to establish. We may note that such a property does not depend on the choice of basis. This is in spite of the fact that a pure state may appear to be a mixed state in another basis! For instance, given |0) assumed a pure state in basis {|0), |1)}, the same state appears to be "mixed" in the basis {l+), |-)} according to the equality |0) = (I+) + --) V But if |0) is a pure state, the system has p = |0) <01 for density operator. It is then an easy exercise to show that p = |0) <0| is actually transformed into p = |+)(+1 in this new basis, which illustrates that pure states remain pure states, independent of basis representation and conserve their properties in any basis transformation. It is important, therefore, not to confuse basis states (as a projection possibility for the system) and pure states (as the only system state possibility). Assume, next, an operator U corresponding to a physical observable (meaning that U is Hermitian or has real, nonzero eigenvalues, see Chapter 17). Let us now take a close look at the operator pU, and its diagonal matrix elements in the basis {|xi}}, namely (pU)ii =(xilpU)x.). From Eq. ((21.3)), we obtain (21.5) It is seen from this result that (pU)ii represents the expectation value (Ui) = of the observable U, should the system be measured in the state /xi) (see Eq. (17.57)), affected by the corresponding probability weight pi of such a measure- ment event. The summation over all event possibilities, which is the trace of pU, takes Quantum information theory the form (21.6) = (U)4. This result shows that given a physical system in state /V/ with associated density matrix p, the expected value (U)v of the observable U is simply given by the trac