perty S(PA) =S(PB). (21.21) Quantum information theory The proof of this property is based on the Schmidt decomposition, as described in Appendix V. According to the Schmidt decomposition (which remains to be familiarized with from this appendix), any pure joint state of a composite system can be expressed in the form (21.22) where Xi (i=1...n) are nonnegative numbers (called Schmidt coefficients) satisfy- ing the property = 1, and where {|i a }}, {|ib)} are some orthonormal basis for the subsystems A, B. Since 1/s ) is a pure state, its density operator is 14/(4/. From Eq. (21.20), we obtain (21.23) We now need to make sense of this result by introducing the notion of partial trace, which applies to composite systems. Let |iA), |ja) be any two basis states in the system A, and likewise, let |kB), lb) be any two basis states in the system B. The partial trace trA(U) of the tensor operator U = (1B/ is then defined as: trAU = (21.24) Likewise, the partial trace tr B(U) is defined as trBU =tBiA)(j0kBlB) = (21.25) tr(i)) It should also be noted that for any pair of orthogonal states |i), |j), and by definition of the tensor operator |i> (jl, we have tr(|i) )(jj)=(i|j)= Sij. We now apply the above definitions and this last property to calculate and trb/u) (V/ based on the result in Eq. (21.23). We obtain: = (iB/ (21.26) = PB, 21.2 Relative, joint, and conditional entropy, and mutual information = x (kb|iB) (21.27) In these two equations, the identification in the right-hand side with the subsystem density operators PA, PB comes from the aforementioned property of the Schmidt coefficients, i.e., = 1, making, de facto, {x2} the corresponding probability distribution. We must now realize that this result is quite interesting. Indeed, the only assumption that a composite system is in a pure state, thus, implies that PA = PB, and hence the property S(PA) = S(pB), as was claimed in Eq. (21.21). The examples described at the end of this section provide more opportunities to apply the technique of partial tracing to the deter