2QP+PAQ+QAQ=AP+PAQ+QAQ Next we can show that PAQ = 0 if A is normal. Indeed, we have AA+12) which shows that 12') = A+12) is also an eigenstate of A and belongs to its eigenspace. Therefore, QA+ = 0, hence, QA+ P = (PAQ)+ PAQ. We, thus, obtain Finally, we can show that QAQ is normal. We have (QAQ)(QA+Q)=QAQQA+Q = =QAQA+Q=QA(I-P)A+Q = QA(I-P)A+Q=QAA+Q-QAPA+Q = 0A+AQ-xQPA+Q=QA+AQ QA+(P+Q)AQ=Q(A+P)AQ+QA+QAQ = QA+QQAQ=(QA+Q)(QAQ), which proves that QAQ is normal. We have found that A is decomposed as the sum A = AP + A', with P being diagonal in the eigenspace 12), and A' = QAQ being normal. Since A' is a linear operator, it must have at least one eigenvalue, say X', thus, it can be decomposed into A'=NP + A", where P' is diagonal in the eigenspace 12'), and A" is normal. By induction, we conclude that there exists an orthonormal basis {12), 12') ...} of the same dimension as A for which the matrix A is diagonal. 17.1 Dirac notation basis representation. Indeed, using the basis transformation in Eq. (17.24), and the third property in Eq. (17.32), we obtain: = = tr(U). (17.33) Density operator or matrix Let Vn = {|xi}} = {|x1), |x2),...,/Xn)} an orthonormal basis for the space of quantum states 1/4/, i.e., satisfying (xi/xj) = Sij. In this basis, the states have a unique decompo- sition, which takes the form = + = (17.34) where x; (i = 1, n) represents the complex coordinates. If the modulus or length of 1/5/> is unity, we have (17.35) As we have seen in Chapter 16, for qubits, the number pi = |xil2 represents the probabil- ity of. finding (or measuring) the state 1/>/ in the basis state /xi). Hence, the coordinates Xi represent complex amplitude probabilities. We can now define the density operator or matrix associated with the state 1/>/ as: (17.36) The density matrix is diagonal, since its elements Pij are given by Pij=(xilplx (17.37) Hence, the density matrix operator takes the diagonal matrix representation: (17.38) Quantum measurements It is immediately noted that the trace of the density matrix is unity, since 