the exclusive OR, noted +). For instance, if x = 00101 and y = 10111, we have /x+y) = |10010). With such a definition at hand, given X E C1 we are able to construct all possible quantum states |x+y) with y E C2 as well as the normalized sum: (24.36) Given the number 2k1=|C1| = of codewords X in C1, how many orthogonal quantum states x + C2> can be, thus, generated? The answer to this question is |C1|/|C2|=2ki-k2, which stems from group theory (GT) and the fine notion of cosets.2 Here, I shall not venture into GT, but leave it as an interesting exercise to establish that the C = (7,4) Hamming code, (described in Chapter 11) forms a group (C, +) under the bit-wise addition + and to make the inventory of its various cosets. A key property is that any element x E C1 must belong to one and only one coset of C2. The same coset x+C2=x+C2corresponds to two different elements X, x' E C1 satisfying the property x - x' E C2 and, hence + C2) = = xx + C2).3 It is clear that if two different elements X, x' E C1 do not belong to the same coset C2 x' + 2 I provide here a simplified description of cosets, which also explains the notation x + C2. First, it is impor- tant to be familiar with, or to quickly revisit the basics of groups and subgroups, for instance through the links: http://en.wikipedia.org/wiki/Group_%28mathematics%29, http://en.wikipedia.org/wiki/Subgroup. Then assume two commutative groups C1 and C2 with additive operation +, such that C2 C C1 is a subgroup of C1. The cosets of C2 in C1, noted Xi + C2, are defined for all elements Xi E C1 as follows Taking an illustrative example, assume C1 = (0, 1, 2, 3), with + representing the addition modulo 4, and which has the only "nontrivial" subgroup C2 (0, ,2). The cosets of C2 in C1 are 0 + C2 = (0,2)= C2 1+C2=(1,3) 2+C2= (2,0)=(0,2)=C2 +C2 = (3,1)=(1,3)=1 + C2. It is seen that there are two distinct cosets of C2 in C1, including C2 itself, which are (0,2) and (1,3). In the general case with groups and subgroups of finite sizes |C1|, |C21, Lagrange's theorem states that 