rmal basis where p is diagonal. The composite system PQ, as described by the tensor product represents the joint state of the system after the preparation of the states Px, based on the definition of partial tracing, i.e., -PxPx Holevo bound = Pp. The third system, R, is a fictitious auxiliary system representing, together with Q, the measuring apparatus of the recipient. We assume that before measurement, R is in the pure state |0), corresponding to PR= |0) <0|. Thus, the initial state of the whole system (or quantum communication channel) is described by the density operator Next, we need to define the effect of the recipient's POVM measurement. As seen in Chapter 17, the POVM measurement is defined as a set of Hermitian operators, {Ey}, which satisfy the completeness relation (Eq. (17.62)), and which correspond to a set of measurement operators, {My==E}, = satisfying the property = Ey. = Applying any single measurement labeled y to a system in state p is a quantum operation that leaves the system in the post-measurement state p'=MypM; Here, we shall use a somewhat different quantum operation. It consists of effecting the POVM measurement in system Q and storing the result in system R. The corresponding quantum operation leaves the composite system QR in the post-measurement state This definition is justified by the partial tracing over system Q to yield, for system R: P'R=tre(POR) = trq Appendix X where (Ey)x = tr(EyPx) represents the expectation value of the measurement Ey, given the input state Px. Note that tr(p'R) # 1. To get the actual final state of R we need to effect the normalization [(Ey)xly)(y or, in the matrix form in base {|y}}: (E1)x (E2)x (X10) (En)x The coefficients in this density-matrix definition, thus, provide the conditional probabil- ity of obtaining (Ey)x as the outcome of the measurement, given the input state Px. This justifies the definition in Eq. (X7) of the proposed quantum operation (not normalized, for simplicity). In summary, our measurement operation leaves the composite system