 by (17.45) =-plogp-(1-p)log(1-p)=f(p). In the result in Eq. (17.45), we recognize the Shannon entropy of a two-event source X2 = {0, 1}, corresponding to the two possible states of a classical information bit. As we have seen, however, the qubit is a superposition of both information states, which we referred to as |0), |1), with corresponding probabilities /a/2, 1B12. We now have the required conceptual tools to analyze the notion of quantum measurement. Quantum measurements and types In this section, I introduce and analyze the concepts associated with different types of quantum measurement. A general definition for quantum measurement operators will first be introduced. This definition will then be applied to measurements in the orthonormal basis, to the projective (or von Neumann) measurements and to the so- called POVM measurements. Through Dirac notation we have made a formal description of the quantum states and their various transformation properties through the action of linear operators. Such a description did not require any quantum-mechanics background, because in Chapters 15 and 16 we obtained a solid view of the world of qubits, as described by 2D complex vectors, and their operator transformations, as described by unitary rotations on the Bloch sphere. The simpler world of qubits, thus, offers a convenient introduction to the greater view of quantum states 1/>/ and their linear operator transformations 1/4') = A/V . In the same spirit of conceptual simplification, we can view a quantum system as being a physical system with which one can associate a quantum state lv), as expressed onto some pure-state basis V"={|x>}, = and a set of linear operators {A}. We are now interested in learning about what can be physically measured in such a system, and how it may possibly be measured. The following will show that there are different approaches for performing physical measurements in quantum systems. Assume, first, that for a quantum system with n pure states, there exist a certain number n of possible me