nonnegative numbers, called Schmidt coefficients, satisfying the property =1, = and where {|i a }}, {|ib)} are some orthonormal basis for the subsystems A, B. The proof that the decomposition in Eq. (V1) exists and is unique proceeds as follows. Let {|j}} and {|k>} be two orthonormal bases for the subsystems A and B, respectively. The joint state can most generally be defined through the expansion where Wjk are the complex coefficients of some matrix S. According to the principle of singular value decomposition, 1 it is possible to express So in the form S = UDV, where U, V are unitary matrices, and D is a diagonal matrix having nonnegative elements. The diagonal elements Xi = dii of D are called the singular values of S. The matrix elements of S2, thus, decompose as follows which we substitute into Eq. (V2) to obtain: See, for instance: http://mathworld.wolfram.com/SingularValueDecomposition.html;I http://en.wikipedia. org/wiki/Singular_value_decomposition. Schmidt decomposition of joint pure states We then introduce the definitions The states |ia),|ib) form orthonormal sets, because of the unitarity of the matrices U, V. Considering |iA), for instance, we have (LAILA) = = (U+U)li = Ili = Sli, which proves the orthonormality of the set {|i a }}} and likewise for {|ib)}. Substituting Eq. (V5) into Eq. (V4) yields Eq. (V1), i.e., the Schmidt decomposition of 1/>/. Because (v) is a pure state, we also have (V/V/) = = 1, which completes the proof. Here are two examples of Schmidt decomposition according to Eq. (V1): with the other choice of orthonormal basis {|, |IB)} = {|0B}, -|1B}} for the subsys- tem B. Appendix W (Chapter 21) State purification This appendix describes the operation of state purification. The operation consists of associating a pure joint state 1VAR) with a system A that is in a mixed state (VA). The benefit of the operation is the possibility of deriving the density operator PA directly from the partial trace of the pure-state density operator PAR The term state purification, thus, means expandi