stem in state |a), it is possible to duplicate this state into a second quantum system, SO as to obtain the joint state |a) X |a). It simply cannot be achieved. This is a result of the noncloning theorem, which is described in Section 16.6. Next, I shall introduce the notion of tensor product for linear operators (which SO far have been referred to as matrices operating on the qubit vector space). Assume an operator A defined on the |a) qubit space and an operator B defined on the |b) qubit space. The operator tensor product A x B applies to the tensor states |a) x |b) and is defined according to the following distribution rule: AB(a)b))=Aa)@Bb) (16.51) An operator A that is tensored with itself n times will be noted Aon, namely, X An times. (16.52) The operator tensor product A x B also satisfies the useful properties according to which conjugation, transposition, and Hermitian conjugation¹ are distributive. Namely: ©B)* = A* O B* (A@B)T=ATQBT (16.53) These three properties stem from the definition of the operator tensor product in Eq. (16.51), in which the operations of transposition and complex or Hermitian conju- gation are clearly distributive. How can we derive the matrix of the tensor operator A x B? The rule, which is referred to as the Kronecker product, is quite simple. Assume that A is represented by a nxm matrix (n lines, m columns), with coefficients Aij 1, , n, j = 1,...,m). We have, by definition (16.54) Thus, in the above reduced form, A x B is an n X m matrix with coefficients AijB. If B is represented by a pxq matrix (p lines, q columns), then A X B is clearly an np X mq matrix. The Kronecker-product rule also applies to single-column matrices, or vectors, 12 To recall, for any operator A of matrix coefficients Aij, the conjugate operator A* has for coefficients Aj (complex conjugate of Aij), the transposed operator AT has for coefficients Aji, and the Hermitian- conjugate (or adjoint) operator A+ has for coefficients A. Also, the transposed and Hermitian conjugate of the product AB are (AB)+ = 